FI4-14. The 2-kg package leaves the conveyor belt at A

F14-1. The spring is placed between the wall and the 10-kg block. If the block is subjected to a force of F = 500 N. determine its velocity when s = 0.5 m. When 5 = 0, the block is at rest and the spring is uncompressed. The contact surface is smooth.

FI4-2. If the motor exerts a constant force of 300 N on the cable, determine the speed of the 20-kg crate when it travels 5=10 m up the plane, starting from rest. The coefficient of kinetic friction between the crate and the plane is /i* = 0.3.

F14-3. If the motor exerts a force of F = (600 + 252) N on the cable, determine the speed of the 100-kg crate when it rises to s = 15 m.The crate is initially at rest on the ground.

F14-4. The 1.8-Mg dragster is traveling at 125 m/s when the engine is shut off and the parachute is released. If the drag force of the parachute can be approximated by the graph, determine the speed of the dragster when it has traveled 400 m.

F14-5. When s = 0.6 m, the spring is unstretched and the 10-kg block has a speed of 5 m/s down the smooth plane. Determine the distance s when the block stops.

The 5-lb collar is pulled by a cord that passes around a small peg at C. If the cord is subjected to a constant force of \(F=10 \mathrm{lb}\), and the collar is at rest when it is at A, determine its speed when it reaches B. Neglect friction.

F14-7. If the contact surface between the 20-kg block and the ground is smooth, determine the power of force F when / = 4 s. Initially, the block is at rest.

FI4-8. If F = (10^)N. where s is in meters, and the contact surface between the block and the ground is smooth, determine the power of force F when s = 5 m. When 5 = 0. the 20-kg block is moving at V = 1 m/s.

F14-9. If the motor winds in the cable with a constant speed of v = 3 ft/s, determine the power supplied to the motor. The load weighs 100 lb and the efficiency of the motor is e = 0.8. Neglect the mass of the pulleys.

FI4-10. The coefficient of kinetic friction between the 20-kg block and the inclined plane is jxk = 0.2. If the block is traveling up the inclined plane with a constant velocity v 5 m/s. determine the power of force F.

F14-11. If the 50-kg load A is hoisted by motor M so that the load has a constant velocity of 1.5 m/s. determine the power input to the motor, which operates at an efficiency e = 0.8.

F14-12. At the instant shown, point P on the cable has a velocity vP = 12 m/s. which is increasing at a rate of aP = 6 m/s2. Determine the power input to motor M at this instant if it operates with an efficiency e = 0.8. The mass of block A is 50 kg.

F14-13. The 2-kg pendulum bob is released from rest when it is at A. Determine the speed of the bob and the tension in the cord when the bob passes through its lowest position. B.

FI4-14. The 2-kg package leaves the conveyor belt at A with a speed of vA = 1 m/s and slides down the smooth ramp. Determine the required speed of the conveyor belt at B so that the package can be delivered without slipping on the belt. Also, find the normal reaction the curved portion of the ramp exerts on the package at B if p/} = 2 m.

F14-15. The 2-kg collar is given a downward velocity of 4 m/s when it is at A. If the spring has an unstretched length of 1 m and a stiffness of k = 30 N/m. determine the velocity of the collar at s = 1 m.

FI4-16. The 5-lb collar is released from rest at A and travels along the frictionless guide. Determine the speed of the collar when it strikes the stop B. The spring has an unstretched

F14-17. The 75-lb block is released from rest 5 ft above the plate. Determine the compression of each spring when the block momentarily comes to rest after striking the plate. Neglect the mass of the plate. The springs are initially unstretched.

F14-18. The 4-kg collar C has a velocity of vA = 2 m/s when it is at A. If the guide rod is smooth, determine the speed of the collar when it is at B. The spring has an unstretched length of /(> = 0.2 m.

14-1. The 20-kg crate is subjected to a force having a constant direction and a magnitude F= 100 N. When s = 15 m, the crate is moving to the right with a speed of 8 m/s. Determine its speed when 5 = 25 m. The coefficient of kinetic friction between the crate and the ground is lik = 0.25.

For protection, the barrel barrier is placed in front of the bridge pier. If the relation between the force and deflection of the barrier is \(F=\left(90(103)x^{1/2}\right)\ \mathrm{lb}\). where x is in ft, determine the car's maximum penetration in the barrier. The car has a weight of 4000 lb and it is traveling with a speed of 75 ft/s just before it hits the barrier.

The crate, which has a mass of 100 kg. is subjected to the action of the two forces. If it is originally at rest, determine the distance it slides in order to attain a speed of 6 m/s. The coefficient of kinetic friction between the crate and the surface is \(\mu_{k}=0.2\).

*14-4. The 2-kg block is subjected to a force having a constant direction and a magnitude F = (300/(1 + $)) N, where s is in meters. When 5 = 4 m. the block is moving to the left with a speed of 8 m/s. Determine its speed when s = 12 m. The coefficient of kinetic friction between the block and the ground is fxk = 0.25.

14-5. When a 7-kg projectile is fired from a cannon barrel that has a length of 2 m, the explosive force exerted on the projectile, while it is in the barrel, varies in the manner shown. Determine the approximate muzzle velocity of the projectile at the instant it leaves the barrel. Neglect the effects of friction inside the barrel and assume the barrel is horizontal.

14-6. The spring in the toy gun has an unstrelched length of 100 mm. It is compressed and locked in the position shown. When the trigger is pulled, the spring unstretches 12.5 mm, and the 20-g ball moves along the barrel. Determine the speed of the ball when it leaves the gun. Neglect friction.

14-7. As indicated by the derivation, the principle of work and energy is valid for observers in any inertial reference frame. Show that this is so, by considering the 10-kg block which rests on the smooth surface and is subjected to a horizontal force of 6 N. If observer A is in a fixed frame x. determine the final speed of the block if it has an initial speed of 5 m/s and travels 10 m, both directed to the right and measured from the fixed frame. Compare the result with that obtained by an observer B. attached to the x' axis and moving at a constant velocity of 2 m/s relative to A. Hint: The distance the block travels will first have to be computed for observer B before applying the principle of work and energy.

*14-8. If the 50-kg crate is subjected to a force of P = 200 N. determine its speed when it has traveled 15 m starting from rest. The coefficient of kinetic friction between the crate and the ground is = 0.3.

14-9. If the 50-kg crate starts from rest and attains a speed of 6 m/s when it has traveled a distance of 15 m. determine the force P acting on the crate. The coefficient of kinetic friction between the crate and the ground is /j* * = 0.3.

1410. The 2-Mg car has a velocity of = 100km/h when the driver sees an obstacle in front of the car. If it takes 0.75 s for him to react and lock the brakes, causing the car to skid, determine the distance the car travels before it stops. The coefficient of kinetic friction between the tires and the road is = 0.25.

14-11. The 2-Mg car has a velocity of = lOOkm/h when the driver secs an obstacle in front of the car. It takes 0.75 s for him to react and lock the brakes, causing the car to skid. If the car slops when it has traveled a distance of 175 m. determine the coefficient of kinetic friction between the tires and the road.

*14-12. Design considerations for the bumper B on the 5-Mg train car require use of a nonlinear spring having the load-deflection characteristics shown in the graph. Select the proper value of k so that the maximum deflection of the spring is limited to 0.2 m when the car. traveling at 4 m/s, strikes the rigid stop. Neglect the mass of the car wheels.

The 2-lb brick slides down a smooth roof, such that when it is at A it has a velocity of 5 ft/s. Determine the speed of the brick just before it leaves the surface at B. the distance d from the wall to where it strikes the ground, and the speed at which it his the ground.

14-14. If the cord is subjected to a constant force of F = 300 N and the 15-kg smooth collar starts from rest at A, determine the velocity of the collar when it reaches point B. Neglect the size of the pulley.

14-15. The crash cushion for a highway barrier consists of a nest of barrels filled with an impact-absorbing material. The barrier stopping force is measured versus the vehicle penetration into the barrier. Determine the distance a car having a weight of 4000 lb will penetrate the barrier if it is originally traveling at 55 ft/s when it strikes the first barrel.

*14-16. Determine the velocity of the 60-lb block A if the two blocks arc released from rest and the 40-lb block B moves 2 ft up the incline. The coefficient of kinetic friction between both blocks and the inclined planes is fxk = 0.10.

14-17. If the cord is subjected to a constant force of F= 30 lb and the smooth 10-lb collar starts from rest at A. determine its speed when it passes point B. Neglect the size of pulley C.

14-18. The two blocks A and B have weights WA = 60 lb and WH = 10 lb. If the kinetic coefficient of friction between the incline and block A is /x* = 0.2, determine the speed of A after it moves 3 ft down the plane starting from rest. Neglect the mass of the cord and pulleys.

14-19. If the 10-lb block passes point A on the smooth track with a speed of vA = 5 ft/s. determine the normal reaction on the block when it reaches point B.

The steel ingot has a mass of \(1800 \mathrm{~kg}\). It travels along the conveyor at a speed \(v=0.5 \mathrm{~m} / \mathrm{s}\) when it collides with the "nested" spring assembly. Determine the maximum deflection in each spring needed to stop the motion of the ingot. Take \(k_A=5 \mathrm{kN} / \mathrm{m}, k_B=3 \mathrm{kN} / \mathrm{m}\).

14-21. The steel ingot has a mass of 1800 kg. It travels along the conveyor at a speed v = 0.5 m/s when it collides with the nested spring assembly. If the stiffness of the outer spring is ^ = 5kN/m, determine the required stiffness kB of the inner spring so that the motion of the ingot is stopped at the moment the front. C, of the ingot is 0.3 m from the wall.

14-22. The 25-lb block has an initial speed of Vq = 10 ft/s when it is midway between springs A and B. After striking spring B. it rebounds and slides across the horizontal plane toward spring A. etc. If the coefficient of kinetic friction between the plane and the block is nk = 0.4. determine the total distance traveled by the block before it comes to rest.

14-23. The train car has a mass of 10 Mg and is traveling at 5 m/s when it reaches A. If the rolling resistance is 1/100 of the weight of the car, determine the compression of each spring when the car is momentarily brought to rest.

*14-24. The 0.5-kg ball is fired up the smooth vertical circular track using the spring plunger. The plunger keeps the spring compressed 0.08 m when s = 0. Determine how far 5 it must be pulled back and released so that the ball will begin to leave the track when 0 = 135.

14-25. The skier starts from rest at A and travels down the ramp. If friction and air resistance can be neglected, determine his speed vB when he reaches B. Also, find the distance s to where he strikes the ground at C. if he makes the jump traveling horizontally at B. Neglect the skiers size. He has a mass of 70 kg.

14-26. The catapulting mechanism is used to propel the 10-kg slider A to the right along the smooth track. The propelling action is obtained by drawing the pulley attached to rod BC rapidly to the left by means of a piston P. If the piston applies a constant force F = 20 kN to rod BC such that it moves it 0.2 m, determine the speed attained by the slider if it was originally at rest. Neglect the mass of the pulleys, cable, piston, and rod BC.

14-27. Block A has a weight of 60 lb and block B has a weight of 10 lb. Determine the distance A must descend from rest before it obtains a speed of 8 ft/s. Also, what is the tension in the cord supporting block A? Neglect the mass of the cord and pulleys.

14-28. The cyclist travels to point A, pedaling until he reaches a speed vA =4 m/s. He then coasts freely up the curved surface. Determine how high he reaches up the surface before he comes to a slop. Also, what are the resultant normal force on the surface at this point and his acceleration? The total mass of the bike and man is 75 kg. Neglect friction, the mass of the wheels, and the size of the bicycle.

14-29. The collar has a mass of 20 kg and slides along the smooth rod. Two springs are attached to it and the ends of the rod as shown. If each spring has an uncompressed length of 1 m and the collar has a speed of 2 m/s when s = 0. determine the maximum compression of each spring due to the back-and-forth (oscillating) motion of the collar.

14-30. The 30-lb box A is released from rest and slides down along the smooth ramp and onto the surface of a cart. If the cart is prevented front moving. determine the distance s from the end of the cart to where the box stops. The coefficient of kinetic friction between the cart and the box is/i* = 0.6.

14-31. Marbles having a mass of 5 g are dropped from rest at A through the smooth glass tube and accumulate in the can at C. Determine the placement R of the can from the end of the tube and the speed at which the marbles fall into the can. Neglect the size of the can.

The cyclist travels to point A, pedaling until he reaches a speed \(v_{A}=8 \mathrm{\ m} / \mathrm{s}\). He then coasts freely up the curved surface. Determine the normal force he exerts on the surface when he reaches point ZJ.The total mass of the bike and man is 75 kg. Neglect friction, the mass of the wheels, and the size of the bicycle.

The man at window A wishes to throw the 30-kg sack on the ground. To do this he allows it to swing from rest at B to point C, when he releases the cord at \(\theta=30^{\circ}\). Determine the speed at which it strikes the ground and the distance R.

14-34. The spring bumper is used to arrest the motion of the 4-lb block, which is sliding toward it at v = 9 ft/s. As shown, the spring is confined by the plate P and wall using cables so that its length is 1.5 ft. If the stiffness of the spring is k = 50 Ib/ft, determine the required unstretched length of the spring so that the plate is not displaced more than 0.2 ft after the block collides into it. Neglect friction, the mass of the plate and spring, and the energy loss between the plate and block during the collision.

14-35. The collar has a mass of 20 kg and is supported on the smooth rod.The attached springs are undeformed when d = 0.5 m. Determine the speed of the collar after the applied force F = 100 N causes it to be displaced so that d = 0.3 in. When d = 0.5 in the collar is at rest.

*14-36. If the force exerted by the motor M on the cable is 250 N. determine the speed of the 100-kg crate when it is hoisted to s = 3 in. The crate is at rest when 5 = 0.

14-37. If the track is to be designed so that the passengers of the roller coaster do not experience a normal force equal to zero or more than 4 times their weight, determine the limiting heights hA and hc so that this does not occur. The roller coaster starts from rest at position A. Neglect friction.

14-38. The 150-lb skater passes point A with a speed of 6 ft/s. Determine his speed when he reaches point B and the normal force exerted on him by the track at this point. Neglect friction.

14-39. The 8-kg cylinder A and 3-kg cylinder B are released from rest. Determine the speed of A after it has moved 2 m starting from rest. Neglect the mass of the cord and pulleys.

*14-40. Cylinder A has a mass of 3 kg and cylinder B has a mass of 8 kg. Determine the speed of A after it has moved 2 m starting from rest. Neglect the mass of the cord and pulleys.

14-41. A 2-lb block rests on the smooth semicylindrical surface. An elastic cord having a stiffness k = 2 lb/ft is attached to the block at B and to the base of the semicylinder at point C. If the block is released from rest at A (ft = 0), determine the unstretched length of the cord so the block begins to leave the semicylinder at the instant 0 = 45. Neglect the size of the block.

14-42. The jeep has a weight of 2500 lb and an engine which transmits a power of 100 hp to all the wheels. Assuming the wheels do not slip on the ground, determine the angle 0 of the largest incline the jeep can climb at a constant speed v = 30ft/s.

14-43. Determine the power input for a motor necessary to lift 300 lb at a constant rate of 5 ft/s. The efficiency of the motor is e = 0.65.

An automobile having a mass of 2 Mg travels up a \(7^{\circ}\) slope at a constant speed of \(v=100 \mathrm{\ km} / \mathrm{h}\). If mechanical friction and wind resistance are neglected, determine the power developed by the engine if the automobile has an efficiency \(\varepsilon=0.65\).

The Milkin Aircraft Co. manufactures a turbojet engine that is placed in a plane having a weight of \(13000 \mathrm{lb}\). If the engine develops a constant thrust of \(5200 \mathrm{lb}\), determine the power output of the plane when it is just ready to take off with a speed of \(600 \mathrm{mi} / \mathrm{h}\).

14-46. To dramatize the loss of energy in an automobile, consider a car having a weight of 5000 lb that is traveling at 35 mi/h. If the car is brought to a stop, determine how long a 100-W light bulb must burn to expend the same amount of energy. (1 mi = 5280 ft.)

14-47. The escalator steps move with a constant speed of 0.6 m/s. If the steps are 125 mm high and 250 mm in length, determine the power of a motor needed to lift an average mass of 150 kg per step. There are 32 steps.

*14-48. If the escalator in Prob. 1446 is not moving, determine the constant speed at which a man having a mass of 80 kg must walk up the steps to generate 100 W of powerthe same amount that is needed to power a standard light bulb.

14-49. The 2-Mg car increases its speed uniformly from rest to 25 m/s in 30 s up the inclined road. Determine the maximum power that must be supplied by the engine, which operates with an efficiency of e = 0.8. Also, find the average power supplied by the engine.

14-50. Tlie crate has a mass of 150 kg and rests on a surface for which the coefficients of static and kinetic friction are /jls = 0.3 and nk = 0.2. respectively. If the motor M supplies a cable force of F = (8r + 20) N. where t is in seconds, determine the power output developed by the motor when / = 5 s.

14-51. The 50-kg crate is hoisted up the 30 incline by the pulley system and motor M. If the crate starts from rest and by constant acceleration attains a speed of 4 m/s after traveling 8 m along the plane, determine the power that must be supplied to the motor at this instant. Neglect friction along the plane. The motor has an efficiency of e = 0.74.

*14-52. The 50-lb load is hoisted by the pulley system and motor M. If the motor exerts a constant force of 30 lb on the cable, determine the power that must be supplied to the motor if the load has been hoisted s = 10 ft starting from rest.The motor has an efficiency of e = 0.76.

14-53. The 10-lb collar starts from rest at A and is lifted by applying a constant vertical force of F = 25 lb to the cord. If the rod is smooth, determine the power developed by the force at the instant 0 = 60.

14-54. The 10-lb collar starts from rest at A and is lifted with a constant speed of 2 ft/s along the smooth rod. Determine the power developed by the force F at the instant shown.

14-55. The elevator E and its freight have a total mass of 400 kg. Hoisting is provided by the motor M and the 60-kg block C. If the motor has an efficiency of e = 0.6. determine the power that must be supplied to the motor when the elevator is hoisted upward at a constant speed of vE = 4 m/s.

*14-56. The sports car has a mass of 2.3 Mg, and while it is traveling at 28 m/s the driver causes it to accelerate at 5 m/s2. If the drag resistance on the car due to the wind is Fn = (0.3V2) N. where v is the velocity in m/s, determine the power supplied to the engine at this instant. The engine has a running efficiency of e = 0.68.

14-57. The sports car has a mass of 2.3 Mg and accelerates at 6 m/s2, starting from rest. If the drag resistance on the car due to the wind is Fn = (10/0 N, where v is the velocity in m/s. determine the power supplied to the engine when / = 5 s. The engine has a running efficiency of e = 0.68.

14-58. The block has a mass of 150 kg and rests on a surface for which the coefficients of static and kinetic friction are /x, = 0.5 and /x* = 0.4. respectively. If a force F = (6012) N. where t is in seconds, is applied to the cable, determine the power developed by the force when / = 5 s. Hint: First determine the time needed for the force to cause motion.

14-59. The rocket sled has a mass of 4 Mg and travels from rest along the horizontal track for which the coefficient of kinetic friction is fk = 0.20. If the engine provides a constant thrust T 150 kN, determine the power output of the engine as a function of time. Neglect the loss of fuel mass and air resistance.

*14-60. A loaded truck weighs 16(10') lb and accelerates uniformly on a level road from 15 ft/s to 30 ft/s during 4 s. If the frictional resistance to motion is 325 lb. determine the maximum power that must be delivered to the wheels.

14-61. If the jet on the dragster supplies a constant thrust of T = 20 kN. determine the power generated by the jet as a function of time. Neglect drag and rolling resistance, and the loss of fuel. The dragster has a mass of 1 Mg and starts from rest.

14-62. An athlete pushes against an exercise machine with a force that varies with time as shown in the first graph. Also, the velocity of the athletes arm acting in the same direction as the force varies with time as shown in the second graph. Determine the power applied as a function of time and the work done in / = 0.3 s.

14-63. An athlete pushes against an exercise machine with a force that varies with time as shown in the first graph. Also, the velocity of the athletes arm acting in the same direction as the force varies with time as shown in the second graph. Determine the maximum power developed during the 0.3-second time period.

*14-64. The 500-kg elevator starts from rest and travels upward with a constant acceleration ac = 2 m/s2. Determine the power output of the motor M when / = 3 s. Neglect the mass of the pulleys and cable.

14-65. The 50-lb block rests on the rough surface for which the coefficient of kinetic friction is fxk = 0.2. A force F = (40 + .v2) lb, where s is in ft, acts on the block in the direction shown. If the spring is originally unstretched (s = 0) and the block is at rest, determine the power developed by the force the instant the block has moved s = 1.5 ft.

14-66. The girl has a mass of 40 kg and center of mass at G. Tf she is swinging to a maximum height defined by 0 = 60, determine the force developed along each of the four supporting posts such as AR at the instant 0 = 0. The swing is centrally located between the posts.

14-67. Two equal-length springs are nested together in order to form a shock absorber. If it is designed to arrest the motion of a 2-kg mass that is dropped .v = 0.5 m above the top of the springs from an at-rest position, and the maximum compression of the springs is to be 0.2 m. determine the required stiffness of the inner spring, kH, if the outer spring has a stiffness kA = 400 N/m.

*14-68. The collar has a weight of 8 lb. If it is pushed down so as to compress the spring 2 ft and then released from rest (/i = 0), determine its speed when it is displaced h = 4.5 ft. The spring is not attached to the collar. Neglect friction.

14-69. The collar has a weight of 8 lb. If it is released from rest at a height of h = 2 ft from the top of the uncompressed spring, determine the speed of the collar after it falls and compresses the spring 0.3 ft.

The 2-kg ball of negligible size is fired from point A with an initial velocity of \(10 \mathrm{~m} / \mathrm{s}\) up the smooth inclined plane. Determine the distance from point C to where it hits the horizontal surface at D. Also, what is its velocity when it strikes the surface?

14-71. The ride at an amusement park consists of a gondola which is lifted to a height of 120 ft at A. If it is released from rest and falls along the parabolic track, determine the speed at the instant y = 20 ft. Also determine the normal reaction of the tracks on the gondola at this instant. The gondola and passenger have a total weight of 500 lb. Neglect the effects of friction and the mass of the wheels.

*14-72. The 2-kg collar is attached to a spring that has an unstretched length of 3 m. If the collar is drawn to point B and released from rest, determine its speed when it arrives at point A.

14-73. The 2-kg collar is attached to a spring that has an unstretched length of 2 m. If the collar is drawn to point B and released from rest, determine its speed when it arrives at point A.

14-74. The 0.5-lb ball is shot from the spring device shown. The spring has a stiffness k = lOlb/in. and the four cords C and plate P keep the spring compressed 2 in. when no load is on the plate.The plate is pushed back 3 in. from its initial position. If it is then released from rest, determine the speed of the ball when it travels 30 in. up the smooth plane.

14-75. The 0.5-lb ball is shot from the spring device shown. Determine the smallest stiffness k which is required to shoot the ball a maximum distance of 30 in. up the smooth plane after the spring is pushed back 3 in. and the ball is released from rest. The four cords C and plate P keep the spring compressed 2 in. when no load is on the plate.

* 14-76. The roller coaster car having a mass m is released from rest at point A. If the track is to be designed so that the car does not leave it at B. determine the required height /?. Also, find the speed of the car when it reaches point C. Neglect friction.

14-77. A 750-mm-long spring is compressed and confined by the plate P. which can slide freely along the vertical 600-mm-long rods. The 40-kg block is given a speed of v = 5 m/s when it is h = 2 m above the plate. Determine how far the plate moves downwards when the block momentarily stops after striking it. Neglect the mass of the plate.

The 2-lb block is given an initial velocity of 20 ft/s when it is at A. If the spring has an unstretched length of 2 ft and a stiffness of k = 100 Ib/ft. determine the velocity of the block when s = 1 ft.

14-79. The block has a weight of 1.5 lb and slides along the smooth chute AB. It is released from rest at A. which has coordinates of A(5 ft.O,10 ft). Determine the speed at which it slides off at B. which has coordinates of B(0.8 ft. 0).

Each of the two elastic rubber bands of the slingshot has an unstretched length of \(200 \mathrm{~mm}\). If they are pulled back to the position shown and released from rest. determine the speed of the \(25-\mathrm{g}\) pellet just after the rubber bands become unstretched. Neglect the mass of the rubber bands. Each rubber band has a stiffness of \(k=50 \mathrm{~N} / \mathrm{m}\).

14-81. Each of the two elastic rubber bands of the slingshot has an unstretched length of 200 mm. If they are pulled back to the position shown and released from rest, determine the maximum height the 25-g pellet will reach if it is fired vertically upward. Neglect the mass of the rubber bands and the change in elevation of the pellet while it is constrained by the rubber bands. Each rubber band has a stiffness k - 50 N/m.

14-82. If the mass of the earth is show that the gravitational potential energy of a body of mass m located a distance r from the center of the earth is Vg = GMem/r. Recall that the gravitational force acting between the earth and the body is F = G{Mcm/r), Eq. 13-1. For the calculation, locate the datum at r* sc. Also.prove that Fis a conservative force.

14-83. A rocket of mass m is fired vertically from the surface of the earth, i.e.. at r = r,. Assuming that no mass is lost as it travels upward, determine the work it must do against gravity to reach a distance r2. The force of gravity is F = GMem/r (Eq. 13-1). where M, is the mass of the earth and r the distance between the rocket and the center of the earth.

The firing mechanism of a pinball machine consists of a plunger P having a mass of 0.25 kg and a spring of stiffness k = 300 N/m. When s = 0, the spring is compressed 50 mm. If the arm is pulled back such that s = 100 mm and released, determine the speed of the 0.3-kg pinball B just before the plunger strikes the stop. i.e.. s = 0. Assume all surfaces of contact to be smooth. The ball moves in the horizontal plane. Neglect friction, the mass of the spring, and the rolling motion of the ball.

A 60-kg satellite travels in free flight along an elliptical orbit such that at A. where\(r_{A}=20\ \mathrm{Mm}\), it has a speed \(v_A=40\ \mathrm{Mm}/\mathrm{h}\). What is the speed of the satellite when it reaches point B. where \(r_B=80\mathrm{\ Mm}?\) Hint: See Prob. 14-82, where \(M_e=5.976\left(10^{24}\right)\mathrm{\ kg}\) and \(G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)\).

14-86. Just for fun, two 150-lb engineering students A and B intend to jump off the bridge from rest using an elastic cord (bungee cord) having a stiffness k = 80 lb/ft. They wish to just reach the surface of the river, when A. attached to the cord, lets go of B at the instant they touch the water. Determine the proper unstretched length of the cord to do the stunt, and calculate the maximum acceleration of student A and the maximum height he reaches above the water after the rebound. From your results, comment on the feasibility of doing this stunt.

14-87. The 20-lb collar slides along the smooth rod. If the collar is released from rest at A. determine its speed when it passes point B.The spring has an unstretched length of 3 ft.

*14-88. Two equal-length springs having a stiffness kA = 300 N/m and kR = 200 N/m are nested together in order to form a shock absorber. If a 2-kg block is dropped from an at-rest position 0.6 m above the top of the springs, determine their deformation when the block momentarily stops.

14-89. When the 6-kg box reaches point A it has a speed of vA = 2 m/s. Determine the angle 0 at which it leaves the smooth circular ramp and the distance 5 to where it falls into the cart. Neglect friction.

14-90. The Raptor is an outside loop roller coaster in which riders are belled into seats resembling ski-lift chairs. Determine the minimum speed v0 at which the cars should coast down from the top of the hill, so that passengers can just make the loop without leaving contact with their seats. Neglect friction, the size of the car and passenger, and assume each passenger and car has a mass m.

14-91. The Raptor is an outside loop roller coaster in which riders are belted into seats resembling ski-lift chairs. If the cars travel at v0 = 4 m/s when they are at the top of the hill, determine their speed when they are at the top of the loop and the reaction of the 70-kg passenger on his seat at this instant.The car has a mass of 50 kg.Take h = I2m.p = 5 m. Neglect friction and the size of the car and passenger.

*14-92. The 75-kg man bungee jumps off the bridge at A with an initial downward speed of 1.5 m/s. Determine the required unstretched length of the elastic cord to which he is attached in order that he stops momentarily just above the surface of the water. The stiffness of the elastic cord is k = 80 N/m. Neglect the size of the man.

14-93. The 10-kg sphere C is released from rest when 6 = 0 and the tension in the spring is 100 N. Determine the speed of the sphere at the instant 0 = 90. Neglect the mass of rod AB and the size of the sphere.

14-94. The double-spring bumper is used to stop the 1500-lb steel billet in the rolling mill. Determine the maximum displacement of the plate A if the billet strikes the plate with a speed of 8 ft/s. Neglect the mass of the springs, rollers and the plates A and 5. Take k} = 3000 lb/ft, k2 = 45 000 lb/ft.

14-95. The 2-lb box has a velocity of 5 ft/s when it begins to slide down the smooth inclined surface at A. Determine the point C (a\ y) where it strikes the lower incline.

*14-96. The 2-lb box has a velocity of 5 ft/s when it begins to slide down the smooth inclined surface at A. Determine its speed just before hitting the surface at C and the time to travel from A to C. The coordinates of point C are x = 17.66 ft, and y = 8.832 ft.

A pan of negligible mass is attached to two identical springs of stiffness k = 250 N/m. If a 10-kg box is dropped from a height of 0.5 m above the pan, determine the maximum vertical displacement d. Initially each spring has a tension of 50 N.

The 20-kg crate is subjected to a force having constant direction and a magnitude F = 100 N. When s = 15 m, the crate is moving to the right with a speed of 8 m/s. Determine its speed when s = 25 m. The coefficient of kinetic friction between the crate and the ground is \(\mu_{k}=0.25\).

P14-2. As the large ring rotates, the operator can apply a breaking mechanism that binds the cars to the ring, which then allows the cars to rotate with the ring. Assuming the passengers arc not belted into the cars, determine the smallest speed of the ring (cars) so that no passenger will fall out. When should the operator release the brake so that the cars can achieve their greatest speed as they slide freely on the ring? Estimate the greatest normal force of the scat on a passenger when this speed is reached. Use numerical values to explain your answer.

PI4-3. The woman pulls the water balloon launcher back, stretching each of the four elastic cords. Estimate the maximum height and the maximum range of a ball placed within the container if it is released from the position shown. Use numerical values and any necessary measurements from the photo. Assume the unstrctched length and stiffness of each cord is known.

P14-4. The girl is momentarily at rest in the position shown. If the unstretched length and stiffness of each of the two elastic cords is known, determine approximately how far the girl descends before she again becomes momentarily at rest. Use numerical values and take any necessary measurements from the photo.