An ideal rope or bicycle-type chain of length L and mass per unit length is resting on a

The system of three particles has the indicated particle masses, velocities, and external forces. Determine , , , T, , and for this two-dimensional system. Problem 4/1

For the particle system of Prob. 4/1, determine and .

The system of three particles has the indicated particle masses, velocities, and external forces. Determine , T, HO, and for this threedimensional system. Problem 4/3

For the particle system of Prob. 4/3, determine HG and . H G

The two 2-kg balls are initially at rest on the horizontal surface when a vertical force F 60 N is applied to the junction of the attached wires as shown. Compute the vertical component ay of the initial acceleration of each ball by considering the system as a whole. Problem 4/5

Three monkeys A, B, and C weighing 20, 25, and 15 lb, respectively, are climbing up and down the rope suspended from D. At the instant represented, A is descending the rope with an acceleration of 5 ft/sec2 , and C is pulling himself up with an acceleration of 3 ft/sec2 . Monkey B is climbing up with a constant speed of 2 ft/sec. Treat the rope and monkeys as a complete system and calculate the tension T in the rope at D. Problem 4/6

The three small spheres are connected by the cords and spring and are supported by a smooth horizontal surface. If a force F 6.4 N is applied to one of the cords, find the acceleration of the mass center of the spheres for the instant depicted. Problem 4/7

The two spheres, each of mass m, are connected by the spring and hinged bars of negligible mass. The spheres are free to slide in the smooth guides up the incline . Determine the acceleration aC of the center C of the spring. Problem 4/8

Calculate the acceleration of the center of mass of the system of the four 10-kg cylinders. Neglect friction and the mass of the pulleys and cables. Problem 4/9

The four systems slide on a smooth horizontal surface and have the same mass m. The configurations of mass in the two pairs are identical. What can be said about the acceleration of the mass center for each system? Explain any difference in the accelerations of the members. Problem 4/10

The total linear momentum of a system of five particles at time t 2.2 s is given by G2.2 3.4i  2.6j 4.6k . At time t 2.4 s, the linear momentum has changed to G2.4 3.7i  2.2j 4.9k . Calculate the magnitude F of the time average of the resultant of the external forces acting on the system during the interval.

The two small spheres, each of mass m, are rigidly connected by a rod of negligible mass and are released from rest in the position shown and slide down the smooth circular guide in the vertical plane. Determine their common velocity v as they reach the horizontal dashed position. Also find the force R between sphere 1 and the supporting surface an instant before the sphere reaches the bottom position A. Problem 4/12

The two small spheres, each of mass m, and their connecting rod of negligible mass are rotating about their mass center G with an angular velocity . At the same instant the mass center has a velocity v in the x-direction. Determine the angular momentum of the assembly at the instant when G has coordinates x and y. Problem 4/13

Each of the five connected particles has a mass of 0.6 kg, with G as the center of mass of the system. At a certain instant the angular momentum of the system about G is 1.20k , and the x- and y-components of the velocity of G are 3 m/s and 4 m/s, respectively. Calculate the angular momentum HO of the system about O for this instant. Problem 4/14

The three identical bars, each weighing 8 lb, are connected by the two freely pinned links of negligible weight and are resting on a smooth horizontal surface. Calculate the initial acceleration a of the center of the middle bar when the 10-lb force is applied to the connecting link as shown.

A centrifuge consists of four cylindrical containers, each of mass m, at a radial distance r from the rotation axis. Determine the time t required to bring the centrifuge to an angular velocity  from rest under a constant torque M applied to the shaft. The diameter of each container is small compared with r, and the mass of the shaft and supporting arms is small compared with m. Problem 4/16

The three small spheres are welded to the light rigid frame which is rotating in a horizontal plane about a vertical axis through O with an angular velocity 20 rad/s. If a couple MO 30 is applied to the frame for 5 seconds, compute the new angular velocity . Problem 4/17

The four 3-kg balls are rigidly mounted to the rotating frame and shaft, which are initially rotating freely about the vertical z-axis at the angular rate of 20 rad/s clockwise when viewed from above. If a constant torque M 30 is applied to the shaft, calculate the time t to reverse the direction of rotation and reach an angular velocity 20 rad/s in the same sense as M. Problem 4/18

Billiard ball A is moving in the y-direction with a velocity of 2 m/s when it strikes ball B of identical size and mass initially at rest. Following the impact, the balls are observed to move in the directions shown. Calculate the velocities vA and vB which the balls have immediately after the impact. Treat the balls as particles and neglect any friction forces acting on the balls compared with the force of impact. Problem 4/19

The 300-kg and 400-kg mine cars are rolling in opposite directions along the horizontal track with the respective speeds of 0.6 m/s and 0.3 m/s. Upon impact the cars become coupled together. Just prior to impact, a 100-kg boulder leaves the delivery chute with a velocity of 1.2 m/s in the direction shown and lands in the 300-kg car. Calculate the velocity v of the system after the boulder has come to rest relative to the car. Would the final velocity be the same if the cars were coupled before the boulder dropped? Problem 4/20

The three freight cars are rolling along the horizontal track with the velocities shown. After the impacts occur, the three cars become coupled together and move with a common velocity v. The weights of the loaded cars A, B, and C are 130,000, 100,000, and 150,000 lb, respectively. Determine v and calculate the percentage loss n of energy of the system due to coupling. Problem 4/21

The man of mass m1 and the woman of mass m2 are standing on opposite ends of the platform of mass m0 which moves with negligible friction and is initially at rest with s 0. The man and woman begin to approach each other. Derive an expression for the displacement s of the platform when the two meet in terms of the displacement x1 of the man relative to the platform.

The woman A, the captain B, and the sailor C weigh 120, 180, and 160 lb, respectively, and are sitting in the 300-lb skiff which is gliding through the water with a speed of 1 knot. If the three people change their positions as shown in the second figure, find the distance x from the skiff to the position where it would have been if the people had not moved. Neglect any resistance to motion afforded by the water. Does the sequence or timing of the change in positions affect the final result? Problem 4/23

The two spheres are rigidly connected to the rod of negligible mass and are initially at rest on the smooth horizontal surface. A force F is suddenly applied to one sphere in the y-direction and imparts an impulse of 10 during a negligibly short period of time. As the spheres pass the dashed position, calculate the velocity of each one. Problem 4/24

The three small spheres, each of mass m, are secured to the light rods to form a rigid unit supported in the vertical plane by the smooth circular surface. The force of constant magnitude P is applied perpendicular to one rod at its midpoint. If the unit starts from rest at 0, determine (a) the minimum force Pmin which will bring the unit to rest at 60 and (b) the common velocity v of spheres 1 and 2 when 60 if P 2Pmin. Problem 4/25

The three small steel balls, each of mass 2.75 kg, are connected by the hinged links of negligible mass and equal length. They are released from rest in the positions shown and slide down the quarter-circular guide in the vertical plane. When the upper sphere reaches the bottom position, the spheres have a horizontal velocity of 1.560 m/s. Calculate the energy loss Q due to friction and the total impulse Ix on the system of three spheres during this interval. Problem 4/26

Two steel balls, each of mass m, are welded to a light rod of length L and negligible mass and are initially at rest on a smooth horizontal surface. A horizontal force of magnitude F is suddenly applied to the rod as shown. Determine (a) the instantaneous acceleration of the mass center G and (b) the corresponding rate at which the angular velocity of the assembly about G is changing with time. Problem 4/27

The small car, which has a mass of 20 kg, rolls freely on the horizontal track and carries the 5-kg sphere mounted on the light rotating rod with r 0.4 m. A geared motor drive maintains a constant angular speed 4 rad/s of the rod. If the car has a velocity v 0.6 m/s when 0, calculate v when 60. Neglect the mass of the wheels and any friction. Problem 4/28

The cars of a roller-coaster ride have a speed of 30 km/h as they pass over the top of the circular track. Neglect any friction and calculate their speed v when they reach the horizontal bottom position. At the top position, the radius of the circular path of their mass centers is 18 m, and all six cars have the same mass. Problem 4/29

The two small spheres, each of mass m, are connected by a cord of length 2b (measured to the centers of the spheres) and are initially at rest on a smooth horizontal surface. A projectile of mass m0 with a velocity v0 perpendicular to the cord hits it in the middle, causing the deflection shown in part b of the figure. Determine the velocity v of m0 as the two spheres near contact, with approaching 90 as indicated in part c of the figure. Also find for this condition. Problem 4/30

The carriage of mass 2m is free to roll along the horizontal rails and carries the two spheres, each of mass m, mounted on rods of length l and negligible mass. The shaft to which the rods are secured is mounted in the carriage and is free to rotate. If the system is released from rest with the rods in the vertical position where 0, determine the velocity vx of the carriage and the angular velocity of the rods for the instant when 180. Treat the carriage and the spheres as particles and neglect any friction. Problem 4/31

The 50,000-lb flatcar supports a 15,000-lb vehicle on a 5 ramp built on the flatcar. If the vehicle is released from rest with the flatcar also at rest, determine the velocity v of the flatcar when the vehicle has rolled s 40 ft down the ramp just before hitting the stop at B. Neglect all friction and treat the vehicle and the flatcar as particles. Problem 4/32 s A

A flexible nonextensible rope of mass  per unit length and length equal to 1/4 of the circumference of the fixed drum of radius r is released from rest in the horizontal dashed position, with end B secured to the top of the drum. When the rope finally comes to rest with end A at C, determine the loss of energy Q of the system. What becomes of the lost energy? Problem 4/33

A horizontal bar of mass and small diameter is suspended by two wires of length l from a carriage of mass which is free to roll along the horizontal rails. If the bar and carriage are released from rest with the wires making an angle with the vertical, determine the velocity of the bar relative to the carriage and the velocity of the carriage at the instant when . Neglect all friction and treat the carriage and the bar as particles in the vertical plane of motion. Problem 4/34

The jet aircraft has a mass of 4.6 Mg and a drag (air resistance) of 32 kN at a speed of 1000 km/h at a particular altitude. The aircraft consumes air at the rate of 106 kg/s through its intake scoop and uses fuel at the rate of 4 kg/s. If the exhaust has a rearward velocity of 680 m/s relative to the exhaust nozzle, determine the maximum angle of elevation  at which the jet can fly with a constant speed of 1000 km/h at the particular altitude in question. Problem 4/35

A jet of air issues from the nozzle with a velocity of 300 ft/sec at the rate of 6.50 ft3 /sec and is deflected by the right-angle vane. Calculate the force F required to hold the vane in a fixed position. The specific weight of the air is 0.0753 lb/ft3 . Problem 4/36

Fresh water issues from the nozzle with a velocity of 30 m/s at the rate of 0.05 m3 /s and is split into two equal streams by the fixed vane and deflected through 60 as shown. Calculate the force F required to hold the vane in place. The density of water is 1000 kg/m3 . Problem 4/37

The jet water ski has reached its maximum velocity of 70 km/h when operating in salt water. The water intake is in the horizontal tunnel in the bottom of the hull, so the water enters the intake at the velocity of 70 km/h relative to the ski. The motorized pump discharges water from the horizontal exhaust nozzle of 50-mm diameter at the rate of 0.082 m3 /s. Calculate the resistance R of the water to the hull at the operating speed. Problem 4/38

The fire tug discharges a stream of salt water (density 1030 kg/m3 ) with a nozzle velocity of 40 m/s at the rate of 0.080 m3 /s. Calculate the propeller thrust T which must be developed by the tug to maintain a fixed position while pumping. Problem 4/39

The figure shows the top view of an experimental rocket sled which is traveling at a speed of 1000 ft/sec when its forward scoop enters a water channel to act as a brake. The water is diverted at right angles relative to the motion of the sled. If the frontal flow area of the scoop is 15 in.2 , calculate the initial braking force. The specific weight of water is 62.4 lb/ft3 . Problem 4/40

A jet-engine noise suppressor consists of a movable duct which is secured directly behind the jet exhaust by cable A and deflects the blast directly upward. During a ground test, the engine sucks in air at the rate of 43 kg/s and burns fuel at the rate of 0.8 kg/s. The exhaust velocity is 720 m/s. Determine the tension T in the cable. Problem 4/41 A 15

The 90 vane moves to the left with a constant velocity of 10 m/s against a stream of fresh water issuing with a velocity of 20 m/s from the 25-mm-diameter nozzle. Calculate the forces Fx and Fy on the vane required to support the motion. Problem 4/42

A jet of fluid with cross-sectional area A and mass density issues from the nozzle with a velocity v and impinges on the inclined trough shown in section. Some of the fluid is diverted in each of the two directions. If the trough is smooth, the velocity of both diverted streams remains v, and the only force which can be exerted on the trough is normal to the bottom surface. Hence, the trough will be held in position by forces whose resultant is F normal to the trough. By writing impulse-momentum equations for the directions along and normal to the trough, determine the force F required to support the trough. Also find the volume rates of flow and for the two streams. Problem 4/43

The 8-oz ball is supported by the vertical stream of fresh water which issues from the 1/2-in.-diameter nozzle with a velocity of 35 ft/sec. Calculate the height h of the ball above the nozzle. Assume that the stream remains intact and there is no energy lost in the jet stream. Problem 4/44

A jet-engine thrust reverser to reduce an aircraft speed of 200 km/h after landing employs folding vanes which deflect the exhaust gases in the direction indicated. If the engine is consuming 50 kg of air and 0.65 kg of fuel per second, calculate the braking thrust as a fraction n of the engine thrust without the deflector vanes. The exhaust gases have a velocity of 650 m/s relative to the nozzle. Problem 4/45

Salt water is being discharged into the atmosphere from the two 30 outlets at the total rate of Each of the discharge nozzles has a flow diameter of 100 mm, and the inside diameter of the pipe at the connecting section A is 250 mm. The pressure of the water at section A-A is 550 kPa. If each of the six bolts at the flange A-A is tightened to a tension of 10 kN, calculate the average pressure p on the flange gasket, which has an area of . The pipe above the flange and the water within it have a mass of 60 kg. Problem 4/46

The axial-flow fan C pumps air through the duct of circular cross section and exhausts it with a velocity v at B. The air densities at A and B are A and B, respectively, and the corresponding pressures are pA and pB. The fixed deflecting blades at D restore axial flow to the air after it passes through the propeller blades C. Write an expression for the resultant horizontal force R exerted on the fan unit by the flange and bolts at A. Problem 4/47 A B C D Dia. = d Dia. = d E

Air is pumped through the stationary duct A with a velocity of 50 ft/sec and exhausted through an experimental nozzle section BC. The average static pressure across section B is 150 lb/in.2 gage, and the specific weight of air at this pressure and at the temperature prevailing is 0.840 lb/ft3 . The average static pressure across the exit section C is measured to be 2 lb/in.2 gage, and the corresponding specific weight of air is 0.0760 lb/ft3 . Calculate the force T exerted on the nozzle flange at B by the bolts and the gasket to hold the nozzle in place. Problem 4/48

One of the most advanced methods for cutting metal plates uses a high-velocity water jet which carries an abrasive garnet powder. The jet issues from the 0.01-in.-diameter nozzle at A and follows the path shown through the thickness t of the plate. As the plate is slowly moved to the right, the jet makes a narrow precision slot in the plate. The water-abrasive mixture is used at the low rate of 1/2 gal/min and has a specific weight of 68 lb/ft3 . Water issues from the bottom of the plate with a velocity which is 60 percent of the impinging nozzle velocity. Calculate the horizontal force F required to hold the plate against the jet. (There are 231 in.3 in 1 gal.) Problem 4/49 F A

The sump pump has a net mass of 310 kg and pumps fresh water against a 6-m head at the rate of 0.125 m3 /s. Determine the vertical force R between the supporting base and the pump flange at A during operation. The mass of water in the pump may be taken as the equivalent of a 200-mm-diameter column 6 m in height. Problem 4/50

In a test of the operation of a cherry-picker fire truck, the equipment is free to roll with its brakes released. For the position shown, the truck is observed to deflect the spring of stiffness k 15 kN/m a distance of 150 mm because of the action of the horizontal stream of water issuing from the nozzle when the pump is activated. If the exit diameter of the nozzle is 30 mm, calculate the velocity v of the stream as it leaves the nozzle. Also determine the added moment M which the joint at A must resist when the pump is in operation with the nozzle in the position shown. Problem 4/51

The experimental ground-effect machine has a total weight of 4200 lb. It hovers 1 or 2 ft off the ground by pumping air at atmospheric pressure through the circular intake duct at B and discharging it horizontally under the periphery of the skirt C. For an intake velocity v of 150 ft/sec, calculate the average air pressure p under the 18-ft-diameter machine at ground level. The specific weight of the air is 0.076 lb/ft3 . Problem 4/52

A commercial aircraft flying horizontally at 500 mi/hr encounters a heavy downpour of rain falling vertically at the rate of 20 ft/sec with an intensity equivalent to an accumulation of 1 in./hr on the ground. The upper surface area of the aircraft projected onto the horizontal plane is 2960 ft2 . Calculate the negligible downward force F of the rain on the aircraft. Problem 4/53

The ducted fan unit of mass m is supported in the vertical position on its flange at A. The unit draws in air with a density  and a velocity u through section A and discharges it through section B with a velocity v. Both inlet and outlet pressures are atmospheric. Write an expression for the force R applied to the flange of the fan unit by the supporting slab.

The 180 return pipe discharges salt water (specific weight 64.4 lb/ft3 ) into the atmosphere at a constant rate of 1.6 ft3 /sec. The static pressure in the water at section A is 10 lb/in.2 above atmospheric pressure. The flow area of the pipe at A is 20 in.2 and that at each of the two outlets is 3.2 in.2 If each of the six flange bolts is tightened with a torque wrench so that it is under a tension of 150 lb, determine the average pressure p on the gasket between the two flanges. The flange area in contact with the gasket is 16 in.2 Also determine the bending moment M in the pipe at section A if the left-hand discharge is blocked off and the flow rate is cut in half. Neglect the weight of the pipe and the water within it. Problem 4/55

The fire hydrant is tested under a high standpipe pressure. The total flow of 10 ft3 /sec is divided equally between the two outlets, each of which has a cross-sectional area of 0.040 ft2 . The inlet crosssectional area at the base is 0.75 ft2 . Neglect the weight of the hydrant and water within it and compute the tension T, the shear V, and the bending moment M in the base of the standpipe at B. The specific weight of water is 62.4 lb/ft3 . The static pressure of the water as it enters the base at B is 120 lb/in.2 Problem 4/56

A rotary snow plow mounted on a large truck eats its way through a snow drift on a level road at a constant speed of 20 km/h. The plow discharges 60 Mg of snow per minute from its 45 chute with a velocity of 12 m/s relative to the plow. Calculate the tractive force P on the tires in the direction of motion necessary to move the plow and find the corresponding lateral force R between the tires and the road. Problem 4/57

The industrial blower sucks in air through the axial opening A with a velocity v1 and discharges it at atmospheric pressure and temperature through the 150-mm-diameter duct B with a velocity v2. The blower handles 16 m3 of air per minute with the motor and fan running at 3450 rev/min. If the motor requires 0.32 kW of power under no load (both ducts closed), calculate the power P consumed while air is being pumped. Problem 4/58

The feasibility of a one-passenger VTOL (vertical takeoff and landing) craft is under review. The preliminary design calls for a small engine with a high power-to-weight ratio driving an air pump that draws in air through the 70 ducts with an inlet velocity v 40 m/s at a static gage pressure of 1.8 kPa across the inlet areas totaling 0.1320 m2 . The air is exhausted vertically down with a velocity u 420 m/s. For a 90-kg passenger, calculate the maximum net mass m of the machine for which it can take off and hover. (See Table D/1 for air density.) Problem 4/59

The military jet aircraft has a gross weight of 24,000 lb and is poised for takeoff with brakes set while the engine is revved up to maximum power. At this condition, air with a specific weight of 0.0753 lb/ft3 is sucked into the intake ducts at the rate of 106 lb/sec with a static pressure of 0.30 lb/in.2 (gage) across the duct entrance. The total cross-sectional area of both intake ducts (one on each side) is 1800 in.2 The airfuel ratio is 18, and the exhaust velocity u is 3100 ft/sec with zero back pressure (gage) across the exhaust nozzle. Compute the initial acceleration a of the aircraft upon release of the brakes. Problem 4/60

The helicopter shown has a mass m and hovers in position by imparting downward momentum to a column of air defined by the slipstream boundary shown. Find the downward velocity v given to the air by the rotor at a section in the stream below the rotor, where the pressure is atmospheric and the stream radius is r. Also find the power P required of the engine. Neglect the rotational energy of the air, any temperature rise due to air friction, and any change in air density . Problem 4/61

The VTOL (vertical takeoff and landing) military aircraft is capable of rising vertically under the action of its jet exhaust, which can be vectored from  0 for takeoff and hovering to 90 for forward flight. The loaded aircraft has a mass of 8600 kg. At full takeoff power, its turbo-fan engine consumes air at the rate of 90 kg/s and has an airfuel ratio of 18. Exhaust-gas velocity is 1020 m/s with essentially atmospheric pressure across the exhaust nozzles. Air with a density of 1.206 kg/m3 is sucked into the intake scoops at a pressure of 2 kPa (gage) over the total inlet area of 1.10 m2 . Determine the angle for vertical takeoff and the corresponding vertical acceleration ay of the aircraft. Problem 4/62

A marine terminal for unloading bulk wheat from a ship is equipped with a vertical pipe with a nozzle at A which sucks wheat up the pipe and transfers it to the storage building. Calculate the x- and y-components of the force R required to change the momentum of the flowing mass in rounding the bend. Identify all forces applied externally to the bend and mass within it. Air flows through the 14-in.-diameter pipe at the rate of 18 tons per hour under a vacuum of 9 in. of mercury ( p 4.42 lb/in.2 ) and carries with it 150 tons of wheat per hour at a speed of 124 ft/sec. Problem 4/63

The sprinkler is made to rotate at the constant angular velocity  and distributes water at the volume rate Q. Each of the four nozzles has an exit area A. Write an expression for the torque M on the shaft of the sprinkler necessary to maintain the given motion. For a given pressure and, thus, flow rate Q, at what speed 0 will the sprinkler operate with no applied torque? Let  be the density of the water. Problem 4/64

A high-speed jet of air issues from the 40-mm-diameter nozzle A with a velocity v of 240 m/s and impinges on the vane OB, shown in its edge view. The vane and its right-angle extension have negligible mass compared with the attached 6-kg cylinder and are freely pivoted about a horizontal axis through O. Calculate the angle assumed by the vane with the horizontal. The air density under the prevailing conditions is 1.206 kg/m3 . State any assumptions. Problem 4/65

An axial section of the suction nozzle A for a bulk wheat unloader is shown here. The outer pipe is secured to the inner pipe by several longitudinal webs which do not restrict the flow of air. A vacuum of 9 in. of mercury ( p 4.42 lb/in.2 gage) is maintained in the inner pipe, and the pressure across the bottom of the outer pipe is atmospheric ( p 0). Air at 0.075 lb/ft3 is drawn in through the space between the pipes at a rate of 18 tons/hr at atmospheric pressure and draws with it 150 tons of wheat per hour up the pipe at a velocity of 124 ft/sec. If the nozzle unit below section A-A weighs 60 lb, calculate the compression C in the connection at A-A. Problem 4/66

In the figure is shown an impulse-turbine wheel for a hydroelectric power plant which is to operate with a static head of water of 300 m at each of its six nozzles and is to rotate at the speed of . Each wheel and generator unit is to develop an output power of . The efficiency of the generator may be taken to be 0.90, and an efficiency of 0.85 for the conversion of the kinetic energy of the water jets to energy delivered by the turbine may be expected. The mean peripheral speed of such a wheel for greatest efficiency will be about 0.47 times the jet velocity. If each of the buckets is to have the shape shown, determine the necessary jet diameter d and wheel diameter D. Assume that the water acts on the bucket which is at the tangent point of each jet stream. Problem 4/67 D 10 10 u v B

A test vehicle designed for impact studies has a mass m 1.4 Mg and is accelerated from rest by the impingement of a high-velocity water jet upon its curved deflector attached to the rear of the vehicle. The jet of fresh water is produced by the air-operated piston and issues from the 140-mm-diameter nozzle with a velocity v 150 m/s. Frictional resistance of the vehicle, treated as a particle, amounts to 10 percent of its weight. Determine the velocity u of the vehicle 3 seconds after release from rest. (Hint: Adapt the results of Sample Problem 4/6.) Problem 4/68 60 u m v

At the instant of vertical launch the rocket expels exhaust at the rate of 220 kg/s with an exhaust velocity of 820 m/s. If the initial vertical acceleration is 6.80 m/s2 , calculate the total mass of the rocket and fuel at launch. Problem 4/69

When the rocket reaches the position in its trajectory shown, it has a mass of 3 Mg and is beyond the effect of the earths atmosphere. Gravitational acceleration is 9.60 m/s2 . Fuel is being consumed at the rate of 130 kg/s, and the exhaust velocity relative to the nozzle is 600 m/s. Compute the n- and t-components of acceleration of the rocket. Problem 4/70

The space shuttle, together with its central fuel tank and two booster rockets, has a total mass of 2.04(106 ) kg at liftoff. Each of the two booster rockets produces a thrust of 11.80(106 ) N, and each of the three main engines of the shuttle produces a thrust of 2.00(106 ) N. The specific impulse (ratio of exhaust velocity to gravitational acceleration) for each of the three main engines of the shuttle is 455 s. Calculate the initial vertical acceleration a of the assembly with all five engines operating and find the rate at which fuel is being consumed by each of the shuttles three engines. Problem 4/71

A tank truck for washing down streets has a total weight of 20,000 lb when its tank is full. With the spray turned on, 80 lb of water per second issue from the nozzle with a velocity of 60 ft/sec relative to the truck at the 30 angle shown. If the truck is to accelerate at the rate of 2 ft/sec2 when starting on a level road, determine the required tractive force P between the tires and the road when (a) the spray is turned on and (b) the spray is turned off. Problem 4/72

A tank, which has a mass of 50 kg when empty, is propelled to the left by a force P and scoops up fresh water from a stream flowing in the opposite direction with a velocity of 1.5 m/s. The entrance area of the scoop is 2000 mm2 , and water enters the scoop at a rate equal to the velocity of the scoop relative to the stream. Determine the force P at a certain instant for which 80 kg of water have been ingested and the velocity and acceleration of the tank are 2 m/s and 0.4 m/s2 , respectively. Neglect the small impact pressure at the scoop necessary to elevate the water in the tank. Problem 4/73

A small rocket of initial mass m0 is fired vertically upward near the surface of the earth ( g constant). If air resistance is neglected, determine the manner in which the mass m of the rocket must vary as a function of the time t after launching in order that the rocket may have a constant vertical acceleration a, with a constant relative velocity u of the escaping gases with respect to the nozzle.

The magnetometer boom for a spacecraft consists of a large number of triangular-shaped units which spring into their deployed configuration upon release from the canister in which they were folded and packed prior to release. Write an expression for the force F which the base of the canister must exert on the boom during its deployment in terms of the increasing length x and its time derivatives. The mass of the boom per unit of deployed length is . Treat the supporting base on the spacecraft as a fixed platform and assume that the deployment takes place outside of any gravitational field. Neglect the dimension b compared with x.

Fresh water issues from the two 30-mm-diameter holes in the bucket with a velocity of 2.5 m/s in the directions shown. Calculate the force P required to give the bucket an upward acceleration of 0.5 m/s2 from rest if it contains 20 kg of water at that time. The empty bucket has a mass of 0.6 kg. Problem 4/76 P 20 2

The upper end of the open-link chain of length L and mass  per unit length is lowered at a constant speed v by the force P. Determine the reading R of the platform scale in terms of x. Problem 4/77

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb/sec with a velocity of 10 ft/sec in the direction shown and is deposited on the moving flatbed truck. The tractive force between the driving wheels and the road is 380 lb, which overcomes the 200 lb of frictional road resistance. Determine the acceleration a of the truck 4 seconds after the hopper is opened over the truck bed, at which instant the truck has a forward speed of 1.5 mi/hr. The empty weight of the truck is 12,000 lb. Problem 4/78

A railroad coal car weighs 54,600 lb empty and carries a total load of 180,000 lb of coal. The bins are equipped with bottom doors which permit discharging coal through an opening between the rails. If the car dumps coal at the rate of 20,000 lb/sec in a downward direction relative to the car, and if frictional resistance to motion is 4 lb per ton of total remaining weight, determine the coupler force P required to give the car an acceleration of 0.15 ft/sec2 in the direction of P at the instant when half the coal has been dumped. Problem 4/79

The figure represents an idealized one-dimensional structure of uniform mass  per unit length moving horizontally with a velocity v0 when its front end collides with an immovable barrier and crushes. The force F required to initiate and maintain an accordionlike deformation is constant. Neglect the length b of the collapsed portion of the structure compared with the movement of s of the undeformed portion following the impact. The undeformed part may be viewed as a body of decreasing mass. Derive the differential equation which relates F to s, and by using Eq. 4/20 carefully. Check your expression by applying Eq. 4/6 to both parts together as a system of constant mass. Problem 4/80

A coil of heavy flexible cable with a total length of 100 m and a mass of 1.2 kg/m is to be laid along a straight horizontal line. The end is secured to a post at A, and the cable peels off the coil and emerges through the horizontal opening in the cart as shown. The cart and drum together have a mass of 40 kg. If the cart is moving to the right with a velocity of 2 m/s when 30 m of cable remain in the drum and the tension in the rope at the post is 2.4 N, determine the force P required to give the cart and drum an acceleration of 0.3 m/s2 . Neglect all friction. Problem 4/81

By lowering a scoop as it skims the surface of a body of water, the aircraft (nicknamed the Super Scooper) is able to ingest 4.5 m3 of fresh water during a 12- second run. The plane then flies to a fire area and makes a massive water drop with the ability to repeat the procedure as many times as necessary. The plane approaches its run with a velocity of 280 km/h and an initial mass of 16.4 Mg. As the scoop enters the water, the pilot advances the throttle to provide an additional 300 hp (223.8 kW) needed to prevent undue deceleration. Determine the initial deceleration when the scooping action starts. (Neglect the difference between the average and the initial rates of water intake.) Problem 4/82

An open-link chain of length L 8 m with a mass of 48 kg is resting on a smooth horizontal surface when end A is doubled back on itself by a force P applied to end A. (a) Calculate the required value of P to give A a constant velocity of 1.5 m/s. (b) Calculate the acceleration a of end A if P 20 N and if v 1.5 m/s when x 4 m. Problem 4/83

A small rocket-propelled vehicle weighs 125 lb, including 20 lb of fuel. Fuel is burned at the constant rate of 2 lb/sec with an exhaust velocity relative to the nozzle of 400 ft/sec. Upon ignition the vehicle is released from rest on the 10 incline. Calculate the maximum velocity v reached by the vehicle. Neglect all friction. Problem 4/84

Determine the force P required to give the open-link chain of total length L a constant velocity v The chain has a mass  per unit length. Also, by applying the impulse-momentum equation to the left-hand portion of the system, verify that the force R supporting the pile of chain equals the weight of the pile. Neglect the small size and mass of the pulley and any friction in the pulley. Problem 4/85

A coal car with an empty mass of 25 Mg is moving freely with a speed of 1.2 m/s under a hopper which opens and releases coal into the moving car at the constant rate of 4 Mg per second. Determine the distance x moved by the car during the time that 32 Mg of coal are deposited in the car. Neglect any frictional resistance to rolling along the horizontal track. Problem 4/86

The cart carries a pile of open-link chain of mass  per unit length. The chain passes freely through the hole in the cart and is brought to rest, link by link, by the tension T in the portion of the chain resting on the ground and secured at its end A. The cart and the chain on it move under the action of the constant force P and have a velocity v0 and mass m0 when x 0. Determine expressions for the acceleration a and velocity v of the cart in terms of x if all friction is neglected. Also find T. Observe that the transition link 2 is decelerated from the velocity v to zero velocity by the tension T transmitted by the last horizontal link 1. Also note that link 2 exerts no force on the following link 3 during the transition. Explain why the term is absent if Eq. 4/20 is applied to this problem. Problem 4/87

The open-link chain of length L and mass  per unit length is released from rest in the position shown, where the bottom link is almost touching the platform and the horizontal section is supported on a smooth surface. Friction at the corner guide is negligible. Determine (a) the velocity v1 of end A as it reaches the corner and (b) its velocity v2 as it strikes the platform. (c) Also specify the total loss Q of energy. Problem 4/88

In the figure is shown a system used to arrest the motion of an airplane landing on a field of restricted length. The plane of mass m rolling freely with a velocity v0 engages a hook which pulls the ends of two heavy chains, each of length L and mass  per unit length, in the manner shown. A conservative calculation of the effectiveness of the device neglects the retardation of chain friction on the ground and any other resistance to the motion of the airplane. With these assumptions, compute the velocity v of the airplane at the instant when the last link of each chain is put in motion. Also determine the relation between the displacement x and the time t after contact with the chain. Assume each link of the chain acquires its velocity v suddenly upon contact with the moving links. Problem 4/89

The free end of the open-link chain of total length L and mass  per unit length is released from rest at x 0. Determine the force R on the fixed end and the tension T1 in the chain at the lower end of the nonmoving part in terms of x. Also find the total loss Q of energy when x L. Problem 4/90

Replace the chain of Prob. 4/90 by a flexible rope or bicycle chain of mass  per unit length and total length L. The free end is released from rest at x 0 and falls under the influence of gravity. Determine the acceleration a of the free end, the force R at the fixed end, and the tension T1 in the rope at the loop, all in terms of x. (Note that a is greater than g. What happens to the energy of the system when x L?)

One end of the pile of chain falls through a hole in its support and pulls the remaining links after it in a steady flow. If the links which are initially at rest acquire the velocity of the chain suddenly and without frictional resistance or interference from the support or from adjacent links, find the velocity v of the chain as a function of x if v 0 when x 0. Also find the acceleration a of the falling chain and the energy Q lost from the system as the last link leaves the platform. (Hint: Apply Eq. 4/20 and treat the product xv as the variable when solving the differential equation. Also note at the appropriate step that dx v dt.) The total length of the chain is L, and its mass per unit length is . Problem 4/92 x

Each of the identical steel balls weighs 4 lb and is fastened to the other two by connecting bars of negligible weight and unequal length. In the absence of friction at the supporting horizontal surface, determine the initial acceleration of the mass center of the assembly when it is subjected to the horizontal force F 20 lb applied to the supporting ball. The assembly is initially at rest in the vertical plane. Can you show that is initially horizontal? Problem 4/93

2-oz bullet is fired horizontally with a velocity v 1000 ft/sec into the slender bar of a 3-lb pendulum initially at rest. If the bullet embeds itself in the bar, compute the resulting angular velocity of the pendulum immediately after the impact. Treat the sphere as a particle and neglect the mass of the rod. Why is the linear momentum of the system not conserved? Problem 4/94

In an operational design test of the equipment of the fire truck, the water cannon is delivering fresh water through its 2-in.-diameter nozzle at the rate of 1400 gal/min at the 20 angle. Calculate the total friction force F exerted by the pavement on the tires of the truck, which remains in a fixed position with its brakes locked. (There are 231 in.3 in 1 gal.) Problem 4/95

A small rocket of initial mass m0 is fired vertically up near the surface of the earth ( g constant), and the mass rate of exhaust m and the relative exhaust velocity u are constant. Determine the velocity v as a function of the time t of flight if the air resistance is neglected and if the mass of the rocket case and machinery is negligible compared with the mass of the fuel carried.

The two balls are attached to the light rigid rod, which is suspended by a cord from the support above it. If the balls and rod, initially at rest, are struck with the force F 12 lb, calculate the corresponding acceleration of the mass center and the rate at which the angular velocity of the bar is changing. Problem 4/97

The rocket shown is designed to test the operation of a new guidance system. When it has reached a certain altitude beyond the effective influence of the earths atmosphere, its mass has decreased to 2.80 Mg, and its trajectory is 30 from the vertical. Rocket fuel is being consumed at the rate of 120 kg/s with an exhaust velocity of 640 m/s relative to the nozzle. Gravitational acceleration is 9.34 m/s2 at its altitude. Calculate the n- and t-components of the acceleration of the rocket. Problem 4/98

A two-stage rocket is fired vertically up and is above the atmosphere when the first stage burns out and the second stage separates and ignites. The second stage carries 1200 kg of fuel and has an empty mass of 200 kg. Upon ignition the second stage burns fuel at the rate of 5.2 kg/s and has a constant exhaust velocity of 3000 m/s relative to its nozzle. Determine the acceleration of the second stage 60 seconds after ignition and find the maximum acceleration and the time t after ignition at which it occurs. Neglect the variation of g and take it to be 8.70 m/s2 for the range of altitude averaging about 400 km

The three identical spheres, each of mass m, are supported in the vertical plane on the 30 incline. The spheres are welded to the two connecting rods of negligible mass. The upper rod, also of negligible mass, is pivoted freely to the upper sphere and to the bracket at A. If the stop at B is suddenly removed, determine the velocity v with which the upper sphere hits the incline. (Note that the corresponding velocity of the middle sphere is v/2.) Explain the loss of energy which has occurred after all motion has ceased. Problem 4/100

A jet of fresh water under pressure issues from the 3/4-in.-diameter fixed nozzle with a velocity v 120 ft/sec and is diverted into the two equal streams. Neglect any energy loss in the streams and compute the force F required to hold the vane in place. Problem 4/101

An ideal rope or bicycle-type chain of length L and mass  per unit length is resting on a smooth horizontal surface when end A is doubled back on itself by a force P applied to end A. End B of the rope is secured to a fixed support. Determine the force P required to give A a constant velocity v. (Hint: The action of the loop can be modeled by inserting a circular disk of negligible mass as shown in the separate sketch and then taking the disk radius as zero. It is easily shown that the tensions in the rope at C, D, and B are all equal to P under the ideal conditions imposed and with constant velocity.) Problem 4/102

In the static test of a jet engine and exhaust nozzle assembly, air is sucked into the engine at the rate of 30 kg/s and fuel is burned at the rate of 1.6 kg/s. The flow area, static pressure, and axial-flow velocity for the three sections shown are as follows: Sec. A Sec. B Sec. C Flow area, m2 0.15 0.16 0.06 Static pressure, kPa 14 140 14 Axial-flow velocity, m/s 120 315 600 Determine the tension T in the diagonal member of the supporting test stand and calculate the force F exerted on the nozzle flange at B by the bolts and gasket to hold the nozzle to the engine housing.

The upper end of the open-link chain of length L and mass  per unit length is released from rest with the lower end just touching the platform of the scale. Determine the expression for the force F read on the scale as a function of the distance x through which the upper end has fallen. (Comment: The chain acquires a free-fall velocity of because the links on the scale exert no force on those above, which are still falling freely. Work the problem in two ways: first, by evaluating the time rate of change of momentum for the entire chain and second, by considering the force F to be composed of the weight of the links at rest on the scale plus the force necessary to divert an equivalent stream of fluid.) Problem 4/104

The open-link chain of total length L and of mass  per unit length is released from rest at x 0 at the same instant that the platform starts from rest at y 0 and moves vertically up with a constant acceleration a. Determine the expression for the total force R exerted on the platform by the chain t seconds after the motion starts. Problem 4/105

The three identical 2-kg spheres are welded to the connecting rods of negligible mass and are hanging by a cord from point A. The spheres are initially at rest when a horizontal force F 16 N is applied to the upper sphere. Calculate the initial acceleration of the mass center of the spheres, the rate at which the angular velocity is increasing, and the initial acceleration a of the top sphere. Problem 4/106

The diverter section of pipe between A and B is designed to allow the parallel pipes to clear an obstruction. The flange of the diverter is secured at C by a heavy bolt. The pipe carries fresh water at the steady rate of 5000 gal/min under a static pressure of 130 lb/in.2 entering the diverter. The inside diameter of the pipe at A and at B is 4 in. The tensions in the pipe at A and B are balanced by the pressure in the pipe acting over the flow area. There is no shear or bending of the pipes at A or B. Calculate the moment M supported by the bolt at C. (Recall that 1 gallon contains 231 in.3 ) Problem 4/107

The chain of length L and mass  per unit length is released from rest on the smooth horizontal surface with a negligibly small overhang x to initiate motion. Determine (a) the acceleration a as a function of x, (b) the tension T in the chain at the smooth corner as a function of x, and (c) the velocity v of the last link A as it reaches the corner. Problem 4/10

A rope or hinged-link bicycle-type chain of length L and mass  per unit length is released from rest with x 0. Determine the expression for the total force R exerted on the fixed platform by the chain as a function of x. Note that the hinged-link chain is a conservative system during all but the last increment of motion. Compare the result with that of Prob. 4/105 if the upward motion of the platform in that problem is taken to be zero. Problem 4/109

The centrifugal pump handles 20 m3 of fresh water per minute with inlet and outlet velocities of 18 m/s. The impeller is turned clockwise through the shaft at O by a motor which delivers 40 kW at a pump speed of 900 rev/min. With the pump filled but not turning, the vertical reactions at C and D are each 250 N. Calculate the forces exerted by the foundation on the pump at C and D while the pump is running. The tensions in the connecting pipes at A and B are exactly balanced by the respective forces due to the static pressure in the water. (Suggestion: Isolate the entire pump and water within it between sections A and B and apply the momentum principle to the entire system.)

Replace the pile of chain in Prob. 4/92 by a coil of rope of mass  per unit length and total length L as shown and determine the velocity of the falling section in terms of x if it starts from rest at x 0. Show that the acceleration is constant at g/2. The rope is considered to be perfectly flexible in bending but inextensible and constitutes a conservative system (no energy loss). Rope elements acquire their velocity in a continuous manner from zero to v in a small transition section of the rope at the top of the coil. For comparison with the chain of Prob. 4/92, this transition section may be considered to have negligible length without violating the requirement that there be no energy loss in the present problem. Also determine the force R exerted by the platform on the coil in terms of x and explain why R becomes zero when x 2L/3. Neglect the dimensions of the coil compared with x. Problem 4/111

The chain of mass  per unit length passes over the small freely turning pulley and is released from rest with only a small imbalance h to initiate motion. Determine the acceleration a and velocity v of the chain and the force R supported by the hook at A, all in terms of h as it varies from essentially zero to H. Neglect the weight of the pulley and its supporting frame and the weight of the small amount of chain in contact with the pulley. (Hint: The force R does not equal two times the equal tensions T in the chain tangent to the pulley.)