0.4-kg particle is located at the position r1 2i 3j k m and has the velocity v1 i j 2k

The 50-kg crate is projected along the floor with an initial speed of 7 m/s at . The coefficient of kinetic friction is 0.40. Calculate the time required for the crate to come to rest and the corresponding distance x traveled. Problem 3/1

The 50-kg crate of Prob. 3/1 is now projected down an incline as shown with an initial speed of 7 m/s. Investigate the time t required for the crate to come to rest and the corresponding distance x traveled if (a) and (b) . Problem 3/2

The 100-lb crate is carefully placed with zero velocity on the incline. Describe what happens if (a) and (b) Problem 3/3

A 60-kg woman holds a 9-kg package as she stands within an elevator which briefly accelerates upward at a rate of g/4. Determine the force R which the elevator floor exerts on her feet and the lifting force L which she exerts on the package during the acceleration interval. If the elevator support cables suddenly and completely fail, what values would R and L acquire? Problem 3/4

During a brake test, the rear-engine car is stopped from an initial speed of 100 km/h in a distance of 50 m. If it is known that all four wheels contribute equally to the braking force, determine the braking force F at each wheel. Assume a constant deceleration for the 1500-kg car. Problem 3/5

What fraction n of the weight of the jet airplane is 15 the net thrust (nozzle thrust T minus air resistance R) required for the airplane to climb at an angle with the horizontal with an acceleration a in the direction of flight? Problem 3/6

The 300-Mg jet airliner has three engines, each of which produces a nearly constant thrust of 240 kN during the takeoff roll. Determine the length s of runway required if the takeoff speed is 220 km/h. Compute s first for an uphill takeoff direction from A to B and second for a downhill takeoff from B to A on the slightly inclined runway. Neglect air and rolling resistance. Problem 3/7

The 180-lb man in the bosuns chair exerts a pull of 50 lb on the rope for a short interval. Find his acceleration. Neglect the mass of the chair, rope, and pulleys. Problem 3/8

A man pulls himself up the incline by the method shown. If the combined mass of the man and cart is 100 kg, determine the acceleration of the cart if the man exerts a pull of 250 N on the rope. Neglect all friction and the mass of the rope, pulleys, and wheels. Problem 3/9

A car is climbing the hill of slope at a constant speed v. If the slope decreases abruptly to at point A, determine the acceleration a of the car just after passing point A if the driver does not change the throttle setting or shift into a different gear. Problem 3/10

Calculate the vertical acceleration a of the 100-lb cylinder for each of the two cases illustrated. Neglect friction and the mass of the pulleys. Problem 3/11

A driver finds that her car will descend the slope at a certain constant speed with no brakes or throttle required. The slope decreases fairly abruptly to at point A. If the driver takes no action but continues to coast, determine the acceleration a of the car just after it passes point A for the conditions (a) and (b) . Problem 3/12

By itself, the 2500-kg pickup truck executes a 0100 km/h acceleration run in 10 s along a level road. What would be the corresponding time when pulling the 500-kg trailer? Assume constant acceleration and neglect all retarding forces. Problem 3/13

Reconsider the pickup-truck/trailer combination of the previous problem. If the unit uniformly accelerates from rest to a speed of 25 m/s in a distance of 150 m, determine the tension T in the towing tongue OA. Neglect all effects of the tongue angle, i.e., assume that OA is horizontal.

A train consists of a 400,000-lb locomotive and one hundred 200,000-lb hopper cars. If the locomotive exerts a friction force of 40,000 lb on the rails in starting the train from rest, compute the forces in couplers 1 and 100. Assume no slack in the couplers and neglect friction associated with the hopper cars. Problem 3/15

The collar A is free to slide along the smooth shaft B mounted in the frame. The plane of the frame is vertical. Determine the horizontal acceleration a of the frame necessary to maintain the collar in a fixed position on the shaft. Problem 3/16

The 5-oz pinewood-derby car is released from rest at the starting line A and crosses the finish line C 2.75 sec later. The transition at B is small and smooth. Assume that the net retarding force is constant throughout the run and find this force. Problem 3/17

The beam and attached hoisting mechanism together weigh 2400 lb with center of gravity at G. If the initial acceleration a of point P on the hoisting cable is 20 ft/sec2 , calculate the corresponding reaction at the support A. Problem 3/18 1000 lb a P G A

The 10-kg steel sphere is suspended from the 15-kg frame which slides down the incline. If the coefficient of kinetic friction between the frame and incline is 0.15, compute the tension in each of the supporting wires A and B. Problem 3/19

The block shown is observed to have a velocity 20 ft/sec as it passes point A and a velocity ft/sec as it passes point B on the incline. Calculate the coefficient of kinetic friction between the block and the incline if ft and . Problem 3/20

Determine the initial acceleration of the 15-kg block if (a) N and (b) N. The system is initially at rest with no slack in the cable, and the mass and friction of the pulleys are negligible. Problem 3/21

The system of the previous problem starts from rest with no slack in the cable. What value of the tension T will result in an initial block acceleration of 0.8 m/s2 to the right?

Small objects are delivered to the 72-in. inclined chute by a conveyor belt A which moves at a speed ft/sec. If the conveyor belt B has a speed ft/sec and the objects are delivered to this belt with no slipping, calculate the coefficient of friction between the objects and the chute. Problem 3/23

If the coefficients of static and kinetic friction between the 20-kg block A and the 100-kg cart B are both essentially the same value of 0.50, determine the acceleration of each part for (a) N and (b) N. Problem 3/24

A simple pendulum is pivoted at O and is free to swing in the vertical plane of the plate. If the plate is given a constant acceleration a up the incline , write an expression for the steady angle assumed by the pendulum after all initial start-up oscillations have ceased. Neglect the mass of the slender supporting rod. Problem 3/25 a O

The tractor-trailer unit is moving down the incline with a speed of 5 mi/hr when the driver brakes the tractor to a stop in a distance of 4 ft. Estimate the percent increase n in the hitch-force component which is parallel to the incline, compared with the force present at steady speed. The cart and its load combined weigh 500 lb. State any assumptions. Problem 3/26

The device shown is used as an accelerometer and consists of a 4-oz plunger A which deflects the spring as the housing of the unit is given an upward acceleration a. Specify the necessary spring stiffness k which will permit the plunger to deflect 1/4 in. beyond the equilibrium position and touch the electrical contact when the steadily but slowly increasing upward acceleration reaches 5g. Friction may be neglected. Problem 3/27

The acceleration of the 50-kg carriage A in its smooth vertical guides is controlled by the tension T exerted on the control cable which passes around the two circular pegs fixed to the carriage. Determine the value of T required to limit the downward acceleration of the carriage to 1.2 m/s2 if the coefficient of friction between the cable and the pegs is 0.20. (Recall the relation between the tensions in a flexible cable which is slipping on a fixed peg: ) Problem 3/28

The system is released from rest with the cable taut. For the friction coefficients and , calculate the acceleration of each body and the tension T in the cable. Neglect the small mass and friction of the pulleys. Problem 3/29

A jet airplane with a mass of 5 Mg has a touchdown speed of 300 km/h, at which instant the braking parachute is deployed and the power shut off. If the total drag on the aircraft varies with velocity as shown in the accompanying graph, calculate the distance x along the runway required to reduce the speed to 150 km/h. Approximate the variation of the drag by an equation of the form , where k is a constant. Problem 3/30

A heavy chain with a mass per unit length is pulled by the constant force P along a horizontal surface consisting of a smooth section and a rough section. The chain is initially at rest on the rough surface with . If the coefficient of kinetic friction between the chain and the rough surface is , determine the velocity v of the chain when . The force P is greater than in order to initiate motion. Problem 3/31

The sliders A and B are connected by a light rigid bar of length m and move with negligible friction in the slots, both of which lie in a horizontal plane. For the position where m, the velocity of A is m/s to the right. Determine the acceleration of each slider and the force in the bar at this instant. Problem 3/32

The sliders A and B are connected by a light rigid bar and move with negligible friction in the slots, both of which lie in a horizontal plane. For the position shown, the hydraulic cylinder imparts a velocity and acceleration to slider A of 0.4 m/s and 2 m/s2 , respectively, both to the right. Determine the acceleration of slider B and the force in the bar at this instant. Problem 3/33 A

The 4-lb collar is released from rest against the light elastic spring, which has a stiffness of 10 lb/in. and has been compressed a distance of 6 in. Determine the acceleration a of the collar as a function of the vertical displacement x of the collar measured in feet from the point of release. Find the velocity v of the collar when ft. Friction is negligible. Problem 3/34

The nonlinear spring has a tensile force-deflection relationship given by , where x is in meters and Fs is in newtons. Determine the acceleration of the 6-kg block if it is released from rest at (a) mm and (b) mm. Problem 3/35 Undeformed spring position 6 kg

Two configurations for raising an elevator are shown. Elevator A with attached hoisting motor and drum has a total mass of 900 kg. Elevator B without motor and drum also has a mass of 900 kg. If the motor supplies a constant torque of 600 to its 250-mm-diameter drum for 2 s in each case, select the configuration which results in the greater upward acceleration and determine the corresponding velocity v of the elevator 1.2 s after it starts from rest. The mass of the motorized drum is small, thus permitting it to be analyzed as though it were in equilibrium. Neglect the mass of cables and pulleys and all friction. Problem 3/36

Compute the acceleration of block A for the instant depicted. Neglect the masses of the pulleys. Problem 3/37

The inclined block A is given a constant rightward acceleration a. Determine the range of values of for which block B will not slip relative to block A, regardless of how large the acceleration a is. The coefficient of static friction between the blocks is . Problem 3/38

A spring-loaded device imparts an initial vertical velocity of 50 m/s to a 0.15-kg ball. The drag force on the ball is , where FD is in newtons when the speed v is in meters per second. Determine the maximum altitude h attained by the ball (a) with drag considered and (b) with drag neglected. Problem 3/39 0.15 k

A shock absorber is a mechanical device which provides resistance to compression or extension given by , where c is a constant and v is the time rate of change of the length of the absorber. An absorber of constant N s/m is shown being tested with a 100-kg cylinder suspended from it. The system is released with the cable taut at and allowed to extend. Determine (a) the steadystate velocity vs of the lower end of the absorber and (b) the time t and displacement y of the lower end when the cylinder has reached 90 percent of its steady-state speed. Neglect the mass of the piston and attached rod. Problem 3/40

The design of a lunar mission calls for a 1200-kg spacecraft to lift off from the surface of the moon and travel in a straight line from point A and pass point B. If the spacecraft motor has a constant thrust of 2500 N, determine the speed of the spacecraft as it passes point B. Use Table D/2 and the gravitational law from Chapter 1 as needed. Problem 3/41

For what value(s) of the angle will the acceleration of the 80-lb block be 26 ft/sec2 to the right? Problem 3/42

With the blocks initially at rest, the force P is increased slowly from zero to 60 lb. Plot the accelerations of both masses as functions of P. Problem 3/43

An object projected vertically up from the surface of the earth with a sufficiently high velocity v0 can escape from the earths gravitational field. Calculate this velocity on the basis of the absence of an atmosphere to offer resistance due to air friction. To eliminate the effect of the earths rotation on the velocity measurement, consider the launch to be from the north or south pole. Use the mean radius of the earth and the absolute value of g as cited in Art. 1/5 and compare your answer with the value cited in Table D/2.

The system is released from rest in the position shown. Calculate the tension T in the cord and the acceleration a of the 30-kg block. The small pulley attached to the block has negligible mass and friction. (Suggestion: First establish the kinematic relationship between the accelerations of the two bodies.) Problem 3/45

The rod of the fixed hydraulic cylinder is moving to the left with a speed of 100 mm/s and this speed is momentarily increasing at a rate of 400 mm/s each second at the instant when Determine the tension in the cord at that instant. The mass of slider B is 0.5 kg, the length of the cord is 1050 mm, and the effects of the radius and friction of the small pulley at A are negligible. Find results for cases (a) negligible friction at slider B and (b) at slider B. The action is in a vertical plane. Problem 3/46 250 mm C B A 0.5 kg

The small 0.6-kg block slides with a small amount of friction on the circular path of radius 3 m in the vertical plane. If the speed of the block is 5 m/s as it passes point A and 4 m/s as it passes point B, determine the normal force exerted on the block by the surface at each of these two locations. Problem 3/47

A 2-lb slider is propelled upward at A along the fixed curved bar which lies in a vertical plane. If the slider is observed to have a speed of 10 ft/sec as it passes position B, determine (a) the magnitude N of the force exerted by the fixed rod on the slider and (b) the rate at which the speed of the slider is decreasing. Assume that friction is negligible. Problem 3/48

The 0.1-kg particle has a speed m/s as it passes the position shown. The coefficient of kinetic friction between the particle and the verticalplane track is . Determine the magnitude of the total force exerted by the track on the particle. What is the deceleration of the particle? Problem 3/49

0 The 4-oz slider has a speed ft/sec as it passes point A of the smooth guide, which lies in a horizontal plane. Determine the magnitude R of the force which the guide exerts on the slider (a) just before it passes point A of the guide and (b) as it passes point B.

Determine the proper bank angle for the airplane flying at 400 mi/hr and making a turn of 2-mile radius. Note that the force exerted by the air is normal to the supporting wing surface. Problem 3/51

The slotted arm rotates about its center in a horizontal plane at the constant angular rate rad/sec and carries a 3.22-lb spring-mounted slider which oscillates freely in the slot. If the slider has a speed of 24 in./sec relative to the slot as it crosses the center, calculate the horizontal side thrust P exerted by the slotted arm on the slider at this instant. Determine which side, A or B, of the slot is in contact with the slider. Problem 3/52

The hollow tube is pivoted about a horizontal axis through point O and is made to rotate in the vertical plane with a constant counterclockwise angular velocity rad/sec. If a 0.2-lb particle is sliding in the tube toward O with a velocity of 4 ft/sec relative to the tube when the position is passed, calculate the magnitude N of the normal force exerted by the wall of the tube on the particle at this instant. Problem 3/53

The member OA rotates about a horizontal axis through O with a constant counterclockwise angular velocity rad/sec. As it passes the position , a small block of mass m is placed on it at a radial distance in. If the block is observed to slip at , determine the coefficient of static friction between the block and the member. Problem 3/54

In the design of a space station to operate outside the earths gravitational field, it is desired to give the structure a rotational speed N which will simulate the effect of the earths gravity for members of the crew. If the centers of the crews quarters are to be located 12 m from the axis of rotation, calculate the necessary rotational speed N of the space station in revolutions per minute. Problem 3/55 1

A swing ride is shown in the figure. Calculate the necessary angular velocity for the swings to assume an angle with the vertical. Neglect the mass of the cables and treat the chair and person as one particle. Problem 3/56

A Formula-1 car encounters a hump which has a circular shape with smooth transitions at either end. (a) What speed vB will cause the car to lose contact with the road at the topmost point B? (b) For a speed km/h, what is the normal force exerted by the road on the 640-kg car as it passes point A? Problem 3/57

In order to simulate a condition of apparent weightlessness experienced by astronauts in an orbiting spacecraft, a jet transport can change its direction at the top of its flight path by dropping its flightpath direction at a prescribed rate for a short interval of time. Specify if the aircraft has a speed km/h. Problem 3/58

The standard test to determine the maximum lateral acceleration of a car is to drive it around a 200-ft-diameter circle painted on a level asphalt surface. The driver slowly increases the vehicle speed until he is no longer able to keep both wheel pairs straddling the line. If this maximum speed is 35 mi/hr for a 3000-lb car, determine its lateral acceleration capability an in gs and compute the magnitude F of the total friction force exerted by the pavement on the car tires. Problem 3/59

The car of Prob. 3/59 is traveling at 25 mi/hr when the driver applies the brakes, and the car continues to move along the circular path. What is the maximum deceleration possible if the tires are limited to a total horizontal friction force of 2400 lb?

The concept of variable banking for racetrack turns is shown in the figure. If the two radii of curvature are ft and ft for cars A and B, respectively, determine the maximum speed for each car. The coefficient of static friction is for both cars. Problem 3/61

The small ball of mass m and its supporting wire become a simple pendulum when the horizontal cord is severed. Determine the ratio k of the tension T in the supporting wire immediately after the cord is cut to that in the wire before the cord is cut. Problem 3/62

A small object is given an initial horizontal velocity v0 at the bottom of a smooth slope. The angle made by the slope with the horizontal varies according to sin , where k is a constant and s is the distance measured along the slope from the bottom. Determine the maximum distance s which the object slides up the slope.

A 3220-lb car enters an S-curve at A with a speed of 60 mi/hr with brakes applied to reduce the speed to 45 mi/hr at a uniform rate in a distance of 300 ft measured along the curve from A to B. The radius of curvature of the path of the car at B is 600 ft. Calculate the total friction force exerted by the road on the tires at B. The road at B lies in a horizontal plane. Problem 3/64

A pilot flies an airplane at a constant speed of 600 km/h in the vertical circle of radius 1000 m. Calculate the force exerted by the seat on the 90-kg pilot at point A and at point B. Problem 3/65

The 30-Mg aircraft is climbing at the angle under a jet thrust T of 180 kN. At the instant represented, its speed is 300 km/h and is increasing at the rate of 1.96 m/s2 . Also is decreasing as the aircraft begins to level off. If the radius of curvature of the path at this instant is 20 km, compute the lift L and drag D. (Lift L and drag D are the aerodynamic forces normal to and opposite to the flight direction, respectively.) Problem 3/66

The hollow tube assembly rotates about a vertical axis with angular velocity rad/s and rad/s2 . A small 0.2-kg slider P moves inside the horizontal tube portion under the control of the string which passes out the bottom of the assembly. If m, m/s, and m/s2 , determine the tension T in the string and the horizontal force exerted on the slider by the tube. Problem 3/

A flatbed truck going 100 km/h rounds a horizontal curve of 300-m radius inwardly banked at . The coefficient of static friction between the truck bed and the 200-kg crate it carries is 0.70. Calculate the friction force F acting on the crate. Problem 3/68

Explain how to utilize the graduated pendulum to measure the speed of a vehicle traveling in a horizontal circular arc of known radius r. Problem 3/69

The bowl-shaped device rotates about a vertical axis with a constant angular velocity . If the particle is observed to approach a steady-state position in the presence of a very small amount of friction, determine . The value of r is 0.2 m. Problem 3/70

The 2-kg slider fits loosely in the smooth slot of the disk, which rotates about a vertical axis through point O. The slider is free to move slightly along the slot before one of the wires becomes taut. If the disk starts from rest at time and has a constant clockwise angular acceleration of 0.5 rad/s2 , plot the tensions in wires 1 and 2 and the magnitude N of the force normal to the slot as functions of time t for the interval . Problem 3/71

A 2-kg sphere S is being moved in a vertical plane by a robotic arm. When the angle is , the angular velocity of the arm about a horizontal axis through O is 50 deg/s clockwise and its angular acceleration is 200 deg/s2 counterclockwise. In addition, the hydraulic element is being shortened at the constant rate of 500 mm/s. Determine the necessary minimum gripping force P if the coefficient of static friction between the sphere and the gripping surfaces is 0.50. Compare P with the minimum gripping force Ps required to hold the sphere in static equilibrium in the position. Problem 3/72

The rocket moves in a vertical plane and is being propelled by a thrust T of 32 kN. It is also subjected to an atmospheric resistance R of 9.6 kN. If the rocket has a velocity of 3 km/s and if the gravitational acceleration is 6 m/s2 at the altitude of the rocket, calculate the radius of curvature of its path for the position described and the time-rate-ofchange of the magnitude v of the velocity of the rocket. The mass of the rocket at the instant considered is 2000 kg. Problem 3/73

The robot arm is elevating and extending simultaneously. At a given instant, , deg/s, deg/s2 , m, m/s, and m/s2 . Compute the radial and transverse forces Fr and that the arm must exert on the gripped part P, which has a mass of 1.2 kg. Compare with the case of static equilibrium in the same position. Problem 3/74

A stretch of highway includes a succession of evenly spaced dips and humps, the contour of which may be represented by the relation . What is the maximum speed at which the car A can go over a hump and still maintain contact with the road? If the car maintains this critical speed, what is the total reaction N under its wheels at the bottom of a dip? The mass of the car is m. Problem 3/75

Determine the speed v at which the race car will have no tendency to slip sideways on the banked track, that is, the speed at which there is no reliance on friction. In addition, determine the minimum and maximum speeds, using the coefficient of static friction . State any assumptions. Problem 3/76 s

Small steel balls, each with a mass of 65 g, enter the semicircular trough in the vertical plane with a horizontal velocity of 4.1 m/s at A. Find the force R exerted by the trough on each ball in terms of and the velocity vB of the balls at B. Friction is negligible. Problem 3/77

The flat circular disk rotates about a vertical axis through O with a slowly increasing angular velocity . Prior to rotation, each of the 0.5-kg sliding blocks has the position mm with no force in its attached spring. Each spring has a stiffness of 400 N/m. Determine the value of x for each spring for a steady speed of 240 rev/min. Also calculate the normal force N exerted by the side of the slot on the block. Neglect any friction between the blocks and the slots, and neglect the mass of the springs. (Hint: Sum forces along and normal to the slot.) Problem 3/78

The spring-mounted 0.8-kg collar A oscillates along the horizontal rod, which is rotating at the constant angular rate rad/s. At a certain instant, r is increasing at the rate of 800 mm/s. If the coefficient of kinetic friction between the collar and the rod is 0.40, calculate the friction force F exerted by the rod on the collar at this instant. Problem 3/79

The slotted arm revolves in the horizontal plane about the fixed vertical axis through point O. The 3-lb slider C is drawn toward O at the constant rate of 2 in./sec by pulling the cord S. At the instant for which in., the arm has a counterclockwise angular velocity rad/sec and is slowing down at the rate of 2 rad/sec2 . For this instant, determine the tension T in the cord and the magnitude N of the force exerted on the slider by the sides of the smooth radial slot. Indicate which side, A or B, of the slot contacts the slider. Problem 3/80

A small coin is placed on the horizontal surface of  30 the rotating disk. If the disk starts from rest and is given a constant angular acceleration , determine an expression for the number of revolutions N through which the disk turns before the coin slips. The coefficient of static friction between the coin and the disk is . Problem 3/81

The rotating drum of a clothes dryer is shown in the figure. Determine the angular velocity of the drum which results in loss of contact between the clothes and the drum at . Assume that the small vanes prevent slipping until loss of contact. Problem 3/82

A body at rest relative to the surface of the earth rotates with the earth and therefore moves in a circular path about the polar axis of the earth considered fixed. Derive an expression for the ratio k of the apparent weight of such a body as measured by a spring scale at the equator (calibrated to read the actual force applied) to the true weight of the body, which is the absolute gravitational attraction to the earth. The absolute acceleration due to gravity at the equator is m/s2 . The radius of the earth at the equator is km, and the angular velocity of the earth is rad/s. If the true weight is 100 N, what is the apparent measured weight ? W

At the instant when , the horizontal guide is given a constant upward velocity m/s. For this instant calculate the force N exerted by the fixed circular slot and the force P exerted by the horizontal slot on the 0.5-kg pin A. The width of the slots is slightly greater than the diameter of the pin, and friction is negligible. Problem 3/84

The particle P is released at time from the position inside the smooth tube with no velocity relative to the tube, which is driven at the constant angular velocity about a vertical axis. Determine the radial velocity vr, the radial position r, and the transverse velocity as functions of time t. Explain why the radial velocity increases with time in the absence of radial forces. Plot the absolute path of the particle during the time it is inside the tube for m, m, and rad/s. Problem 3/85

The small 5-oz slider A moves without appreciable friction in the hollow tube, which rotates in a horizontal plane with a constant angular speed rad/sec. The slider is launched with an initial speed ft/sec relative to the tube at the inertial coordinates in. and . Determine the magnitude P of the horizontal force exerted on the slider by the tube just before the slider exits the tube. Problem 3/86

The two 0.2-kg sliders A and B move without friction in the horizontal-plane circular slot. Determine the acceleration of each slider and the normal reaction force exerted on each when the system starts from rest in the position shown and is acted upon by the 4-N force P. Also find the tension in the inextensible connecting cord AB. Problem 3/87

Repeat the questions of the previous problem for the revised system configuration shown in the figure. Problem 3/88

The 3000-lb car is traveling at 60 mi/hr on the straight portion of the road, and then its speed is reduced uniformly from A to C, at which point it comes to rest. Compute the magnitude F of the total friction force exerted by the road on the car (a) just before it passes point B, (b) just after it passes point B, and (c) just before it stops at point C. Problem 3/89

The spacecraft P is in the elliptical orbit shown. At the instant represented, its speed is ft/sec. Determine the corresponding values of , , , and . Use ft/sec2 as the acceleration of gravity on the surface of the earth and mi as the radius of the earth. Problem 3/90

The slotted arm OA rotates about a horizontal axis through point O. The 0.2-kg slider P moves with negligible friction in the slot and is controlled by the inextensible cable BP. For the instant under consideration, , rad/s, , and m. Determine the corresponding values of the tension in cable BP and the force reaction R perpendicular to the slot. Which side of the slot contacts the slider? Problem 3/91

The small pendulum of mass m is suspended from a trolley which runs on a horizontal rail. The trolley and pendulum are initially at rest with . If the trolley is given a constant acceleration , determine the maximum angle through which the pendulum swings. Also find the tension T in the cord in terms of . Problem 3/92

A small object is released from rest at A and slides with friction down the circular path. If the coeffi- cient of friction is 0.20, determine the velocity of the object as it passes B. (Hint: Write the equations of motion in the n- and t-directions, eliminate N, and substitute . The resulting equation is a linear nonhomogeneous differential equation of the form , the solution of which is well known.) Problem 3/93 3 m m k = 0.20 A B

The slotted arm OB rotates in a horizontal plane about point O of the fixed circular cam with constant angular velocity rad/s. The spring has a stiffness of 5 kN/m and is uncompressed when . The smooth roller A has a mass of 0.5 kg. Determine the normal force N which the cam exerts on A and also the force R exerted on A by the sides of the slot when . All surfaces are smooth. Neglect the small diameter of the roller. Problem 3/94

A small collar of mass m is given an initial velocity of magnitude v0 on the horizontal circular track fabricated from a slender rod. If the coefficient of kinetic friction is , determine the distance traveled before the collar comes to rest. (Hint: Recognize that the friction force depends on the net normal force.) Problem 3/95

The small cart is nudged with negligible velocity from its horizontal position at A onto the parabolic path, which lies in a vertical plane. Neglect friction and show that the cart maintains contact with the path for all values of k. Problem 3/96

The spring is unstretched when . If the body moves from the initial position to the final position , (a) determine the work done by the spring on the body and (b) determine the work done on the body by its weight. Problem 3/97

The small body has a speed at point A. Neglecting friction, determine its speed at point B after it has risen 0.8 m. Is knowledge of the shape of the track necessary? Problem 3/98

The 64.4-lb crate slides down the curved path in the vertical plane. If the crate has a velocity of 3 ft/sec down the incline at A and a velocity of 25 ft/sec at B, compute the work U done on the crate by friction during the motion from A to B. Problem 3/99

The 1.5-lb collar slides with negligible friction on the fixed rod in the vertical plane. If the collar starts from rest at A under the action of the constant 2-lb horizontal force, calculate its velocity v as it hits the stop at B. Problem 3/100

In the design of a spring bumper for a 3500-lb car, it is desired to bring the car to a stop from a speed of 5 mi/hr in a distance equal to 6 in. of spring deformation. Specify the required stiffness k for each of the two springs behind the bumper. The springs are undeformed at the start of impact. Problem 3/101

A two-engine jet transport has a loaded weight of 90,000 lb and a forward thrust of 9800 lb per engine during takeoff. If the transport requires 4800 ft of level runway starting from rest to become airborne at a speed of 140 knots , determine the average resistance R to motion over the runway length due to drag (air resistance) and mechanical retardation by the landing gear.

The small collar of mass m is released from rest at A and slides down the curved rod in the vertical plane with negligible friction. Express the velocity v of the collar as it strikes the base at B in terms of the given conditions. Problem 3/103

For the sliding collar of Prob. 3/103, if , , and , and if the velocity of the collar as it strikes the base B is 4.70 m/s after release of the collar from rest at A, calculate the work Q of friction. What happens to the energy which is lost?

The two small 0.2-kg sliders are connected by a light rigid bar and are constrained to move without friction in the circular slot. The force is constant in magnitude and direction and is applied to the moving slider A. The system starts from rest in the position shown. Determine the speed of slider A as it passes the initial position of slider B if (a) the circular track lies in a horizontal plane and if (b) the circular track lies in a vertical plane. The value of R is 0.8 m. Problem 3/105

The man and his bicycle together weigh 200 lb. What power P is the man developing in riding up a 5-percent grade at a constant speed of 15 mi/hr? Problem 3/106 5

The system is released from rest with no slack in the cable and with the spring unstretched. Determine the distanced s traveled by the 10-kg cart before it comes to rest (a) if m approaches zero and (b) if . Assume no mechanical interference. Problem 3/107

The system is released from rest with no slack in the cable and with the spring stretched 200 mm. Determine the distance s traveled by the 10-kg cart before it comes to rest (a) if m approaches zero and (b) if . Assume no mechanical interference. Problem 3/108

The 2-kg collar is released from rest at A and slides down the inclined fixed rod in the vertical plane. The coefficient of kinetic friction is 0.40. Calculate (a) the velocity v of the collar as it strikes the spring and (b) the maximum deflection x of the spring. Problem 3/109

Each of the two systems is released from rest. Calculate the velocity v of each 50-lb cylinder after the 40-lb cylinder has dropped 6 ft. The 20-lb cylinder of case (a) is replaced by a 20-lb force in case (b). Problem 3/110 50 lb 40 lb 20 lb 50 lb 40 lb 20 lb (a) (b)

The 120-lb woman jogs up the flight of stairs in 5 seconds. Determine her average power output. Convert all given information to SI units and repeat your calculation. Problem 3/111

The 4-kg ball and the attached light rod rotate in the vertical plane about the fixed axis at O. If the assembly is released from rest at and moves under the action of the 60-N force, which is maintained normal to the rod, determine the velocity v of the ball as approaches . Treat the ball as a particle. Problem 3/112

An escalator handles a steady load of 30 people per minute in elevating them from the first to the second floor through a vertical rise of 24 ft. The average person weighs 140 lb. If the motor which drives the unit delivers 4 hp, calculate the mechanical efficiency e of the system. Problem 3/113

A 1200-kg car enters an 8-percent downhill grade at a speed of 100 km/h. The driver applies her brakes to bring the car to a speed of 25 km/h in a distance of 0.5 km measured along the road. Calculate the energy loss Q dissipated from the brakes in the form of heat. Neglect any friction losses from other causes such as air resistance

The 15-lb cylindrical collar is released from rest in the position shown and drops onto the spring. Calculate the velocity v of the cylinder when the spring has been compressed 2 in. Problem 3/115 18 A 15 lb B k = 80 lb/in

Determine the constant force P required to cause the 0.5-kg slider to have a speed at position 2. The slider starts from rest at position 1 and the unstretched length of the spring of modulus is 200 mm. Neglect friction. Problem 3/116

In a design test of penetration resistance, a 12-g bullet is fired through a 400-mm stack of fibrous plates with an entering velocity of 600 m/s. If the bullet emerges with a velocity of 300 m/s, calculate the average resistance R to penetration. What is the loss of energy and where does it go? Problem 3/117

The motor unit A is used to elevate the 300-kg cylinder at a constant rate of 2 m/s. If the power meter B registers an electrical input of 2.20 kW, calculate the combined electrical and mechanical efficiency e of the system. Problem 3/118

A 1700-kg car starts from rest at position A and accelerates uniformly up the incline, reaching a speed of 100 km/h at position B. Determine the power required just before the car reaches position B and also the power required when the car is halfway between positions A and B. Calculate the net tractive force F required. Problem 3/119

Two 425,000-lb locomotives pull 50 200,000-lb coal hoppers. The train starts from rest and accelerates uniformly to a speed of 40 mi/hr over a distance of 8000 ft on a level track. The constant rolling resistance of each car is 0.005 times its weight. Neglect all other retarding forces and assume that each locomotive contributes equally to the tractive force. Determine (a) the tractive force exerted by each locomotive at 20 mi/hr, (b) the power required from each locomotive at 20 mi/hr, (c) the power required from each locomotive as the train speed approaches 40 mi/hr, and (d) the power required from each locomotive if the train cruises at a steady 40 mi/hr.

The 0.6-lb slider moves freely along the fixed curved rod from A to B in the vertical plane under the action of the constant 1.3-lb tension in the cord. If the slider is released from rest at A, calculate its velocity v as it reaches B. Problem 3/121

A projectile is launched from the north pole with an initial vertical velocity . What value of will result in a maximum altitude of R/2? Neglect aerodynamic drag and use as the surfacelevel acceleration due to gravity. Problem 3/122

The spring is compressed an amount and the system is released from rest. Determine the power supplied by the spring to the 4-kg cart (a) just after release, (b) as the cart passes the position for which the spring is compressed an amount /2, and (c) as the cart passes the equilibrium position. Problem 3/123

In a test to determine the crushing characteristics of a packing material, a steel cone of mass \(m\) is released, falls a distance \(h\), and then penetrates the material. The radius of the cone is proportional to the square of the distance from its tip. The resistance \(R\) of the material to penetration depends on the cross-sectional area of the penetrating object and thus is proportional to the fourth power of the cone penetration distance \(x\), or \(R=k x^4\). If the cone comes to rest at a distance \(x = d\), determine the constant \(k\) in terms of the test conditions and results. Utilize a single application of the work energy equation.

The small slider of mass m is released from rest while in position A and then slides along the vertical-plane track. The track is smooth from A to D and rough (coefficient of kinetic friction ) from point D on. Determine (a) the normal force NB exerted by the track on the slider just after it passes point B, (b) the normal force NC exerted by the track on the slider as it passes the bottom point C, and (c) the distance s traveled along the incline past point D before the slider stops. Problem 3/125

The 0.5-kg collar slides with negligible friction along the fixed spiral rod, which lies in the vertical plane. The rod has the shape of the spiral , where r is in meters and is in radians. The collar is released from rest at A and slides to B under the action of a constant radial force . Calculate the velocity v of the slider as it reaches B. Problem 3/126

The 300-lb carriage has an initial velocity of 9 ft/sec down the incline at A, when a constant force of 110 lb is applied to the hoisting cable as shown. Calculate the velocity of the carriage when it reaches B. Show that in the absence of friction this velocity is independent of whether the initial velocity of the carriage at A was up or down the incline. Problem 3/127

Each of the sliders A and B has a mass of 2 kg and moves with negligible friction in its respective guide, with y being in the vertical direction. A 20-N horizontal force is applied to the midpoint of the connecting link of negligible mass, and the assembly is released from rest with . Calculate the velocity with which A strikes the horizontal guide when . Problem 3/128

The ball is released from position A with a velocity of 3 m/s and swings in a vertical plane. At the bottom position, the cord strikes the fixed bar at B, and the ball continues to swing in the dashed arc. Calculate the velocity of the ball as it passes position C. Problem 3/129

The two 0.2-kg sliders A and B are connected by a light rigid bar of length . If the system is released from rest while in the position shown with the spring undeformed, determine the maximum compression of the spring. Note the presence of a constant 0.14-MPa air pressure acting on one 500-mm2 side of slider A. Neglect friction. The motion occurs in a vertical plane. Problem 3/130

Once under way at a steady speed, the 1000-kg elevator A rises at the rate of 1 story (3 m) per second. Determine the power input Pin into the motor unit M if the combined mechanical and electrical efficiency of the system is . Problem 3/131

The 7-kg collar A slides with negligible friction on the fixed vertical shaft. When the collar is released from rest at the bottom position shown, it moves up the shaft under the action of the constant force applied to the cable. Calculate the stiffness k which the spring must have if its maximum compression is to be limited to 75 mm. The position of the small pulley at B is fixed. Problem 3/132

Calculate the horizontal velocity v with which the 48-lb carriage must strike the spring in order to compress it a maximum of 4 in. The spring is known as a hardening spring, since its stiffness increases with deflection as shown in the accompanying graph. Problem 3/133

The spring attached to the 10-kg mass is nonlinear, having the forcedeflection relationship shown in the figure, and is unstretched when . If the mass is moved to the position and released from rest, determine its velocity v when . Determine the corresponding velocity if the spring were linear according to , where x is in meters and the force F is in kilonewtons. Problem 3/134

The 6-kg cylinder is released from rest in the position shown and falls on the spring, which has been initially precompressed 50 mm by the light strap and restraining wires. If the stiffness of the spring is 4 kN/m, compute the additional deflection of the spring produced by the falling cylinder before it rebounds. Problem 3/135

Extensive testing of an experimental 2000-lb automobile reveals the aerodynamic drag force FD and the total nonaerodynamic rolling-resistance force FR to be as shown in the plot. Determine (a) the power required for steady speeds of 30 and 60 mi/hr on a level road, (b) the power required for a steady speed of 60 mi/hr both up and down a 6-percent incline, and (c) the steady speed at which no power is required going down the 6-percent incline. Problem 3/136

The three springs of equal moduli are unstretched when the cart is released from rest in the position . If and , determine (a) the speed v of the cart when , (b) the maximum displacement xmax of the cart, and (c) the steady-state displacement xss that would exist after all oscillations cease. Problem 3/137

The 50-lb slider in the position shown has an initial velocity on the inclined rail and slides under the influence of gravity and friction. The coefficient of kinetic friction between the slider and the rail is 0.50. Calculate the velocity of the slider as it passes the position for which the spring is compressed a distance . The spring offers a compressive resistance C and is known as a hardening spring, since its stiffness increases with deflection as shown in the accompanying graph. Problem 3/138

The 2-lb collar is released from rest at A and slides freely up the inclined rod, striking the stop at B with a velocity v. The spring of stiffness lb/ft has an unstretched length of 15 in. Calculate v. Problem 3/139

The 4-kg slider is released from rest at A and slides with negligible friction down the circular rod in the vertical plane. Determine (a) the velocity v of the slider as it reaches the bottom at B and (b) the maximum deformation x of the spring. Problem 3/140

The 1.2-kg slider is released from rest in position A and slides without friction along the vertical-plane guide shown. Determine (a) the speed of the slider as it passes position B and (b) the maximum deflection of the spring. Problem 3/141

The 1.2-kg slider of the system of Prob. 3/141 is released from rest in position A and slides without friction along the vertical-plane guide. Determine the normal force exerted by the guide on the slider (a) just before it passes point C, (b) just after it passes point C, and (c) just before it passes point E.

Point P on the 2-kg cylinder has an initial velocity as it passes position A. Neglect the mass of the pulleys and cable and determine the distance y of point P below A when the 3-kg cylinder has acquired an upward velocity of 0.6 m/s. Problem 3/143

The spring of constant k is unstretched when the slider of mass m passes position B. If the slider is released from rest in position A, determine its speeds as it passes points B and C. What is the normal force exerted by the guide on the slider at position C? Neglect friction between the mass and the circular guide, which lies in a vertical plane. Problem 3/144

It is desired that the 100-lb container, when released from rest in the position shown, have no velocity after dropping 7 ft to the platform below. Specify the proper weight W of the counterbalancing cylinder. Problem 3/145

The system is released from rest with the spring initially stretched 3 in. Calculate the velocity v of the cylinder after it has dropped 0.5 in. The spring has a stiffness of 6 lb/in. Neglect the mass of the small pulley. Problem 3/146

The projectile of Prob. 3/122 is repeated here. By the method of this article, determine the vertical launch velocity which will result in a maximum altitude of R/2. The launch is from the north pole and aerodynamic drag can be neglected. Use as the surface-level acceleration due to gravity. Problem 3/147

The 1.5-kg slider C moves along the fixed rod under the action of the spring whose unstretched length is 0.3 m. If the velocity of the slider is 2 m/s at point A and 3 m/s at point B, calculate the work U done by friction between these two points. Also, determine the average friction force acting on the slider between A and B if the length of the path is 0.70 m. The x-y plane is horizontal. Problem 3/148

The light rod is pivoted at O and carries the 5- and 10-lb particles. If the rod is released from rest at and swings in the vertical plane, calculate (a) the velocity v of the 5-lb particle just before it hits the spring in the dashed position and (b) the maximum compression x of the spring. Assume that x is small so that the position of the rod when the spring is compressed is essentially horizontal. Problem 3/149

The 0.8-kg particle is attached to the system of two light rigid bars, all of which move in a vertical plane. The spring is compressed an amount b/2 when , and the length . The system is released from rest in a position slightly above that for . (a) If the maximum value of is observed to be , determine the spring constant k. (b) For , determine the speed of the particle when . Also find the corresponding value of . Problem 3/150

The two springs, each of stiffness , are of equal length and undeformed when . If the mechanism is released from rest in the position , determine its angular velocity when . The mass m of each sphere is 3 kg. Treat the spheres as particles and neglect the masses of the light rods and springs. Problem 3/151

If the system is released from rest, determine the speeds of both masses after B has moved 1 m. Neglect friction and the masses of the pulleys. Problem 3/152

The 3-lb ball is given an initial velocity in the vertical plane at position A, where the two horizontal attached springs are unstretched. The ball follows the dashed path shown and crosses point B, which is 5 in. directly below A. Calculate the velocity of the ball at B. Each spring has a stiffness of 10 lb/in. Problem 3/153

The 0.75-kg particle is attached to the light slender rod OA which pivots freely about a horizontal axis through point O. The system is released from rest while in the position where the spring is unstretched. If the particle is observed to stop momentarily in the position , determine the spring constant k. For your computed value of k, what is the particle speed v at the position ? Problem 3/154

The spring has an unstretched length of 25 in. If the system is released from rest in the position shown, determine the speed v of the ball (a) when it has dropped a vertical distance of 10 in. and (b) when the rod has rotated . Problem 3/155

The two 1.5-kg spheres are released from rest and gently nudged outward from the position and then rotate in a vertical plane about the fixed centers of their attached gears, thus maintaining the same angle for both rods. Determine the velocity v of each sphere as the rods pass the position . The spring is unstretched when , and the masses of the two identical rods and the two gear wheels may be neglected. Problem 3/156

A rocket launches an unpowered space capsule at point A with an absolute velocity at an altitude of 25 mi. After the capsule has traveled a distance of 250 mi measured along its absolute space trajectory, its velocity at B is 7600 mi/hr and its altitude is 50 mi. Determine the average resistance P to motion in the rarified atmosphere. The earth weight of the capsule is 48 lb, and the mean radius of the earth is 3959 mi. Consider the center of the earth fixed in space. Problem 3/157

The collar has a mass of 2 kg and is attached to the light spring, which has a stiffness of 30 N/m and an unstretched length of 1.5 m. The collar is released from rest at A and slides up the smooth rod under the action of the constant 50-N force. Calculate the velocity v of the collar as it passes position B.

The shank of the 5-lb vertical plunger occupies the dashed position when resting in equilibrium against the spring of stiffness . The upper end of the spring is welded to the plunger, and the lower end is welded to the base plate. If the plunger is lifted in. above its equilibrium position and released from rest, calculate its velocity v as it strikes the button A. Friction is negligible. Problem 3/159

Upon its return voyage from a space mission, the spacecraft has a velocity of 24 000 km/h at point A, which is 7000 km from the center of the earth. Determine the velocity of the spacecraft when it reaches point B, which is 6500 km from the center of the earth. The trajectory between these two points is outside the effect of the earths atmosphere. Problem 3/160

The 5-kg cylinder is released from rest in the position shown and compresses the spring of stiffness . Determine the maximum compression xmax of the spring and the maximum velocity of the cylinder along with the corresponding deflection x of the spring. Problem 3/161

A 175-lb pole vaulter carrying a uniform 16-ft, 10-lb pole approaches the jump with a velocity v and manages to barely clear the bar set at a height of 18 ft. As he clears the bar, his velocity and that of the pole are essentially zero. Calculate the minimum possible value of required for him to make the jump. Both the horizontal pole and the center of gravity of the vaulter are 42 in. above the ground during the approach. Problem 3/162

The cylinder of mass m is attached to the collar bracket at A by a spring of stiffness k. The collar fits loosely on the vertical shaft, which is lowering both the collar and the suspended cylinder with a constant velocity v. When the collar strikes the base B, it stops abruptly with essentially no rebound. Determine the maximum additional deflection of the spring after the impact.

The cars of an amusement-park ride have a speed at the lowest part of the track. Determine their speed at the highest part of the track. Neglect energy loss due to friction. (Caution: Give careful thought to the change in potential energy of the system of cars.) Problem 3/164

The two right-angle rods with attached spheres are released from rest in the position . If the system is observed to momentarily come to rest when , determine the spring constant k. The spring is unstretched when . Treat the spheres as particles and neglect friction. Problem 3/165 60 mm 180 mm 2 kg 2 kg k

Calculate the maximum velocity of slider B if the system is released from rest with . Motion is in the vertical plane. Assume that friction is negligible. The sliders have equal masses, and the motion is restricted to . Problem 3/166

The mechanism is released from rest with where the uncompressed spring of stiffness is just touching the underside of the 4-kg collar. Determine the angle corresponding to the maximum compression of the spring. Motion is in the vertical plane, and the mass of the links may be neglected. Problem 3/167

A particle of mass m is attached to one end of a light slender rod which pivots about a horizontal axis through point O. The spring constant and the distance . If the system is released from rest in the horizontal position shown where the spring is unstretched, the bar is observed to deflect a maximum of clockwise. Determine (a) the particle mass m and (b) the particle speed v after a displacement of from the position shown. Neglect friction. Problem 3/168

The 3-kg sphere is carried by the parallelogram linkage where the spring is unstretched when . If the mechanism is released from rest at , calculate the velocity v of the sphere when the position is passed. The links are in the vertical plane, and their mass is small and may be neglected. Problem 3/169

The system is at rest with the spring unstretched when . The 3-kg particle is then given a slight nudge to the right. (a) If the system comes to momentary rest at , determine the spring constant k. (b) For the value , find the speed of the particle when . Use the value m throughout and neglect friction. Problem 3/170 k C B m A b

The system is released from rest with the angle r/2 . Determine when reaches . Use the values , , and m. Neglect friction and the mass of bar OB, and treat the body B as a particle. Problem 3/171

The flexible bicycle-type chain of length and mass per unit length is released from rest with in the smooth circular channel and falls through the hole in the supporting surface. Determine the velocity v of the chain as the last link leaves the slot. Problem 3/172

The rubber mallet is used to drive a cylindrical plug into the wood member. If the impact force varies with time as shown in the plot, determine the magnitude of the linear impulse delivered by the mallet to the plug. Problem 3/173

The 1500-kg car has a velocity of 30 km/h up the 10-percent grade when the driver applies more power for 8 s to bring the car up to a speed of 60 km/h. Calculate the time average F of the total force tangent to the road exerted on the tires during the 8 s. Treat the car as a particle and neglect air resistance. Problem 3/174

A 0.2-kg particle is moving with a velocity m/s at time If the single force F (5 3t)i (2 t 2 )j 3k N acts on the particle, determine its velocity at time .

A 75-g projectile traveling at 600 m/s strikes and becomes embedded in the 50-kg block, which is initially stationary. Compute the energy lost during the impact. Express your answer as an absolute value and as a percentage n of the original system energy E. Problem 3/176

A jet-propelled airplane with a mass of 10 Mg is flying horizontally at a constant speed of 1000 km/h under the action of the engine thrust T and the equal and opposite air resistance R. The pilot ignites two rocket-assist units, each of which develops a forward thrust T0 of 8 kN for 9 s. If the velocity of the airplane in its horizontal flight is 1050 km/h at the end of the 9 s, calculate the timeaverage increase in air resistance. The mass of the rocket fuel used is negligible compared with that of the airplane. Problem 3/177

A 60-g bullet is fired horizontally with a velocity into the 3-kg block of soft wood initially at rest on the horizontal surface. The bullet emerges from the block with the velocity , and the block is observed to slide a distance of 2.70 m before coming to rest. Determine the coefficient of kinetic friction between the block and the supporting surface. Problem 3/178

At time , the velocity of cylinder A is 0.3 m/s down. By the methods of this article, determine the velocity of cylinder B at time Assume no mechanical interference and neglect all friction. Problem 3/179

The resistance to motion of a certain racing toboggan is 2 percent of the normal force on its runners. Calculate the time t required for the toboggan to reach a speed of 100 km/h down the slope if it starts from rest.

Freight car A with a gross weight of 150,000 lb is moving along the horizontal track in a switching yard at 2 mi/hr. Freight car B with a gross weight of 120,000 lb and moving at 3 mi/hr overtakes car A and is coupled to it. Determine (a) the common velocity v of the two cars as they move together after being coupled and (b) the loss of energy due to the impact. Problem 3/181

The 90-kg man dives from the 40-kg canoe. The velocity indicated in the figure is that of the man relative to the canoe just after loss of contact. If the man, woman, and canoe are initially at rest, determine the horizontal component of the absolute velocity of the canoe just after separation. Neglect drag on the canoe, and assume that the 60-kg woman remains motionless relative to the canoe. Problem 3/182

An experimental rocket sled weighs 5200 lb and is propelled by six rocket motors each with an impulse rating of 8600 lb-sec. The rockets are fired at 1-sec intervals, and the duration of each rocket firing is 2 sec. If the velocity of the sled 10 sec from the start is 200 mi/hr, determine the time average R of the total aerodynamic and mechanical resistance to motion. Neglect the loss of mass due to exhausted fuel compared with the mass of the sled. Problem 3/183

The 200-kg lunar lander is descending onto the moons surface with a velocity of 6 m/s when its retro-engine is fired. If the engine produces a thrust T for 4 s which varies with time as shown and then cuts off, calculate the velocity of the lander when assuming that it has not yet landed. Gravitational acceleration at the moons surface is Problem 3/184

The slider of mass m1 is released from rest in the position shown and then slides down the right side of the contoured body of mass m2. For the conditions m1 0.50 kg, m2 3 kg, and r 0.25 m, determine the absolute velocities of both masses at the instant of separation. Neglect friction. Problem 3/185

A supertanker with a total displacement (weight) of 140(103 ) long tons (one long ton equals 2240 lb) is moving forward at a speed of 2 knots when the engines are reversed to produce a rearward propeller thrust of 90,000 lb. How long would it take the tanker to acquire a speed of 2 knots in the reverse direction? Can you justify neglecting the impulse of water resistance of the hull? (Recall 1 knot 1.151 mi/hr.)

The 20-lb block is moving to the right with a velocity of 2 ft/sec on a horizontal surface when a force P is applied to it at time t 0. Calculate the velocity v of the block when t 0.4 sec. The coefficient of kinetic friction is 0.30. Problem 3/187

The initially stationary 12-kg block is subjected to the time-varying force whose magnitude P is shown in the plot. The angle remains constant. Determine the block speed at (a) t 1 s and (b) t 4 s. Problem 3/188

The tow truck with attached 1200-kg car accelerates uniformly from 30 km/h to 70 km/h over a 15-s interval. The average rolling resistance for the car over this speed interval is 500 N. Assume that the 60 angle shown represents the time average configuration and determine the average tension in the tow cable. Problem 3/189

The 140-g projectile is fired with a velocity of 600 m/s and picks up three washers, each with a mass of 100 g. Find the common velocity v of the projectile and washers. Determine also the loss of energy during the interaction. Problem 3/190

The spring of modulus k 200 N/m is compressed a distance of 300 mm and suddenly released with the system at rest. Determine the absolute velocities of both masses when the spring is unstretched. Neglect friction. Problem 3/191

The 4-kg cart, at rest at time t 0, is acted on by a horizontal force which varies with time t as shown. Neglect friction and determine the velocity of the cart at t 1 s and at t 3 s. Problem 3/192

The space shuttle launches an 800-kg satellite by ejecting it from the cargo bay as shown. The ejection mechanism is activated and is in contact with the satellite for 4 s to give it a velocity of 0.3 m/s in the z-direction relative to the shuttle. The mass of the shuttle is 90 Mg. Determine the component of velocity v of the shuttle in the minus z-direction resulting from the ejection. Also find the time average Fav of the ejection force. Problem 3/193

The initially stationary 100-lb block is subjected to the time-varying force whose magnitude P is shown in the plot. Determine the speed v of the block at times t 1, 3, 5, and 7 sec. Note that the force P is zero after t 6 sec. Problem 3/194

The 900-kg motorized unit A is designed to raise and lower the 600-kg bucket B of concrete. Determine the average force R which supports unit A during the 6 seconds required to slow the descent of the bucket from 3 m/s to 0.5 m/s. Analyze the entire system as a unit without finding the tension in the cable. Problem 3/195

The cart of mass m is subjected to the exponentially decreasing force F, which represents a shock or blast loading. If the cart is stationary at time t 0, determine its velocity v and displacement s as functions of time. What is the value of v for large values of t? Problem 3/196

Determine the time required by a diesel-electric locomotive, which produces a constant drawbar pull of 60,000 lb, to increase the speed of an 1800- ton freight train from 20 mi/hr to 30 mi/hr up a 1-percent grade. Train resistance is 10 lb per ton.

The 450-kg ram of a pile driver falls 1.4 m from rest and strikes the top of a 240-kg pile embedded 0.9 m in the ground. Upon impact the ram is seen to move with the pile with no noticeable rebound. Determine the velocity v of the pile and ram immediately after impact. Can you justify using the principle of conservation of momentum even though the weights act during the impact? Problem 3/198

The cart is moving down the incline with a velocity v0 20 m/s at t 0, at which time the force P begins to act as shown. After 5 seconds the force continues at the 50-N level. Determine the velocity of the cart at time t 8 s and calculate the time t at which the cart velocity is zero. Problem 3/199

Car B is initially stationary and is struck by car A moving with initial speed v1 20 mi/hr. The cars become entangled and move together with speed after the collision. If the time duration of the collision is 0.1 sec, determine (a) the common final speed , (b) the average acceleration of each car during the collision, and (c) the magnitude R of the average force exerted by each car on the other car during the impact. All brakes are released during the collision. Problem 3/200

The 12-Mg truck drives onto the 350-Mg barge from the dock at 20 km/h and brakes to a stop on the deck. The barge is free to move in the water, which offers negligible resistance to motion at low speeds. Calculate the speed of the barge after the truck has come to rest on it. Problem 3/201 v 2

An 8-Mg truck is resting on the deck of a barge which displaces 240 Mg and is at rest in still water. If the truck starts and drives toward the bow at a speed relative to the barge vrel 6 km/h, calculate the speed v of the barge. The resistance to the motion of the barge through the water is negligible at low speeds. Problem 3/202

Car B weighing 3200 lb and traveling west at 30 mi/hr collides with car A weighing 3400 lb and traveling north at 20 mi/hr as shown. If the two cars become entangled and move together as a unit after the crash, compute the magnitude v of their common velocity immediately after the impact and the angle made by the velocity vector with the north direction. Problem 3/203

A 16.1-lb body is traveling in a horizontal straight line with a velocity of 12 ft/sec when a horizontal force P is applied to it at right angles to the initial direction of motion. If P varies according to the accompanying graph, remains constant in direction, and is the only force acting on the body in its plane of motion, find the magnitude of the velocity of the body when t 2 sec and the angle which the velocity makes with the direction of P. Problem 3/204

The force P, which is applied to the 10-kg block initially at rest, varies linearly with time as indicated. If the coefficients of static and kinetic friction between the block and the horizontal surface are 0.60 and 0.40, respectively, determine the velocity of the block when t 4 s. Problem 3/205

The 10-kg block is at rest on the rough incline at time t 0 and then it is subjected to the force of constant direction and time-varying magnitude P given in the plot. Determine the velocity of the block at times t 1, 3, 5, and 7 s. Note that the force P is zero after t 6 s. Problem 3/206

The 1.62-oz golf ball is struck by the five-iron and acquires the velocity shown in a time period of 0.001 sec. Determine the magnitude R of the average force exerted by the club on the ball. What acceleration magnitude a does this force cause, and what is the distance d over which the launch velocity is achieved, assuming constant acceleration? Problem 3/207

The 580-ton tug is towing the 1200-ton coal barge at a steady speed of 6 knots. For a short period of time, the stern winch takes in the towing cable at the rate of 2 ft/sec. Calculate the reduced speed v of the tug during this interval. Assume the tow cable to be horizontal. (Recall 1 knot 1.688 ft/sec) Problem 3/208

The cylindrical plug A of mass mA is released from rest at B and slides down the smooth circular guide. The plug strikes the block C and becomes embedded in it. Write the expression for the distance s which the block and plug slide before coming to rest. The coefficient of kinetic friction between the block and the horizontal surface is . Problem 3/209

The baseball is traveling with a horizontal velocity of 85 mi/hr just before impact with the bat. Just after the impact, the velocity of the -oz ball is 130 mi/hr directed at to the horizontal as shown. Determine the x- and y-components of the average force R exerted by the bat on the baseball during the 0.005-sec impact. Comment on the treatment of the weight of the baseball (a) during the impact and (b) over the first few seconds after impact. Problem 3/210

A tennis player strikes the tennis ball with her racket while the ball is still rising. The ball speed before impact with the racket is v1 15 m/s and after impact its speed is v2 22 m/s, with directions as shown in the figure. If the 60-g ball is in contact with the racket for 0.05 s, determine the magnitude of the average force R exerted by the racket on the ball. Find the angle made by R with the horizontal. Comment on the treatment of the ball weight during impact. Problem 3/211

The 400-kg ram of a pile driver is designed to fall 1.5 m from rest and strike the top of a 300-kg pile partially embedded in the ground. The deeper the penetration, the greater is the tendency for the ram to rebound as a result of the impact. Calculate the velocity v of the pile immediately after impact for the following three conditions: (a) initial resistance to penetration is small at the outset, and the ram is observed to move with the pile immediately after impact; (b) resistance to penetration has increased, and the ram is seen to have zero velocity immediately after impact; (c) resistance to penetration is high, and the ram is seen to rebound to a height of 100 mm above the point of impact. Why is it permissible to neglect the impulse of the weight of the ram during impact? Problem 3/212

The simple pendulum A of mass mA and length l is suspended from the trolley B of mass mB. If the system is released from rest at 0, determine the velocity vB of the trolley when Friction is negligible. Problem 3/213

Two barges, each with a displacement (mass) of 500 Mg, are loosely moored in calm water. A stunt driver starts his 1500-kg car from rest at A, drives along the deck, and leaves the end of the ramp at a speed of 50 km/h relative to the barge and ramp. The driver successfully jumps the gap and brings his car to rest relative to barge 2 at B. Calculate the velocity v2 imparted to barge 2 just after the car has come to rest on the barge. Neglect the resistance of the water to motion at the low velocities involved

Determine the magnitude HO of the angular momentum of the 2-kg sphere about point O (a) by using the vector definition of angular momentum and (b) by using an equivalent scalar approach. The center of the sphere lies in the x-y plane. Problem 3/215

The 3-kg sphere moves in the x-y plane and has the indicated velocity at a particular instant. Determine its (a) linear momentum, (b) angular momentum about point O, and (c) kinetic energy. Problem 3/216

A particle with a mass of 4 kg has a position vector in meters given by r 3t 2 i 2tj 3tk, where t is the time in seconds. For t 3 s determine the magnitude of the angular momentum of the particle and the magnitude of the moment of all forces on the particle, both about the origin of coordinates.

0.4-kg particle is located at the position r1 2i 3j k m and has the velocity v1 i j 2k m/s at time t 0. If the particle is acted upon by a single force which has the moment MO (4 2t)i j 5k about the origin O of the coordinate system in use, determine the angular momentum about O of the particle when t 4 s.

At a certain instant, the particle of mass m has the position and velocity shown in the figure, and it is acted upon by the force F. Determine its angular momentum about point O and the time rate of change of this angular momentum. Problem 3/219

The small spheres, which have the masses and initial velocities shown in the figure, strike and become attached to the spiked ends of the rod, which is freely pivoted at O and is initially at rest. Determine the angular velocity of the assembly after impact. Neglect the mass of the rod. Problem 3/220

The particle of mass m is gently nudged from the equilibrium position A and subsequently slides along the smooth circular path which lies in a vertical plane. Determine the magnitude of its angular momentum about point O as it passes (a) point B and (b) point C. In each case, determine the time rate of change of HO. Problem 3/221

The assembly starts from rest and reaches an angular speed of 150 rev/min under the action of a 20-N force T applied to the string for t seconds. Determine t. Neglect friction and all masses except those of the four 3-kg spheres, which may be treated as particles. Problem 3/223

Just after launch from the earth, the space-shuttle orbiter is in the 37 137mi orbit shown. At the apogee point A, its speed is 17,290 mi/hr. If nothing were done to modify the orbit, what would its speed be at the perigee P? Neglect aerodynamic drag. (Note that the normal practice is to add speed at A, which raises the perigee altitude to a value that is well above the bulk of the atmosphere.) Problem 3/224

A small 4-oz particle is projected with a horizontal velocity of 6 ft/sec into the top A of the smooth circular guide fixed in the vertical plane. Calculate the time rate of change B of angular momentum about point B when the particle passes the bottom of the guide at C. Problem 3/225

The small particle of mass m and its restraining cord are spinning with an angular velocity on the horizontal surface of a smooth disk, shown in section. As the force F is slightly relaxed, r increases and changes. Determine the rate of change of with respect to r and show that the work done by F during a movement dr equals the change in kinetic energy of the particle. Problem 3/226 r m F

7 The 6-kg sphere and 4-kg block (shown in section) are secured to the arm of negligible mass which rotates in the vertical plane about a horizontal axis at O. The 2-kg plug is released from rest at A and falls into the recess in the block when the arm has reached the horizontal position. An instant before engagement, the arm has an angular velocity rad/s. Determine the angular velocity of the arm immediately after the plug has wedged itself in the block.

The two spheres of equal mass m are able to slide along the horizontal rotating rod. If they are initially latched in position a distance r from the rotating axis with the assembly rotating freely with an angular velocity , determine the new angular velocity after the spheres are released and finally assume positions at the ends of the rod at a radial distance of 2r. Also find the fraction n of the initial kinetic energy of the system which is lost. Neglect the small mass of the rod and shaft. Problem 3/228

The speed of Mercury at its point A of maximum distance from the sun is 38 860 m/s. Determine its speeds at points B and P. Problem 3/229

A small 0.1-kg particle is given a velocity of 2 m/s on the horizontal x-y plane and is guided by the fixed curved rail. Friction is negligible. As the particle crosses the y-axis at A, its velocity is in the x-direction, and as it crosses the x-axis at B, its velocity makes a 60 angle with the x-axis. The radius of curvature of the path at B is 500 mm. Determine the time rate of change of the angular momentum of the particle about the z-axis through O at both A and B. Problem 3/230

Determine the magnitude HO of the angular momentum about the launch point O of the projectile of mass m, which is launched with speed v0 at the angle as shown (a) at the instant of launch and (b) at the instant of impact. Qualitatively account for the two results. Neglect atmospheric resistance. Problem 3/231

The particle of mass m is launched from point O with a horizontal velocity u at time t 0. Determine its angular momentum HO relative to point O as a function of time. Problem 3/232

The particle of mass m is launched from point O with a horizontal velocity u at time t 0. Determine its angular momentum HO relative to point O as a function of time. Problem 3/232

3 A particle of mass m is released from rest in position A and then slides down the smooth verticalplane track. Determine its angular momentum about both points A and D (a) as it passes position B and (b) as it passes position C. Problem 3/233

At the point A of closest approach to the sun, a comet has a velocity vA 188,500 ft/sec. Determine the radial and transverse components of its velocity vB at point B, where the radial distance from the sun is 75(106 ) mi. Problem 3/234

A pendulum consists of two 3.2-kg concentrated masses positioned as shown on a light but rigid bar. The pendulum is swinging through the vertical position with a clockwise angular velocity 6 rad/s when a 50-g bullet traveling with velocity 300 m/s in the direction shown strikes the lower mass and becomes embedded in it. Calculate the angular velocity which the pendulum has immediately after impact and find the maximum angular deflection of the pendulum.

The 1.5-lb sphere moves in a horizontal plane and is controlled by a cord which is reeled in and out below the table in such a way that the center of the sphere is confined to the path given by where x and y are in feet. If the speed of the sphere is vA 8 ft/sec as it passes point A, determine the tension TB in the cord as the sphere passes point B. Friction is negligible. Problem 3/236

A particle is launched with a horizontal velocity v0 0.55 m/s from the position shown and then slides without friction along the funnel-like surface. Determine the angle which its velocity vector makes with the horizontal as the particle passes level O-O. The value of r is 0.9 m. Problem 3/237

The assembly of two 5-kg spheres is rotating freely about the vertical axis at 40 rev/min with . If the force F which maintains the given position is increased to raise the base collar and reduce to , determine the new angular velocity . Also determine the work U done by F in changing the configuration of the system. Assume that the mass of the arms and collars is negligible. Problem 3/238 100 mm 300 m

Tennis balls are usually rejected if they fail to rebound to waist level when dropped from shoulder level. If a ball just passes the test as indicated in the figure, determine the coefficient of restitution e and the percentage n of the original energy lost during the impact. Problem 3/239

If the tennis ball of Prob. 3/239 has a coefficient of restitution e 0.8 during impact with the court surface, determine the velocity v0 with which the ball must be thrown downward from the 1600-mm shoulder level if it is return to the same level after bouncing once on the court surface.

Compute the final velocities and after collision of the two cylinders which slide on the smooth horizontal shaft. The coefficient of restitution is e 0.6. Problem 3/241 v1 = 7 m/s m1 = 2 kg v2 = 5 m/s m

The two bodies have the masses and initial velocities shown in the figure. The coefficient of restitution for the collision is e 0.3, and friction is negligible. If the time duration of the collision is 0.025 s, determine the average impact force which is exerted on the 3-kg body. Problem 3/242

The sphere of mass m1 travels with an initial velocity v1 directed as shown and strikes the sphere of mass m2. For a given coefficient of restitution e, determine the mass ratio m1/m2 which results in m1 being motionless after the impact. Problem 3/243

Three identical steel cylinders are free to slide on the fixed horizontal shaft. Cylinders 2 and 3 are at rest and are approached by cylinder 1 at a speed u. Express the final speed v of cylinder 3 in terms of u and the coefficient of restitution e. Problem 3/244

Cylinder A is moving to the right with speed v when it impacts the initially stationary cylinder B. Both cylinders have mass m, and the coefficient of restitution for the collision is e. Determine the maximum deflection of the spring of modulus k. Neglect friction. Problem 3/245

Car B is initially stationary and is struck by car A, which is moving with speed v. The mass of car B is pm, where m is the mass of car A and p is a positive constant. If the coefficient or restitution is e 0.1, express the speeds and of the two cars at the end of the impact in terms of p and v. Evaluate your expressions for p 0.5. Problem 3/246

Determine the coefficient of restitution e for a steel ball dropped from rest at a height h above a heavy horizontal steel plate if the height of the second rebound is h2. Problem 3/247

If the center of the ping-pong ball is to clear the net as shown, at what height h should the ball be horizontally served? Also determine h2. The coeffi- cient of restitution for the impacts between ball and table is e 0.9, and the radius of the ball is r 0.75 in. Problem 3/248

In the selection of the ram of a pile driver, it is desired that the ram lose all of its kinetic energy at each blow. Hence, the velocity of the ram is zero immediately after impact. The mass of each pile to be driven is 300 kg, and experience has shown that a coefficient of restitution of 0.3 can be expected. What should be the mass m of the ram? Compute the velocity v of the pile immediately after impact if the ram is dropped from a height of 4 m onto the pile. Also compute the energy loss due to impact at each blow. Problem 3/249 4 m m 300 kg

Freight car A of mass mA is rolling to the right when it collides with freight car B of mass mB initially at rest. If the two cars are coupled together at impact, show that the fractional loss of energy equals mB/(mA mB). Problem 3/250

Pool ball B is to be shot into the side pocket D by banking it off the cushion at C. Specify the location x of the cushion impact for coefficients of restitution (a) e 1 and (b) e 0.8. Problem 3/251

Determine the value of the coefficient of restitution e which results in the final velocity being perpendicular to the initial velocity v. Problem 3/252

Determine the value of the coefficient of restitution e for which the outgoing angle is one-half of the incoming angle as shown. Evaluate your general expression for Problem 3/253

The figure shows n spheres of equal mass m suspended in a line by wires of equal length so that the spheres are almost touching each other. If sphere 1 is released from the dashed position and strikes sphere 2 with a velocity v1, write an expression for the velocity vn of the nth sphere immediately after being struck by the one adjacent to it. The common coefficient of restitution is e. Problem 3/254

The ball is released from position A and drops 0.75 m to the incline. If the coefficient of restitution in the impact is e 0.85, determine the slant range R. Problem 3/255

A projectile is launched from point A and has a horizontal range L1 as shown. If the coefficient of restitution at B is e, determine the distance L2. Problem 3/256

A basketball traveling with the velocity shown in the figure strikes the backboard at A. If the coeffi- cient of restitution for this impact is e 0.84, determine the required distance h above the hoop if the ball is to arrive at the center B of the hoop. Carry out two solutions: (a) an approximate solution obtained by neglecting the effects of gravity from A to B and (b) a solution which accounts for gravity from A to B. Neglect the diameter of the ball compared with h. Problem 3/257

The two cars collide at right angles in the intersection of two icy roads. Car A has a mass of 1200 kg and car B has a mass of 1600 kg. The cars become entangled and move off together with a common velocity in the direction indicated. If car A was traveling 50 km/h at the instant of impact, compute the corresponding velocity of car B just before impact. Problem 3/258

The two identical steel balls moving with initial velocities vA and vB collide as shown. If the coefficient of restitution is determine the velocity of each ball just after impact and the percentage loss n of system kinetic energy. Problem 3/259 y x A B vB = 8 ft/sec

A 0.1-kg meteor and a 1000-kg spacecraft have the indicated absolute velocities just before colliding. The meteor punches a hole entirely through the spacecraft. Instruments indicate that the velocity of the meteor relative to the spacecraft just after the collision is vm/s 1880i 6898j m/s. Determine the direction of the absolute velocity of the spacecraft after the collision. Problem 3/260

Two identical hockey pucks moving with initial velocities vA and vB collide as shown. If the coefficient of restitution is , determine the velocity (magnitude and direction with respect to the positive x-axis) of each puck just after impact. Also calculate the percentage loss n of system kinetic energy. Problem 3/261

Sphere A collides with sphere B as shown in the figure. If the coefficient of restitution is e 0.5, determine the x- and y-components of the velocity of each sphere immediately after impact. Motion is confined to the x-y plane. Problem 3/262

Determine the coefficient of restitution e which will allow the ball to bounce down the steps as shown. The tread and riser dimensions, d and h, respectively, are the same for every step, and the ball bounces the same distance above each step. What horizontal velocity vx is required so that the ball lands in the center of each tread? Problem 3/263

During a pregame warmup period, two basketballs collide above the hoop when in the positions shown. Just before impact, ball 1 has a velocity v1 which makes a 30 angle with the horizontal. If the velocity v2 of ball 2 just before impact has the same magnitude as v1, determine the two possible values of the angle , measured from the horizontal, which will cause ball 1 to go directly through the center of the basket. The coefficient of restitution is e 0.8. Problem 3/264 30 2 1 v2 v1 v

The 0.5-kg cylinder A is released from rest from the position shown and drops the distance h1 0.6 m. It then collides with the 0.4-kg block B; the coeffi- cient of restitution is e 0.8. Determine the maximum downward displacement h2 of block B. Neglect all friction and assume that block B is initially held in place by a hidden mechanism until the collision begins. The two springs of modulus k 500 N/m are initially unstretched, and the distance d 0.8 m. Problem 3/265

A child throws a ball from point A with a speed of 50 ft/sec. It strikes the wall at point B and then returns exactly to point A. Determine the necessary angle if the coefficient of restitution in the wall impact is e 0.5. Problem 3/266

Determine the speed v of the earth in its orbit about the sun. Assume a circular orbit of radius 93(106 ) miles.

What velocity v must the space shuttle have in order to release the Hubble space telescope in a circular earth orbit 590 km above the surface of the earth? Problem 3/268

Show that the path of the moon is concave toward the sun at the position shown. Assume that the sun, earth, and moon are in the same line. Problem 3/269

A spacecraft is orbiting the earth in a circular orbit of altitude H. If its rocket engine is activated to produce a sudden burst of speed, determine the increase necessary to allow the spacecraft to escape from the earths gravity field. Calculate if H 200 mi.

Determine the apparent velocity vrel of a satellite moving in a circular equatorial orbit 200 mi above the earth as measured by an observer on the equator (a) for a west-to-east orbit and (b) for an eastto-west orbit. Why is the west-to-east orbit more easily achieved?

A spacecraft is in an initial circular orbit with an altitude of 350 km. As it passes point P, onboard thrusters give it a velocity boost of 25 m/s. Determine the resulting altitude gain at point A. Problem 3/272

If the perigee altitude of an earth satellite is 240 km and the apogee altitude is 400 km, compute the eccentricity e of the orbit and the period of one complete orbit in space.

In one of the orbits of the Apollo spacecraft about the moon, its distance from the lunar surface varied from 60 mi to 180 mi. Compute the maximum velocity of the spacecraft in this orbit.

A satellite is in a circular earth orbit of radius 2R, where R is the radius of the earth. What is the minimum velocity boost necessary to reach point B, which is a distance 3R from the center of the earth? At what point in the original circular orbit should the velocity increment be added? Problem 3/275

The Mars orbiter for the Viking mission was designed to make one complete trip around the planet in exactly the same time that it takes Mars to revolve once about its own axis. This time is 24 h, 37 min, 23 s. In this way, it is possible for the orbiter to pass over the landing site of the lander capsule at the same time in each Martian day at the orbiters minimum (periapsis) altitude. For the Viking I mission, the periapsis altitude of the orbiter was 1508 km. Make use of the data in Table D/2 in Appendix D and compute the maximum (apoapsis) altitude ha for the orbiter in its elliptical path. Problem 3/276 ha h

Determine the speed v required of an earth satellite at point A for (a) a circular orbit, (b) an elliptical orbit of eccentricity e 0.1, (c) an elliptical orbit of eccentricity e 0.9, and (d) a parabolic orbit. In cases (b), (c), and (d), A is the orbit perigee. Problem 3/277

Initially in the 240-km circular orbit, the spacecraft S receives a velocity boost at P which will take it to with no speed at that point. Determine the required velocity increment v at point P and also determine the speed when r 2rP. At what value of does r become 2rP? Problem 3/278

Satellite A moving in the circular orbit and satellite B moving in the elliptical orbit collide and become entangled at point C. If the masses of the satellites are equal, determine the maximum altitude hmax of the resulting orbit. Problem 3/279

If the earth were suddenly deprived of its orbital velocity around the sun, find the time t which it would take for the earth to fall to the location of the center of the sun. (Hint: The time would be one-half the period of a degenerate elliptical orbit around the sun with the semiminor axis approaching zero.) Refer to Table D/2 for the exact period of the earth around the sun.

Just after launch from the earth, the space-shuttle orbiter is in the 37 137-mi orbit shown. The first time that the orbiter passes the apogee A, its two orbital-maneuvering-system (OMS) engines are fired to circularize the orbit. If the weight of the orbiter is 175,000 lb and the OMS engines have a thrust of 6000 lb each, determine the required time duration t of the burn. Problem 3/281

After launch from the earth, the 85 000-kg spaceshuttle orbiter is in the elliptical orbit shown. If the orbit is to be circularized at the apogee altitude of 320 km, determine the necessary time duration during which its two orbital-maneuveringsystem (OMS) engines, each of which has a thrust of 30 kN, must be fired when the apogee position C is reached. Problem 3/282

Just before separation of the lunar module, the Apollo 17 command module was in the lunar orbit shown in the figure. Determine the spacecraft speeds at points P and A, which are called perilune and apolune, respectively. Later in the mission, with the lunar module on the surface of the moon, the orbit of the command module was to be circularized. Determine the speed increment required if circularization is to be performed at A. Problem 3/283 P A 28 km 1

Determine the required velocity vB in the direction  indicated so that the spacecraft path will be tangent to the circular orbit at point C. What must be the distance b so that this path is possible? Problem 3/284

An earth satellite A is in a circular west-to-east equatorial orbit a distance 300 km above the surface of the earth as indicated. An observer B on the equator who sees the satellite directly overhead will see it directly overhead in the next orbit at position because of the rotation of the earth. The radial line to the satellite will have rotated through the angle and the observer will measure the apparent period as a value slightly greater than the true period . Calculate and Problem 3/285 Equator A N O B A B  .

Determine the angle made by the velocity vector v with respect to the -direction for an earth satellite traveling in an elliptical orbit of eccentricity e. Express in terms of the angle measured from perigee. Problem 3/286

Two satellites B and C are in the same circular orbit of altitude 500 miles. Satellite B is 1000 mi ahead of satellite C as indicated. Show that C can catch up to B by putting on the brakes. Specifi- cally, by what amount should the circular-orbit velocity of C be reduced so that it will rendezvous with B after one period in its new elliptical orbit? Check to see that C does not strike the earth in the elliptical orbit. Problem 3/287

Determine the necessary amount by which the circular-orbit velocity of satellite C should be reduced if the catch-up maneuver of Prob. 3/287 is to be accomplished with not one but two periods in a new elliptical orbit.

The spacecraft S is to be injected into a circular orbit of altitude 400 km. Because of equipment malfunction, the injection speed v is correct for the circular orbit, but the initial velocity v makes an angle with the intended direction. What is the maximum permissible error in order that the spacecraft not strike the earth? Neglect atmospheric resistance. Problem 3/289

The 175,000-lb space-shuttle orbiter is in a circular orbit of altitude 200 miles. The two orbitalmaneuvering-system (OMS) engines, each of which has a thrust of 6000 lb, are fired in retrothrust for 150 seconds. Determine the angle which locates the intersection of the shuttle trajectory with the earths surface. Assume that the shuttle position B corresponds to the completion of the OMS burn and that no loss of altitude occurs during the burn. Problem 3/290

Compare the orbital period of the moon calculated with the assumption of a fixed earth with the period calculated without this assumption.

A satellite is placed in a circular polar orbit a distance H above the earth. As the satellite goes over the north pole at A, its retro-rocket is activated to produce a burst of negative thrust which reduces its velocity to a value which will ensure an equatorial landing. Derive the expression for the required reduction of velocity at A. Note that A is the apogee of the elliptical path. Problem 3/292

A spacecraft moving in a west-to-east equatorial orbit is observed by a tracking station located on the equator. If the spacecraft has a perigee altitude H 150 km and velocity v when directly over the station and an apogee altitude of 1500 km, determine an expression for the angular rate p (relative to the earth) at which the antenna dish must be rotated when the spacecraft is directly overhead. Compute p. The angular velocity of the earth is 0.7292( ) rad/s.

Sometime after launch from the earth, a spacecraft S is in the orbital path of the earth at some distance from the earth at position P. What velocity boost at P is required so that the spacecraft arrives at the orbit of Mars at A as shown? Problem 3/294

A spacecraft with a mass of 800 kg is traveling in a circular orbit 6000 km above the earth. It is desired to change the orbit to an elliptical one with a perigee altitude of 3000 km as shown. The transition is made by firing the retro-engine at A with a reverse thrust of 2000 N. Calculate the required time t for the engine to be activated. Problem 3/295

In 1995 a spacecraft called the Solar and Heliospheric Observatory (SOHO) was placed into a circular orbit about the sun and inside that of the earth as shown. Determine the distance h so that the period of the spacecraft orbit will match that of the earth, with the result that the spacecraft will remain between the earth and the sun in a halo orbit. Problem 3/296

A space vehicle moving in a circular orbit of radius r1 transfers to a larger circular orbit of radius r2 by means of an elliptical path between A and B. (This transfer path is known as the Hohmann transfer ellipse.) The transfer is accomplished by a burst of speed at A and a second burst of speed at B. Write expressions for and in terms of the radii shown and the value of g of the acceleration due to gravity at the earths surface. If each is positive, how can the velocity for path 2 be less than the velocity for path 1? Compute each if r1 (6371 500) km and r2 (6371 35 800) km. Note that r2 has been chosen as the radius of a geosynchronous orbit. Problem 3/297

At the instant represented in the figure, a small experimental satellite A is ejected from the shuttle orbiter with a velocity vr 100 m/s relative to the shuttle, directed toward the center of the earth. The shuttle is in a circular orbit of altitude h 200 km. For the resulting elliptical orbit of the satellite, determine the semimajor axis a and its orientation, the period , eccentricity e, apogee speed , perigee speed vp, rmax, and rmin. Sketch the satellite orbit. Problem 3/298

If the spring of constant k is compressed a distance as indicated, calculate the acceleration arel of the block of mass m1 relative to the frame of mass m2 upon release of the spring. The system is initially stationary. Problem 3/299

The flatbed truck is traveling at the constant speed of 60 km/h up the 15-percent grade when the 100- kg crate which it carries is given a shove which imparts to it an initial relative velocity 3 m/s toward the rear of the truck. If the crate slides a distance x 2 m measured on the truck bed before coming to rest on the bed, compute the coefficient of kinetic friction k between the crate and the truck bed. Problem 3/300

The cart with attached x-y axes moves with an absolute speed v 2 m/s to the right. Simultaneously, the light arm of length l 0.5 m rotates about point B of the cart with angular velocity 2 rad/s. The mass of the sphere is m 3 kg. Determine the following quantities for the sphere when 0: G, Grel, T, Trel, HO, where the subscript rel indicates measurement relative to the x-y axes. Point O is an inertially fixed point coincident with point B at the instant under consideration. Problem 3/301

The aircraft carrier is moving at a constant speed and launches a jet plane with a mass of 3 Mg in a distance of 75 m along the deck by means of a steam-driven catapult. If the plane leaves the deck with a velocity of 240 km/h relative to the carrier and if the jet thrust is constant at 22 kN during takeoff, compute the constant force P exerted by the catapult on the airplane during the 75-m travel of the launch carriage. Problem 3/302

The 4000-lb van is driven from position A to position B on the barge, which is towed at a constant speed v0 10 mi/hr. The van starts from rest relative to the barge at A, accelerates to v 15 mi/hr relative to the barge over a distance of 80 ft, and then stops with a deceleration of the same magnitude. Determine the magnitude of the net force F between the tires of the van and the barge during this maneuver. Problem 3/303

The launch catapult of the aircraft carrier gives the 7-Mg jet airplane a constant acceleration and launches the airplane in a distance of 100 m measured along the angled takeoff ramp. The carrier is moving at a steady speed vC 16 m/s. If an absolute aircraft speed of 90 m/s is desired for takeoff, determine the net force F supplied by the catapult and the aircraft engines. Problem 3/304

The coefficients of friction between the flatbed of the truck and crate are s 0.80 and k 0.70. The coefficient of kinetic friction between the truck tires and the road surface is 0.90. If the truck stops from an initial speed of 15 m/s with maximum braking (wheels skidding), determine where on the bed the crate finally comes to rest or the velocity vrel relative to the truck with which the crate strikes the wall at the forward edge of the bed.

A boy of mass m is standing initially at rest relative to the moving walkway, which has a constant horizontal speed u. He decides to accelerate his progress and starts to walk from point A with a steadily increasing speed and reaches point B with a speed v relative to the walkway. During his acceleration he generates an average horizontal force F between his shoes and the walkway. Write the work-energy equations for his absolute and relative motions and explain the meaning of the term muv. Problem 3/306

The block of mass m is attached to the frame by the spring of stiffness k and moves horizontally with negligible friction within the frame. The frame and block are initially at rest with x x0, the uncompressed length of the spring. If the frame is given a constant acceleration a0, determine the maximum velocity (vrel)max of the block relative to the frame. Problem 3/307

The slider A has a mass of 2 kg and moves with negligible friction in the 30 slot in the vertical sliding plate. What horizontal acceleration a0 should be given to the plate so that the absolute acceleration of the slider will be vertically down? What is the value of the corresponding force R exerted on the slider by the slot? Problem 3/308

The ball A of mass 10 kg is attached to the light rod of length l 0.8 m. The mass of the carriage alone is 250 kg, and it moves with an acceleration aO as shown. If 3 rad/s when  90, find the kinetic energy T of the system if the carriage has a velocity of 0.8 m/s (a) in the direction of aO and (b) in the direction opposite to aO. Treat the ball as a particle. Problem 3/309

Consider the system of Prob. 3/309 where the mass of the ball is m 10 kg and the length of the light rod is l 0.8 m. The ballrod assembly is free to rotate about a vertical axis through O. The carriage, rod, and ball are initially at rest with  0 when the carriage is given a constant acceleration aO 3 m/s2 . Write an expression for the tension T in the rod as a function of  and calculate T for the position  /2.

A simple pendulum is placed on an elevator, which accelerates upward as shown. If the pendulum is displaced an amount 0 and released from rest relative to the elevator, find the tension T0 in the supporting light rod when  0. Evaluate your result for 0 /2.

A boy of mass m is standing initially at rest relative to the moving walkway inclined at the angle  and moving with a constant speed u. He decides to accelerate his progress and starts to walk from point A with a steadily increasing speed and reaches point B with a speed vr relative to the walkway. During his acceleration he generates a constant average force F tangent to the walkway between his shoes and the walkway surface. Write the work-energy equations for the motion between A and B for his absolute motion and his relative motion and explain the meaning of the term muvr. If the boy weighs 150 lb and if u 2 ft/sec, s 30 ft, and  10, calculate the power Prel developed by the boy as he reaches the speed of 2.5 ft/sec relative to the walkway. Problem 3/312

A ball is released from rest relative to the elevator at a distance h1 above the floor. The speed of the elevator at the time of ball release is v0. Determine the bounce height h2 of the ball (a) if v0 is constant and (b) if an upward elevator acceleration a g/4 begins at the instant the ball is released. The coeffi- cient of restitution for the impact is e. Problem 3/313

The small slider A moves with negligible friction down the tapered block, which moves to the right with constant speed v v0. Use the principle of work-energy to determine the magnitude vA of the absolute velocity of the slider as it passes point C if it is released at point B with no velocity relative to the block. Apply the equation, both as an observer fixed to the block and as an observer fixed to the ground, and reconcile the two relations. h2 h1 v0 a = g 4 Problem 3/314

When a particle is dropped from rest relative to the surface of the earth at a latitude , the initial apparent acceleration is the relative acceleration due to gravity grel. The absolute acceleration due to gravity g is directed toward the center of the earth. Derive an expression for grel in terms of g, R, , and , where R is the radius of the earth treated as a sphere and is the constant angular velocity of the earth about the polar axis considered fixed. (Although axes x-y-z are attached to the earth and hence rotate, we may use Eq. 3/50 as long as the particle has no velocity relative to x-y-z). (Hint: Use the first two terms of the binomial expansion for the approximation.) Problem 3/315

The figure represents the space shuttle S, which is (a) in a circular orbit about the earth and (b) in an elliptical orbit where P is its perigee position. The exploded views on the right represent the cabin space with its x-axis oriented in the direction of the orbit. The astronauts conduct an experiment by applying a known force F in the x-direction to a small mass m. Explain why F does or does not hold in each case, where x is measured within the spacecraft. Assume that the shuttle is between perigee and apogee in the elliptical orbit so that the orbital speed is changing with time. Note that the t- and x-axes are tangent to the path, and the -axis is normal to the radial r-direction. O O S S r r r x x t, , t r y (a) (b) y S S t t Elliptical Orbit C Problem 3/316

The 4-kg slider is released from rest in position A and slides down the vertical-plane guide. If the maximum compression of the spring is observed to be 40 mm, determine the work U done by friction. Problem 3/317

The crate is at rest at point A when it is nudged down the incline. If the coefficient of kinetic friction between the crate and the incline is 0.30 from A to B and 0.22 from B to C, determine its speeds at points B and C. Problem 3/318

An 88-kg sprinter starts from rest and reaches his maximum speed of 11 m/s in 2.5 s with uniform acceleration. What is his power output when his speed is 5 m/s? Comment on the conditions stated in this problem.

Collar A is free to slide with negligible friction on the circular guide mounted in the vertical frame. Determine the angle assumed by the collar if the frame is given a constant horizontal acceleration a to the right. Problem 3/320

The position of the small \(0.5-\mathrm{kg}\) blocks in the smooth radial slots in the disk which rotates about a vertical axis at \(O\) is used to activate a speed control mechanism. If each block moves from \(r= 150 \mathrm{~mm}\) to \(r=175 \mathrm{~mm}\) while the speed of the disk changes slowly from \(300 \mathrm{rev} / \mathrm{min}\) to \(400 \mathrm{rev} / \mathrm{min}\), design the spring by calculating the spring constant \(k\) of each spring. The springs are attached to the inner ends of the slots and to the blocks.

The simple 2-kg pendulum is released from rest in the horizontal position. As it reaches the bottom position, the cord wraps around the smooth fixed pin at B and continues in the smaller arc in the vertical plane. Calculate the magnitude of the force R supported by the pin at B when the pendulum passes the position 30 . Problem 3/322

For the elliptical orbit of a spacecraft around the earth, determine the speed vA at point A which results in a perigee altitude at B of 200 km. What is the eccentricity e of the orbit? Problem 3/323

The spring of stiffness k is compressed and suddenly released, sending the particle of mass m sliding along the track. Determine the minimum spring compression for which the particle will not lose contact with the loop-the-loop track. The sliding surface is smooth except for the rough portion of length s equal to R, where the coefficient of kinetic friction is k. Problem 3/324

The last two appearances of Comet Halley were in 1910 and 1986. The distance of its closest approach to the sun averages about one-half of the distance between the earth and the sun. Determine its maximum distance from the sun. Neglect the gravitational effects of the planets.

A small sphere of mass m is connected by a string to a swivel at O and moves in a circle of radius r on the smooth plane inclined at an angle with the horizontal. If the sphere has a velocity u at the top position A, determine the tension in the string as the sphere passes the 90 position B and the bottom position C. Problem 3/326

The quarter-circular hollow tube of circular cross section starts from rest at time t 0 and rotates about point O in a horizontal plane with a constant counterclockwise angular acceleration 2 rad/s2 . At what time t will the 0.5-kg particle P slip relative to the tube? The coefficient of static friction between the particle and the tube is s 0.80. Problem 3/327

A person rolls a small ball with speed u along the floor from point A. If x 3R, determine the required speed u so that the ball returns to A after rolling on the circular surface in the vertical plane from B to C and becoming a projectile at C. What is the minimum value of x for which the game could be played if contact must be maintained to point C? Neglect friction. Problem 3/328

A 3600-lb car is traveling with a speed of 60 mi/hr as it approaches point A. Beginning at A, it decelerates uniformly to a speed of 25 mi/hr as it passes point C of the horizontal and unbanked ramp. Determine the total horizontal force F exerted by the road on the car just after it passes point B. Problem 3/329

After release from rest at B, the 2-lb cylindrical plug A slides down the smooth path and embeds itself in the 4-lb block C. Determine the velocity v of the block and embedded plug immediately after engagement and find the maximum deflection x of the spring. Neglect any friction under block C. What fraction n of the original energy of the system is lost? Problem 3/330

The pickup truck is used to hoist the 40-kg bale of hay as shown. If the truck has reached a constant velocity v 5 m/s when x 12 m, compute the corresponding tension T in the rope. Problem 3/331

A slider C has a speed of 3 m/s as it passes point A of the guide, which lies in a horizontal plane. The coefficient of kinetic friction between the slider and the guide is Compute the tangential deceleration at of the slider just after it passes point A if (a) the slider hole and guide cross section are both circular and (b) the slider hole and guide cross section are both square. In case (b), the sides of the square are vertical and horizontal. Assume a slight clearance between the slider and the guide. Problem 3/332

The frame of mass 6m is initially at rest. A particle of mass m is attached to the end of the light rod, which pivots freely at A. If the rod is released from rest in the horizontal position shown, determine the velocity vrel of the particle with respect to the frame when the rod is vertical. Problem 3/333

The object of the pinball-type game is to project the particle so that it enters the hole at E. When the spring is compressed and suddenly released, the particle is projected along the track, which is smooth except for the rough portion between points B and C, where the coefficient of kinetic friction is The particle becomes a projectile at point D. Determine the correct spring compression so that the particle enters the hole at E. State any necessary conditions relating the lengths d and .

The 2-lb piece of putty is dropped 6 ft onto the 60 18-lb block initially at rest on the two springs, each with a stiffness k 3 lb/in. Calculate the additional deflection of the springs due to the impact of the putty, which adheres to the block upon contact. Problem 3/335

A baseball pitcher delivers a fastball with a nearhorizontal velocity of 90 mi/hr. The batter hits a home run over the center-field fence. The 5-oz ball travels a horizontal distance of 350 ft, with an initial velocity in the 45 direction shown. Determine the magnitude Fav of the average force exerted by the bat on the ball during the 0.005 seconds of contact between the bat and the ball. Neglect air resistance during the flight of the ball. Problem 3/336

The 3-kg block A is released from rest in the position shown and subsequently strikes the 1-kg cart B. If the coefficient of restitution for the collision is e 0.7, determine the maximum displacement s of cart B beyond point C. Neglect friction. Problem 3/337

One of the functions of the space shuttle is to release communications satellites at low altitude. A booster rocket is fired at B, placing the satellite in an elliptical transfer orbit, the apogee of which is at the altitude necessary for a geosynchronous orbit. (A geosynchronous orbit is an equatorial-plane circular orbit whose period is equal to the absolute rotational period of the earth. A satellite in such an orbit appears to remain stationary to an earth-fixed observer.) A second booster rocket is then fired at C, and the final circular orbit is achieved. On one of the early space-shuttle missions, a 1500-lb satellite was released from the shuttle at B, where h1 170 miles. The booster rocket was to fire for t 90 seconds, forming a transfer orbit with h2 22,300 miles. The rocket failed during its burn. Radar observations determined the apogee altitude of the transfer orbit to be only 700 miles. Determine the actual time which the rocket motor operated before failure. Assume negligible mass change during the booster rocket firing. Problem 3/338

The system is released from rest while in the position shown. If m1 0.5 kg, m2 4 kg, d 0.5 m, and , determine the speeds of both bodies just after the block leaves the incline (before striking the horizontal surface). Neglect all friction. Problem 3/339

The retarding forces which act on the race car are the drag force FD and a nonaerodynamic force FR. The drag force is where CD is the drag coefficient, is the air density, v is the car speed, and S 30 ft2 is the projected frontal area of the car. The nonaerodynamic force FR is constant at 200 lb. With its sheet metal in good condition, the race car has a drag coefficient CD 0.3 and it has a corresponding top speed v 200 mi/hr. After a minor collision, the damaged front-end sheet metal causes the drag coefficient to be CD 0.4. What is the corresponding top speed of the race car? Problem 3/340

Extensive wind-tunnel and coast-down studies of a 2000-lb automobile reveal the aerodynamic drag force FD and the total nonaerodynamic rolling resistance force FR to vary with speed as shown in the plot. Determine (a) the power P required for steady speeds of 30 mi/hr and 60 mi/hr and (b) the time t and the distance s required for the car to coast down to a speed of 5 mi/hr from an initial speed of 60 mi/hr. Assume a straight, level road and no wind. Problem 3/341

The hollow tube rotates with a constant angular velocity about a horizontal axis through end O. At time t 0 the tube passes the vertical position 0, at which instant the small ball of mass m is released with r essentially zero. Determine r as a function of . Problem 3/342

The bowl-shaped device from Prob. 3/70 rotates about a vertical axis with a constant angular velocity 6 rad/s. The value of r is 0.2 m. Determine the range of the position angle for which a stationary value is possible if the coefficient of static friction between the particle and the surface is 0.20. Problem 3/343

If the vertical frame starts from rest with a constant acceleration a and the smooth sliding collar A is initially at rest in the bottom position 0, plot as a function of and find the maximum position angle max reached by the collar. Use the values a g/2 and r 0.3 m. Problem 3/344

The system of Prob. 3/130 is repeated here. The two 0.2-kg sliders are connected by a light rigid bar of length L 0.5 m. If the system is released from rest in the position shown with the spring unstretched, plot the speeds of A and B as functions of the displacement of B (with zero being the initial position). The 0.14-MPa air pressure acting on one 500-mm2 side of slider A is constant. The motion occurs in a vertical plane. Neglect friction. State the maximum values of vA and vB and the position of B at which each occurs. Problem 3/345

The square plate is at rest in position A at time t 0 and subsequently translates in a vertical circle according to kt2 , where k 1 rad/s2 , the displacement is in radians, and time t is in seconds. A small 0.4-kg instrument P is temporarily fixed to the plate with adhesive. Plot the required shear force F vs. time t for . If the adhesive fails when the shear force F reaches 30 N, determine the time t and angular position when failure occurs. Problem 3/346

The system of Prob. 3/171 is repeated here. The system is released from rest with . Determine and plot as a function of . Determine the maximum magnitude of in the ensuing motion and the value of at which it occurs. Also find the minimum value of . Use the values m1 1 kg, m2 1.25 kg, and b 0.4 m. Neglect friction and the mass of bar OB, and treat the body B as a particle. Problem 3/347

The 26-in. drum rotates about a horizontal axis with a constant angular velocity 7.5 rad/sec. The small block A has no motion relative to the drum surface as it passes the bottom position 0. Determine the coefficient of static friction which would result in block slippage at an angular position ; plot your expression for Determine the minimum required coefficient value min which would allow the block to remain fixed relative to the drum throughout a full revolution. For a friction coefficient slightly less than min, at what angular position would slippage occur? Problem 3/348

A 20-lb sphere A is held at the 60 angle shown and released. It strikes the 10-lb sphere B. The coeffi- cient of restitution for this collision is e 0.75. Sphere B is attached to the end of a light rod that pivots freely about point O. If the spring of constant k 100 lb/ft is initially unstretched, determine the maximum rotation angle of the light rod after impact. Problem 3/349

A particle of mass m is introduced with zero velocity at r 0 when 0. It slides outward through the smooth hollow tube, which is driven at the constant angular velocity 0 about a horizontal axis through point O. If the length l of the tube is 1 m and 0 0.5 rad/s, determine the time t after release and the angular displacement for which the particle exits the tube. Problem 3/350

The tennis player practices by hitting the ball against the wall at A. The ball bounces off the court surface at B and then up to its maximum height at C. For the conditions shown in the figure, plot the location of point C for values of the coeffi- cient of restitution in the range 0.5 e 0.9. (The value of e is common to both A and B.) For what value of e is x 0 at point C, and what is the corresponding value of y? Problem 3/351

The system of Prob. 3/154 is repeated here. If the 0.75-kg particle is released from rest when in the position 0, where the spring is unstretched, determine and plot its speed v as a function of over the range where max is the value of at which the system momentarily comes to rest. The value of the spring modulus k is 100 N/m, and friction can be neglected. State the maximum speed and the angle at which it occurs. Problem 3/352

The simple pendulum of length l 0.5 m has an angular velocity 0.2 rad/s at time t 0 when 0. Derive an integral expression for the time t required to reach an arbitrary angle . Plot t vs. for and state the value of t for Problem 3/353

A 1.8-lb particle P is given an initial velocity v0 1 ft/sec at the position 0 and subsequently slides along the circular path of radius r 1.5 ft. A drag force of magnitude kv acts in the direction opposite to the velocity. If the drag parameter k 0.2 lb-sec/ft, determine and plot the particle speed v and the normal force N exerted on the particle by the surface as functions of over the range . Determine the maximum values of v and N and the values of at which these maxima occur. Neglect friction between the particle and the circular surface. Problem 3/354