Determine the constant angular velocity u # of the

The 10-kg block is subjected to the forces shown. In each case, determine its velocity when t = 2 s if v = 0 when t = 0.

The 10-kg block is subjected to the forces shown. In each case, determine its velocity at s = 8 m if v = 3 m/s at s = 0. Motion occurs to the right.

Determine the initial acceleration of the 10-kg smooth collar. The spring has an unstretched length of 1 m.

Write the equations of motion in the x and y directions for the 10-kg block.

Set up the n, t axes and write the equations of motion for the 10-kg block along each of these axes.

Set up the n, b, t axes and write the equations of motion for the 10-kg block along each of these axes.

The motor winds in the cable with a constant acceleration, such that the 20-kg crate moves a distance s = 6 m in 3 s, starting from rest. Determine the tension developed in the cable. The coefficient of kinetic friction between the crate and the plane is \(\mu_{k}=0.3\).

If motor M exerts a force of \(F=\left(10t^2+100\right)\ \mathrm{N}\) on the cable, where t is in seconds, determine the velocity of the 25-kg crate when t = 4 s. The coefficients of static and kinetic friction between the crate and the plane are \(\mu_{s}=0.3\) and \(\mu_{k}=0.25\), respectively. The crate is initially at rest.

A spring of stiffness k = 500 N/m is mounted against the 10-kg block. If the block is subjected to the force of F = 500 N, determine its velocity at s = 0.5 m. When s = 0, the block is at rest and the spring is uncompressed. The contact surface is smooth.

The 2-Mg car is being towed by a winch. If the winch exerts a force of T = 100(s + 1) N on the cable, where s is the displacement of the car in meters, determine the speed of the car when s = 10 m, starting from rest. Neglect rolling resistance of the car.

The spring has a stiffness k = 200 N/m and is unstretched when the 25-kg block is at A. Determine the acceleration of the block when s = 0.4 m. The contact surface between the block and the plane is smooth.

Block B rests upon a smooth surface. If the coefficients of static and kinetic friction between A and B are \(\mu_{s}=0.4\) and \(\mu_{k}=0.3\), respectively, determine the acceleration of each block if P = 6 lb.

The block rests at a distance of 2 m from the center of the platform. If the coefficient of static friction between the block and the platform is \(\mu_{s}=0.3\), determine the maximum speed which the block can attain before it begins to slip. Assume the angular motion of the disk is slowly increasing.

Determine the maximum speed that the jeep can travel over the crest of the hill and not lose contact with the road.

A pilot weighs 150 lb and is traveling at a constant speed of 120 ft/s. Determine the normal force he exerts on the seat of the plane when he is upside down at A. The loop has a radius of curvature of 400 ft.

The sports car is traveling along a \(30^{\circ}\) banked road having a radius of curvature of \(\rho=500\mathrm{\ ft}\). If the coefficient of static friction between the tires and the road is \(\mu_{s}=0.2\), determine the maximum safe speed so no slipping occurs. Neglect the size of the car.

If the 10-kg ball has a velocity of 3 m/s when it is at the position A, along the vertical path, determine the tension in the cord and the increase in the speed of the ball at this position.

The motorcycle has a mass of 0.5 Mg and a negligible size. It passes point A traveling with a speed of 15 m/s, which is increasing at a constant rate of \(1.5\mathrm{\ m}/\mathrm{s}^2\). Determine the resultant frictional force exerted by the road on the tires at this instant.

Determine the constant angular velocity u # of the vertical shaft of the amusement ride if f = 45. Neglect the mass of the cables and the size of the passengers. 1

he 0.2-kg ball is blown through the smooth vertical circular tube whose shape is defined by r = (0.6 sin u) m, where u is in radians. If u = (p t 2 ) rad, where t is in seconds, determine the magnitude of force F exerted by the blower on the ball when t = 0.5 s.

The 2-Mg car is traveling along the curved road described by r = (50e2u ) m, where u is in radians. If a camera is located at A and it rotates with an angular velocity of u # = 0.05 rad>s and an angular acceleration of u $ = 0.01 rad>s 2 at the instant u = p 6 rad, determine the resultant friction force developed between the tires and the road at this instant.

The 0.2-kg pin P is constrained to move in the smooth curved slot, which is defined by the lemniscate r = (0.6 cos 2u) m. Its motion is controlled by the rotation of the slotted arm OA, which has a constant clockwise angular velocity of u # = -3 rad>s. Determine the force arm OA exerts on the pin P when u = 0. Motion is in the vertical plane.

The 6-lb particle is subjected to the action of its weight and forces \(\mathbf{F}_{1}=\{2 \mathbf{i}+6 \mathbf{j}-2 t \mathbf{k}\} \mathrm{\ lb}\), \(\mathbf{F}_2=\left\{t^2\mathbf{i}-4t\mathbf{j}-1\mathbf{k}\right\}\ \mathrm{lb}\), and \(\mathbf{F}_{3}=\{-2 t\mathbf{i}\} \mathrm{\ lb}\), where t is in seconds. Determine the distance the ball is from the origin 2 s after being released from rest.

The two boxcars A and B have a weight of 20 000 lb and 30 000 lb, respectively. If they are freely coasting down the incline when the brakes are applied to all the wheels of car A, determine the force in the coupling C between the two cars. The coefficient of kinetic friction between the wheels of A and the tracks is \(\mu_{k}=0.5\). The wheels of car B are free to roll. Neglect their mass in the calculation. Suggestion: Solve the problem by representing single resultant normal forces acting on A and B, respectively.

If the coefficient of kinetic friction between the 50-kg crate and the ground is \(\mu_{k}=0.3\), determine the distance the crate travels and its velocity when t = 3 s. The crate starts from rest, and P = 200 N.

If the 50-kg crate starts from rest and achieves a velocity of v = 4 m/s when it travels a distance of 5 m to the right, determine the magnitude of force P acting on the crate. The coefficient of kinetic friction between the crate and the ground is \(\mu_{k}=0.3\).

If blocks A and B of mass 10 kg and 6 kg respectively, are placed on the inclined plane and released, determine the force developed in the link. The coefficients of kinetic friction between the blocks and the inclined plane are \(\mu_{A}=0.1\) and \(\mu_{B}=0.3\). Neglect the mass of the link.

The 10-lb block has a speed of 4 ft/s when the force of \(F=\left(8t^2\right)\ \mathrm{lb}\) is applied. Determine the velocity of the block when t = 2 s. The coefficient of kinetic friction at the surface is \(\mu_{k}=0.2\).

The 10-lb block has a speed of 4 ft/s when the force of \(F=\left(8t^2\right)\ \mathrm{lb}\) is applied. Determine the velocity of the block when it moves s = 30 ft. The coefficient of kinetic friction at the surface is \(\mu_{s}=0.2\).

The speed of the 3500-lb sports car is plotted over the 30-s time period. Plot the variation of the traction force F needed to cause the motion.

The conveyor belt is moving at 4 m/s. If the coefficient of static friction between the conveyor and the 10-kg package B is \(\mu_{s}=0.2\), determine the shortest time the belt can stop so that the package does not slide on the belt.

The conveyor belt is designed to transport packages of various weights. Each 10-kg package has a coefficient of kinetic friction \(\mu_{k}=0.15\). If the speed of the conveyor is 5 m/s, and then it suddenly stops, determine the distance the package will slide on the belt before coming to rest.

Determine the time needed to pull the cord at B down 4 ft starting from rest when a force of 10 lb is applied to the cord. Block A weighs 20 lb. Neglect the mass of the pulleys and cords.

Cylinder B has a mass m and is hoisted using the cord and pulley system shown. Determine the magnitude of force F as a function of the block’s vertical position y so that when F is applied the block rises with a constant acceleration \(\mathbf{a}_{B}\). Neglect the mass of the cord and pulleys.

Block A has a weight of 8 lb and block B has a weight of 6 lb. They rest on a surface for which the coefficient of kinetic friction is \(\mu_{k}=0.2\). If the spring has a stiffness of k = 20 lb/ft, and it is compressed 0.2 ft, determine the acceleration of each block just after they are released.

The 2-Mg truck is traveling at 15 m/s when the brakes on all its wheels are applied, causing it to skid for a distance of 10 m before coming to rest. Determine the constant horizontal force developed in the coupling C, and the frictional force developed between the tires of the truck and the road during this time. The total mass of the boat and trailer is 1 Mg.

The motor lifts the 50-kg crate with an acceleration of \(6\mathrm{\ m}/\mathrm{s}^2\). Determine the components of force reaction and the couple moment at the fixed support A.

The 75-kg man pushes on the 150-kg crate with a horizontal force F. If the coefficients of static and kinetic friction between the crate and the surface are \(\mu_{s}=0.3\) and \(\mu_{k}=0.2\), and the coefficient of static friction between the man’s shoes and the surface is \(\mu_{s}=0.8\), show that the man is able to move the crate. What is the greatest acceleration the man can give the crate?

Determine the acceleration of the blocks when the system is released. The coefficient of kinetic friction is \(\mu_{k}\), and the mass of each block is m. Neglect the mass of the pulleys and cord.

A 40-lb suitcase slides from rest 20 ft down the smooth ramp. Determine the point where it strikes the ground at C. How long does it take to go from A to C?

Solve Prob. 13–18 if the suitcase has an initial velocity down the ramp of \(v_A=10\ \mathrm{ft}/\mathrm{s}\) and the coefficient of kinetic friction along AB is \(\mu_{k}=0.2\).

The conveyor belt delivers each 12-kg crate to the ramp at A such that the crate’s speed is \(v_A=2.5\mathrm{\ m}/\mathrm{s}\), directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_{k}=0.3\), determine the speed at which each crate slides off the ramp at B. Assume that no tipping occurs. Take \(\theta=30^{\circ}\).

The conveyor belt delivers each 12-kg crate to the ramp at A such that the crate’s speed is \(v_A=2.5\mathrm{\ m}/\mathrm{s}\), directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_{k}=0.3\), determine the smallest incline \(\theta\) of the ramp so that the crates will slide off and fall into the cart.

The 50-kg block A is released from rest. Determine the velocity of the 15-kg block B in 2 s.

If the supplied force F = 150 N, determine the velocity of the 50-kg block A when it has risen 3 m, starting from rest.

A 60-kg suitcase slides from rest 5 m down the smooth ramp. Determine the distance R where it strikes the ground at B. How long does it take to go from A to B?

Solve Prob. 13–24 if the suitcase has an initial velocity down the ramp of \(v_A=2\mathrm{\ m}/\mathrm{s}\), and the coefficient of kinetic friction along AC is \(\mu_{k}=0.2\).

The 1.5 Mg sports car has a tractive force of F = 4.5 kN. If it produces the velocity described by v-t graph shown, plot the air resistance R versus t for this time period.

The conveyor belt is moving downward at 4 m/s. If the coefficient of static friction between the conveyor and the 15-kg package B is \(\mu_{3}=0.8\), determine the shortest time the belt can stop so that the package does not slide on the belt.

At the instant shown the 100-lb block A is moving down the plane at 5 ft/s while being attached to the 50-lb block B. If the coefficient of kinetic friction between the block and the incline is \(\mu_{k}=0.2\), determine the acceleration of A and the distance A slides before it stops. Neglect the mass of the pulleys and cables.

The force exerted by the motor on the cable is shown in the graph. Determine the velocity of the 200-lb crate when t = 2.5 s.

The force of the motor M on the cable is shown in the graph. Determine the velocity of the 400-kg crate A when t = 2 s.

The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right at a constant speed of 4 m/s, determine the tension in the rope when \(s_A=5\mathrm{\ m}\). When \(s_{A}=0\), \(s_{B}=0\).

The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right with an acceleration of \(3\mathrm{\ m}/\mathrm{s}^2\) and has a velocity of 4 m/s at the instant \(s_A=5\mathrm{\ m}\), determine the tension in the rope at this instant. When \(s_{A}=0\), \(s_{B}=0\).

Block A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that it will not slide on A. Also, what is the corresponding acceleration? The coefficient of static friction between A and B is \(\mu_{s}\). Neglect any friction between A and the horizontal surface.

The 4-kg smooth cylinder is supported by the spring having a stiffness of \(k_{AB}=120\mathrm{\ N}/\mathrm{m}\). Determine the velocity of the cylinder when it moves downward s = 0.2 m from its equilibrium position, which is caused by the application of the force F = 60 N.

The coefficient of static friction between the 200-kg crate and the flat bed of the truck is \(\mu_{3}=0.3\). Determine the shortest time for the truck to reach a speed of 60 km/h, starting from rest with constant acceleration, so that the crate does not slip.

The 2-lb collar C fits loosely on the smooth shaft. If the spring is unstretched when s = 0 and the collar is given a velocity of 15 ft/s, determine the velocity of the collar when s = 1 ft.

The 10-kg block A rests on the 50-kg plate B in the position shown. Neglecting the mass of the rope and pulley, and using the coefficients of kinetic friction indicated, determine the time needed for block A to slide 0.5 m on the plate when the system is released from rest.

The 300-kg bar B, originally at rest, is being towed over a series of small rollers. Determine the force in the cable when t = 5 s, if the motor M is drawing in the cable for a short time at a rate of \(v=\left(0.4t^2\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds \((0\le t\le6\mathrm{\ s})\). How far does the bar move in 5 s? Neglect the mass of the cable, pulley, and the rollers.

An electron of mass m is discharged with an initial horizontal velocity of \(\mathbf{v}_{0}\). If it is subjected to two fields of force for which \(F_{x}=F_{0}\) and \(F_{y}=0.3 F_{0}\), where \(F_{0}\) is constant, determine the equation of the path, and the speed of the electron at any time t.

The 400-lb cylinder at A is hoisted using the motor and the pulley system shown. If the speed of point B on the cable is increased at a constant rate from zero to \(\mathrm{v}_B=10\mathrm{\ ft}/\mathrm{s}\) in t = 5 s, determine the tension in the cable at B to cause the motion.

Block A has a mass \(m_{A}\) and is attached to a spring having a stiffness k and unstretched length \(l_{0}\). If another block B, having a mass \(m_{B}\), is pressed against A so that the spring deforms a distance d, determine the distance both blocks slide on the smooth surface before they begin to separate. What is their velocity at this instant?

Block A has a mass \(m_{A}\) and is attached to a spring having a stiffness k and unstretched length \(l_{0}\). If another block B, having a mass \(m_{B}\), is pressed against A so that the spring deforms a distance d, show that for separation to occur it is necessary that \(d>2 \mu_{k} g\left(m_{A}+m_{B}\right) / k\), where \(\mu_{k}\) is the coefficient of kinetic friction between the blocks and the ground. Also, what is the distance the blocks slide on the surface before they separate?

A parachutist having a mass m opens his parachute from an at-rest position at a very high altitude. If the atmospheric drag resistance is \(F_{D}=k v^{2}\), where k is a constant, determine his velocity when he has fallen for a time t. What is his velocity when he lands on the ground? This velocity is referred to as the terminal velocity, which is found by letting the time of fall \(t \rightarrow \infty\).

If the motor draws in the cable with an acceleration of \(3\mathrm{\ m}/\mathrm{s}^2\), determine the reactions at the supports A and B. The beam has a uniform mass of 30 kg/m, and the crate has a mass of 200 kg. Neglect the mass of the motor and pulleys.

If the force exerted on cable AB by the motor is \(F=\left(100t^{3/2}\right)\ \mathrm{N}\), where t is in seconds, determine the 50-kg crate’s velocity when t = 5 s. The coefficients of static and kinetic friction between the crate and the ground are \(\mu_{s}=0.4\) and \(\mu_{k}=0.3\), respectively. Initially the crate is at rest.

Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not move relative to B. All surfaces are smooth.

Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to B so that A will not slip on B. The coefficient of static friction between A and B is \(\mu_{s}\). Neglect any friction between B and C.

The smooth block B of negligible size has a mass m and rests on the horizontal plane. If the board AC pushes on the block at an angle \(\theta\) with a constant acceleration \(\mathbf{a}_{0}\), determine the velocity of the block along the board and the distance s the block moves along the board as a function of time t. The block starts from rest when s = 0, t = 0.

If a horizontal force P = 12 lb is applied to block A determine the acceleration of the block B. Neglect friction.

A freight elevator, including its load, has a mass of 1 Mg. It is prevented from rotating due to the track and wheels mounted along its sides. If the motor M develops a constant tension T = 4 kN in its attached cable, determine the velocity of the elevator when it has moved upward 6 m starting from rest. Neglect the mass of the pulleys and cables.

The block A has a mass \(m_{A}\) and rests on the pan B, which has a mass \(m_{B}\). Both are supported by a spring having a stiffness k that is attached to the bottom of the pan and to the ground. Determine the distance d the pan should be pushed down from the equilibrium position and then released from rest so that separation of the block will take place from the surface of the pan at the instant the spring becomes unstretched.

A girl, having a mass of 15 kg, sits motionless relative to the surface of a horizontal platform at a distance of r = 5 m from the platform’s center. If the angular motion of the platform is slowly increased so that the girl’s tangential component of acceleration can be neglected, determine the maximum speed which the girl will have before she begins to slip off the platform. The coefficient of static friction between the girl and the platform is \(\mu=0.2\).

The 2-kg block B and 15-kg cylinder A are connected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of v = 10 m/s, determine the radius r of the circular path along which it travels.

The 2-kg block B and 15-kg cylinder A are connected to a light cord that passes through a hole in the center of the smooth table. If the block travels along a circular path of radius r = 1.5 m, determine the speed of the block.

Determine the maximum constant speed at which the pilot can travel around the vertical curve having a radius of curvature \(\rho=800\mathrm{\ m}\), so that he experiences a maximum acceleration \(a_n=8g=78.5\mathrm{\ m}/\mathrm{s}^2\). If he has a mass of 70 kg, determine the normal force he exerts on the seat of the airplane when the plane is traveling at this speed and is at its lowest point.

Cartons having a mass of 5 kg are required to move along the assembly line at a constant speed of 8 m/s. Determine the smallest radius of curvature, \(\rho\), for the conveyor so the cartons do not slip. The coefficients of static and kinetic friction between a carton and the conveyor are \(\mu_{s}=0.7\) and \(\mu_{k}=0.5\), respectively.

The collar A, having a mass of 0.75 kg, is attached to a spring having a stiffness of k = 200 N/m. When rod BC rotates about the vertical axis, the collar slides outward along the smooth rod DE. If the spring is unstretched when s = 0, determine the constant speed of the collar in order that s = 100 mm. Also, what is the normal force of the rod on the collar? Neglect the size of the collar.

The 2-kg spool S fits loosely on the inclined rod for which the coefficient of static friction is \(\mu_{s}=0.2\). If the spool is located 0.25 m from A, determine the minimum constant speed the spool can have so that it does not slip down the rod.

The 2-kg spool S fits loosely on the inclined rod for which the coefficient of static friction is \(\mu_{s}=0.2\). If the spool is located 0.25 m from A, determine the maximum constant speed the spool can have so that it does not slip up the rod.

At the instant \(\theta=60^{\circ}\), the boy’s center of mass G has a downward speed \(v_G=15\ \mathrm{ft}/\mathrm{s}\). Determine the rate of increase in his speed and the tension in each of the two supporting cords of the swing at this instant. The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords.

At the instant \(\theta=60^{\circ}\), the boy’s center of mass G is momentarily at rest. Determine his speed and the tension in each of the two supporting cords of the swing when \(\theta=90^{\circ}\). The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords.

A girl having a mass of 25 kg sits at the edge of the merry-go-round so her center of mass G is at a distance of 1.5 m from the axis of rotation. If the angular motion of the platform is slowly increased so that the girl’s tangential component of acceleration can be neglected, determine the maximum speed which she can have before she begins to slip off the merry-go-round. The coefficient of static friction between the girl and the merry-go-round is \(\mu_{s}=0.3\).

The pendulum bob B has a weight of 5 lb and is released from rest in the position shown, \(\theta=0^{\circ}\). Determine the tension in string BC just after the bob is released, \(\theta=0^{\circ}\), and also at the instant the bob reaches \(\theta=45^{\circ}\). Take r = 3 ft.

The pendulum bob B has a mass m and is released from rest when \(\theta=0^{\circ}\). Determine the tension in string BC immediately afterwards, and also at the instant the bob reaches the arbitrary position \(\theta\).

Determine the constant speed of the passengers on the amusement-park ride if it is observed that the supporting cables are directed at \(\theta=30^{\circ}\) from the vertical. Each chair including its passenger has a mass of 80 kg. Also, what are the components of force in the n, t, and b directions which the chair exerts on a 50-kg passenger during the motion?

A motorcyclist in a circus rides his motorcycle within the confines of the hollow sphere. If the coefficient of static friction between the wheels of the motorcycle and the sphere is \(\mu_{s}=0.4\), determine the minimum speed at which he must travel if he is to ride along the wall when \(\theta=90^{\circ}\). The mass of the motorcycle and rider is 250 kg, and the radius of curvature to the center of gravity is \(\rho=20 \mathrm{ft}\). Neglect the size of the motorcycle for the calculation.

The vehicle is designed to combine the feel of a motorcycle with the comfort and safety of an automobile. If the vehicle is traveling at a constant speed of 80 km/h along a circular curved road of radius 100 m, determine the tilt angle \(\theta\) of the vehicle so that only a normal force from the seat acts on the driver. Neglect the size of the driver.

The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 m/s, determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point A. Neglect the size of the car.

The 0.8-Mg car travels over the hill having the shape of a parabola. When the car is at point A, it is traveling at 9 m/s and increasing its speed at \(3\mathrm{\ m}/\mathrm{s}^2\). Determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at this instant. Neglect the size of the car.

The package has a weight of 5 lb and slides down the chute. When it reaches the curved portion AB, it is traveling at \(8\ \mathrm{ft}/\mathrm{s}\ \left(\theta=0^{\circ}\right)\). If the chute is smooth, determine the speed of the package when it reaches the intermediate point \(C\ \left(\theta=30^{\circ}\right)\) and when it reaches the horizontal plane \(\left(\theta=45^{\circ}\right)\). Also, find the normal force on the package at C.

The 150-lb man lies against the cushion for which the coefficient of static friction is \(\mu_{s}=0.5\). Determine the resultant normal and frictional forces the cushion exerts on him if, it due to rotation about the z axis, he has a constant speed v = 20 ft/s. Neglect the size of the man. Take \(\theta=60^{\circ}\).

The 150-lb man lies against the cushion for which the coefficient of static friction is \(\mu_{s}=0.5\). If he rotates about the z axis with a constant speed v = 30 ft/s, determine the smallest angle \(\theta\) of the cushion at which he will begin to slip off.

Determine the maximum speed at which the car with mass m can pass over the top point A of the vertical curved road and still maintain contact with the road. If the car maintains this speed, what is the normal reaction the road exerts on the car when it passes the lowest point B on the road?

Determine the maximum constant speed at which the 2-Mg car can travel over the crest of the hill at A without leaving the surface of the road. Neglect the size of the car in the calculation.

The box has a mass m and slides down the smooth chute having the shape of a parabola. If it has an initial velocity of \(v_{0}\) at the origin determine its velocity as a function of x. Also, what is the normal force on the box, and the tangential acceleration as a function of x?

Prove that if the block is released from rest at point B of a smooth path of arbitrary shape, the speed it attains when it reaches point A is equal to the speed it attains when it falls freely through a distance h; i.e., \(v=\sqrt{2 g h}\).

The cylindrical plug has a weight of 2 lb and it is free to move within the confines of the smooth pipe. The spring has a stiffness k = 14 lb/ft and when no motion occurs the distance d = 0.5 ft. Determine the force of the spring on the plug when the plug is at rest with respect to the pipe. The plug is traveling with a constant speed of 15 ft/s, which is caused by the rotation of the pipe about the vertical axis.

When crossing an intersection, a motorcyclist encounters the slight bump or crown caused by the intersecting road. If the crest of the bump has a radius of curvature \(\rho=50\mathrm{\ ft}\), determine the maximum constant speed at which he can travel without leaving the surface of the road. Neglect the size of the motorcycle and rider in the calculation. The rider and his motorcycle have a total weight of 450 lb.

The airplane, traveling at a constant speed of 50 m/s, is executing a horizontal turn. If the plane is banked at \(\theta = 15^{\circ}\), when the pilot experiences only a normal force on the seat of the plane, determine the radius of curvature \(\rho\) of the turn. Also, what is the normal force of the seat on the pilot if he has a mass of 70 kg.

The 2-kg pendulum bob moves in the vertical plane with a velocity of 8 m>s when u = 0. Determine the initial tension in the cord and also at the instant the bob reaches u = 30. Neglect the size of the bob.

The 2-kg pendulum bob moves in the vertical plane with a velocity of 6 m>s when u = 0. Determine the angle u where the tension in the cord becomes zero.

The 8-kg sack slides down the smooth ramp. If it has a speed of 1.5 m>s when y = 0.2 m, determine the normal reaction the ramp exerts on the sack and the rate of increase in the speed of sack at this instant.

The ball has a mass m and is attached to the cord of length l. The cord is tied at the top to a swivel and the ball is given a velocity v0. Show that the angle u which the cord makes with the vertical as the ball travels around the circular path must satisfy the equation tan u sin u = v2 0>gl. Neglect air resistance and the size of the ball.

The 2-lb block is released from rest at A and slides down along the smooth cylindrical surface. If the attached spring has a stiffness \(k = 2\ lb/ft\), determine its unstretched length so that it does not allow the block to leave the surface until \(\theta = 60^{\circ}\).

The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and z = (0.1 sin 2u) ft. If the cam is rotating at a constant rate of 6 rad>s, determine the force at the end A of the follower when u = 45. In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C.

Determine the magnitude of the resultant force acting on a 5-kg particle at the instant t = 2 s, if the particle is moving along a horizontal path defined by the equations r = (2t + 10) m and u = (1.5t 2 - 6t) rad, where t is in seconds.

The path of motion of a 5-lb particle in the horizontal plane is described in terms of polar coordinates as r = (2t + 1) ft and u = (0.5t 2 - t) rad, where t is in seconds. Determine the magnitude of the unbalanced force acting on the particle when t = 2 s.

Rod OA rotates counterclockwise with a constant angular velocity of u . = 5 rad>s. The double collar B is pinconnected together such that one collar slides over the rotating rod and the other slides over the horizontal curved rod, of which the shape is described by the equation r = 1.5(2 - cos u) ft. If both collars weigh 0.75 lb, determine the normal force which the curved rod exerts on one collar at the instant u = 120. Neglect friction. r

The boy of mass 40 kg is sliding down the spiral slide at a constant speed such that his position, measured from the top of the chute, has components r = 1.5 m, u = (0.7t) rad, and z = (-0.5t) m, where t is in seconds. Determine the components of force Fr, Fu, and Fz which the slide exerts on him at the instant t = 2 s. Neglect the size of the boy.

The 40-kg boy is sliding down the smooth spiral slide such that z = -2 m>s and his speed is 2 m>s. Determine the r, u, z components of force the slide exerts on him at this instant. Neglect the size of the boy.

Using a forked rod, a 0.5-kg smooth peg P is forced to move along the vertical slotted path r = (0.5 u) m, where u is in radians. If the angular position of the arm is u = ( p 8 t 2 ) rad, where t is in seconds, determine the force of the rod on the peg and the normal force of the slot on the peg at the instant t = 2 s. The peg is in contact with only one edge of the rod and slot at any instant.

The arm is rotating at a rate of u # = 4 rad>s when u $ = 3 rad>s 2 and u = 180. Determine the force it must exert on the 0.5-kg smooth cylinder if it is confined to move along the slotted path. Motion occurs in the horizontal plane.

If arm OA rotates with a constant clockwise angular velocity of u # = 1.5 rad>s, determine the force arm OA exerts on the smooth 4-lb cylinder B when u = 45.

Determine the normal and frictional driving forces that the partial spiral track exerts on the 200-kg motorcycle at the instant u = 5 3p rad, u # = 0.4 rad>s, u $ = 0.8 rad>s 2 . Neglect the size of the motorcycle.

A smooth can C, having a mass of 3 kg, is lifted from a feed at A to a ramp at B by a rotating rod. If the rod maintains a constant angular velocity of u # = 0.5 rad>s, determine the force which the rod exerts on the can at the instant u = 30. Neglect the effects of friction in the calculation and the size of the can so that r = (1.2 cos u) m. The ramp from A to B is circular, having a radius of 600 mm.

The spring-held follower AB has a mass of 0.5 kg and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.15 m and z = (0.02 cos 2u) m. If the cam is rotating at a constant rate of 30 rad>s, determine the force component Fz at the end A of the follower when u = 30. The spring is uncompressed when u = 90. Neglect friction at the bearing C.

The spring-held follower AB has a mass of 0.5 kg and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.15 m and z = (0.02 cos 2u) m. If the cam is rotating at a constant rate of 30 rad>s, determine the maximum and minimum force components Fz the follower exerts on the cam if the spring is uncompressed when u = 90.

The particle has a mass of 0.5 kg and is confined to move along the smooth vertical slot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the slot on the particle when u = 30. The rod is rotating with a constant angular velocity u . = 2 rad>s. Assume the particle contacts only one side of the slot at any instant.

A car of a roller coaster travels along a track which for a short distance is defined by a conical spiral, r = 3 4z, u = -1.5z, where r and z are in meters and u in radians. If the angular motion u # = 1 rad>s is always maintained, determine the r, u, z components of reaction exerted on the car by the track at the instant z = 6 m. The car and passengers have a total mass of 200 kg.

The 0.5-lb ball is guided along the vertical circular path r = 2rc cos u using the arm OA. If the arm has an angular velocity u . = 0.4 rad>s and an angular acceleration u $ = 0.8 rad>s 2 at the instant u = 30, determine the force of the arm on the ball. Neglect friction and the size of the ball. Set rc = 0.4ft

The ball of mass m is guided along the vertical circular path r = 2rc cos u using the arm OA. If the arm has a constant angular velocity u . 0, determine the angle u 45 at which the ball starts to leave the surface of the semicylinder. Neglect friction and the size of the ball.

Using a forked rod, a smooth cylinder P, having a mass of 0.4 kg, is forced to move along the vertical slotted path \(r = (0.6\ \theta)\ m\), where \(\theta\) is in radians. If the cylinder has a constant speed of \(v_C = 2\ m/s\), determine the force of the rod and the normal force of the slot on the cylinder at the instant \(\theta = \pi\ rad\). Assume the cylinder is in contact with only one edge of the rod and slot at any instant. Hint: To obtain the time derivatives necessary to compute the cylinder’s acceleration components \(a_r\) and \(a_{\theta}\), take the first and second time derivatives of \(r = 0.6\ \theta\). Then, for further information, use Eq. 12–26 to determine \(\dot{\theta}\). Also, take the time derivative of Eq. 12–26, noting that \(\quad \dot{v} = 0\) to determine \(\dot{\theta}\).

The pilot of the airplane executes a vertical loop which in part follows the path of a cardioid, r = 200(1 + cosu) m, where u is in radians. If his speed at A is a constant vp = 85 m>s, determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at A. He has a mass of 80 kg. Hint: To determine the time derivatives necessary to calculate the acceleration components ar and au, take the first and second time derivatives of r = 200(1 + cosu). Then, for further information, use Eq. 1226 to determine u # .

The collar has a mass of 2 kg and travels along the smooth horizontal rod defined by the equiangular spiral r = (eu ) m, where u is in radians. Determine the tangential force F and the normal force N acting on the collar when u = 45, if the force F maintains a constant angular motion u = 2 rad>s . .

The particle has a mass of 0.5 kg and is confined to move along the smooth horizontal slot due to the rotation of the arm OA. Determine the force of the rod on the particle and the normal force of the slot on the particle when \(\theta = 30^{\circ}\). The rod is rotating with a constant angular velocity \(\dot{\theta} = 2\ rad/s\). Assume the particle contacts only one side of the slot at any instant.

Solve Prob. 13105 if the arm has an angular acceleration of u $ = 3 rad>s 2 when u # = 2 rad>s at u = 30.

The forked rod is used to move the smooth 2-lb particle around the horizontal path in the shape of a limaon, r = (2 + cos u) ft. If u = (0.5t 2 ) rad, where t is in seconds, determine the force which the rod exerts on the particle at the instant t = 1 s. The fork and path contact the particle on only one side.

The collar, which has a weight of 3 lb, slides along the smooth rod lying in the horizontal plane and having the shape of a parabola \(r = 4/(1 - cos\ \theta)\), where \(\theta\) is in radians and r is in feet. If the collar’s angular rate is constant and equals \(\dot{\theta} = 4\ rad/s\), determine the tangential retarding force P needed to cause the motion and the normal force that the collar exerts on the rod at the instant \(\theta = 90^{\circ}\).

Rod OA rotates counterclockwise at a constant angular rate u . = 4 rad>s. The double collar B is pinconnected together such that one collar slides over the rotating rod and the other collar slides over the circular rod described by the equation r = (1.6 cos u) m. If both collars have a mass of 0.5 kg, determine the force which the circular rod exerts on one of the collars and the force that OA exerts on the other collar at the instant u = 45. Motion is in the horizontal plane.

Solve Prob. 13109 if motion is in the vertical plane.

A 0.2-kg spool slides down along a smooth rod. If the rod has a constant angular rate of rotation u # = 2 rad>s in the vertical plane, show that the equations of motion for the spool are r $ - 4r - 9.81 sin u = 0 and 0.8r # + Ns - 1.962 cos u = 0, where Ns is the magnitude of the normal force of the rod on the spool. Using the methods of differential equations, it can be shown that the solution of the first of these equations is r = C1e-2t + C2e2t - (9.81>8) sin 2t. If r, r # , and u are zero when t = 0, evaluate the constants C1 and C2 determine r at the instant u = p>4 rad.

The pilot of an airplane executes a vertical loop which in part follows the path of a four-leaved rose, r = (-600cos 2u) ft, where u is in radians. If his speed is a constant vP = 80 ft>s, determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at A. He weights 130 lb. Hint: To determine the time derivatives necessary to compute the acceleration components ar, and a0, take the first and second time derivatives of r = 400(1 + cosu). Then, for further information, use Eq. 1226 to determine u # . Also, take the time derivative of Eq. 1226, noting that vP # = 0 to determine u # .

The earth has an orbit with eccentricity 0.0167 around the sun. Knowing that the earths minimum distance from the sun is 146(106 ) km, find the speed at which the earth travels when it is at this distance. Determine the equation in polar coordinates which describes the earths orbit about the sun

A communications satellite is in a circular orbit above the earth such that it always remains directly over a point on the earths surface. As a result, the period of the satellite must equal the rotation of the earth, which is approximately 24 hours. Determine the satellites altitude h above the earths surface and its orbital speed.

The speed of a satellite launched into a circular orbit about the earth is given by Eq. 13–25. Determine the speed of a satellite launched parallel to the surface of the earth so that it travels in a circular orbit 800 km from the earth’s surface.

The rocket is in circular orbit about the earth at an altitude of 20 Mm. Determine the minimum increment in speed it must have in order to escape the earth’s gravitational field.

Prove Kepler’s third law of motion. Hint: Use Eqs. 13–19, 13–28, 13–29, and 13–31.

The satellite is moving in an elliptical orbit with an eccentricity e = 0.25. Determine its speed when it is at its maximum distance A and minimum distance B from the earth.

The rocket is traveling in free flight along the elliptical orbit. The planet has no atmosphere, and its mass is 0.60 times that of the earth. If the rocket has the orbit shown, determine the rockets speed when it is at A and at B.

Determine the constant speed of satellite S so that it circles the earth with an orbit of radius r = 15 Mm. Hint: Use Eq. 131.

The rocket is in free flight along an elliptical trajectory AA. The planet has no atmosphere, and its mass is 0.70 times that of the earth. If the rocket has an apoapsis and periapsis as shown in the figure, determine the speed of the rocket when it is at point A.

The Viking Explorer approaches the planet Mars on a parabolic trajectory as shown. When it reaches point A its velocity is 10 Mm/h. Determine \(r_0\) and the required change in velocity at A so that it can then maintain a circular orbit as shown. The mass of Mars is 0.1074 times the mass of the earth.

The rocket is initially in free-flight circular orbit around the earth. Determine the speed of the rocket at A. What change in the speed at A is required so that it can move in an elliptical orbit to reach point A’?

The rocket is in free-flight circular orbit around the earth. Determine the time needed for the rocket to travel from the innner orbit at A to the outer orbit at A.

A satellite is launched at its apogee with an initial velocity v0 = 2500 mi>h parallel to the surface of the earth. Determine the required altitude (or range of altitudes) above the earths surface for launching if the free-flight trajectory is to be (a) circular, (b) parabolic, (c) elliptical, with launch at apogee, and (d) hyperbolic. Take G = 34.4(10-9 )(lb # ft2 )>slug2 , Me = 409(1021) slug, the earths radius re = 3960 mi, and 1 mi = 5280 ft.

The rocket is traveling around the earth in free flight along the elliptical orbit. If the rocket has the orbit shown, determine the speed of the rocket when it is at A and at B.

An elliptical path of a satellite has an eccentricity e = 0.130. If it has a speed of 15 Mm>h when it is at perigee, P, determine its speed when it arrives at apogee, A. Also, how far is it from the earths surface when it is at A?

A rocket is in free-flight elliptical orbit around the planet Venus. Knowing that the periapsis and apoapsis of the orbit are 8 Mm and 26 Mm, respectively, determine (a) the speed of the rocket at point A, (b) the required speed it must attain at A just after braking so that it undergoes an 8-Mm free-flight circular orbit around Venus, and (c) the periods of both the circular and elliptical orbits. The mass of Venus is 0.816 times the mass of the earth.

The rocket is traveling in a free flight along an elliptical trajectory AA. The planet has no atmosphere, and its mass is 0.60 times that of the earth. If the rocket has the orbit shown, determine the rockets velocity when it is at point A.

If the rocket is to land on the surface of the planet, determine the required free-flight speed it must have at A so that the landing occurs at B. How long does it take for the rocket to land, going from A to B? The planet has no atmosphere, and its mass is 0.6 times that of the earth.

The rocket is traveling around the earth in free flight along an elliptical orbit AC. If the rocket has the orbit shown, determine the rockets velocity when it is at point A.

The rocket is traveling around the earth in free flight along the elliptical orbit AC. Determine its change in speed when it reaches A so that it travels along the elliptical orbit AB.

If the box is released from rest at A, use numerical values to show how you would estimate the time for it to arrive at B. Also, list the assumptions for your analysis.

The tugboat has a known mass and its propeller provides a known maximum thrust. When the tug is fully powered you observe the time it takes for the tug to reach a speed of known value starting from rest. Show how you could determine the mass of the barge. Neglect the drag force of the water on the tug. Use numerical values to explain your answer.

Determine the smallest speed of each car A and B so that the passengers do not lose contact with the seat while the arms turn at a constant rate. What is the largest normal force of the seat on each passenger? Use numerical values to explain your answer.

Each car is pin connected at its ends to the rim of the wheel which turns at a constant speed. Using numerical values, show how to determine the resultant force the seat exerts on the passenger located in the top car A. The passengers are seated toward the center of the wheel. Also, list the assumptions for your analysis.

The van is traveling at 20 km/h when the coupling of the trailer at A fails. If the trailer has a mass of 250 kg and coasts 45 m before coming to rest, determine the constant horizontal force F created by rolling friction which causes the trailer to stop.

The motor M pulls in its attached rope with an acceleration \(a_{p} = 6 \ m/s^{2}\). Determine the towing force exerted by M on the rope in order to move the 50-kg crate up the inclined plane. The coefficient of kinetic friction between the crate and the plane is \(\mu_{k} = 0.3\). Neglect the mass of the pulleys and rope.

Block B rests on a smooth surface. If the coefficients of friction between A and B are \(\mu_{s} = 0.4\) and \(\mu_{k} = 0.3\), determine the acceleration of each block if F = 50 lb.

If the motor draws in the cable at a rate of \(v = (0 .05 \ s^{3/2}) \ m/s\), where s is in meters, determine the tension developed in the cable when s = 10 m. The crate has a mass of 20 kg, and the coefficient of kinetic friction between the crate and the ground is \(\mu_{k} = 0.2\).

The ball has a mass of 30 kg and a speed v = 4 m/s at the instant it is at its lowest point, \(\theta = 0^{\circ}\). Determine the tension in the cord and the rate at which the ball’s speed is decreasing at the instant \(\theta = 20^{\circ}\). Neglect the size of the ball.

The bottle rests at a distance of 3 ft from the center of the horizontal platform. If the coefficient of static friction between the bottle and the platform is \(\mu_{s} = 0.3\), determine the maximum speed that the bottle can attain before slipping. Assume the angular motion of the platform is slowly increasing.

The 10-lb suitcase slides down the curved ramp for which the coefficient of kinetic friction is \(\mu_{k} = 0.2\). If at the instant it reaches point A it has a speed of 5 ft/s, determine the normal force on the suitcase and the rate of increase of its speed.

The spool, which has a mass of 4 kg, slides along the rotating rod. At the instant shown, the angular rate of rotation of the rod is \(\dot{\theta} = 6 \ rad/s\) and this rotation is increasing at \(\ddot{\theta} = 2 \ rad/s^{2}\). At this same instant, the spool has a velocity of 3 m/s and an acceleration of \(1 \ m/s^{2}\), both measured relative to the rod and directed away from the center O when r = 0.5 m. Determine the radial frictional force and the normal force, both exerted by the rod on the spool at this instant.