A rocket of mass m is fired vertically from the surface of the earth, i.e., at r = r1

Determine the work of the force when it displaces 2 m.

Determine the kinetic energy of the 10-kg block.

Determine the potential energy of the block that has a weight of 100 N.

Determine the potential energy in the spring that has an unstretched length of 4 m.

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 s = 0, the block is at rest and the spring is uncompressed. The contact surface is smooth.

If the motor exerts a constant force of 300 N on the cable, determine the speed of the 20-kg crate when it travels s = 10 m up the plane, starting from rest. The coefficient of kinetic friction between the crate and the plane is \(\mu_{k} = 0.3\).

If the motor exerts a force of \(F = (600 + 2s^{2}) \ N\) 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.

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.

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 lb, and the collar is at rest when it is at A, determine its speed when it reaches B. Neglect friction.

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

If F = (10 s) 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 s = 0, the 20-kg block is moving at v = 1 m/s.

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 \(\varepsilon = 0.8\). Neglect the mass of the pulleys.

The coefficient of kinetic friction between the 20-kg block and the inclined plane is \(\mu_{k} = 0.2\). If the block is traveling up the inclined plane with a constant velocity v = 5 m/s, determine the power of force F.

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 \(\varepsilon = 0.8\).

At the instant shown, point P on the cable has a velocity \(v_{P} = 12 \ m/s\), which is increasing at a rate of \(a_{P} = 6 \ m/s^{2}\). Determine the power input to motor M at this instant if it operates with an efficiency \(\varepsilon = 0.8\). The mass of block A is 50 kg.

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.

The 2-kg package leaves the conveyor belt at A with a speed of \(v_{A} = 1 \ m/s\) 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 \(\rho_{B} = 2 \ m\).

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.

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 length of 0.5 ft.

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.

The 4-kg collar C has a velocity of \(v_{A} = 2 \ m/s\) 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 \(l_{0} = 0.2 \ m\).

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 s = 25 m. The coefficient of kinetic friction between the crate and the ground is \(\mu_{k} = 0.25\).

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 \(F = (90(10^{3})x^{1/2}) \ lb\), 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 \(\mu_{k} = 0.2\).

The 100-kg crate is subjected to the forces shown. If it is originally at rest, determine the distance it slides in order to attain a speed of v = 8 m/s. The coefficient of kinetic friction between the crate and the surface is \(\mu_{k} = 0.2\).

Determine the required height h of the roller coaster so that when it is essentially at rest at the crest of the hill A it will reach a speed of 100 km/h when it comes to the bottom B. Also, what should be the minimum radius of curvature \(\rho\) for the track at B so that the passengers do not experience a normal force greater than 4mg = (39.24m) N? Neglect the size of the car and passenger.

When the driver applies the brakes of a light truck traveling 40 km/h, it skids 3 m before stopping. How far will the truck skid if it is traveling 80 km/h when the brakes are applied?

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 \(x^{\prime} \ axis\) 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.

A force of F = 250 N is applied to the end at B. Determine the speed of the 10-kg block when it has moved 1.5 m, starting from rest.

The “air spring” A is used to protect the support B and prevent damage to the conveyor-belt tensioning weight C in the event of a belt failure D. The force developed by the air spring as a function of its deflection is shown by the graph. If the block has a mass of 20 kg and is suspended a height d = 0.4 m above the top of the spring, determine the maximum deformation of the spring in the event the conveyor belt fails. Neglect the mass of the pulley and belt.

The force F, acting in a constant direction on the 20-kg block, has a magnitude which varies with the position s of the block. Determine how far the block must slide before its velocity becomes 15 m/s. When s = 0 the block is moving to the right at v = 6 m/s. The coefficient of kinetic friction between the block and surface is \(\mu_{k} = 0.3\).

The force of F = 50 N is applied to the cord when s = 2 m. If the 6-kg collar is originally at rest, determine its velocity at s = 0. Neglect friction.

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 hits the ground.

Block A has a weight of 60 lb and block B has a weight of 10 lb. Determine the speed of block A after it moves 5 ft down the plane, starting from rest. Neglect friction and the mass of the cord and pulleys.

The two blocks A and B have weights \(W_{A} = 60 \ lb\) and \(W_{B} = 10 \ lb\). If the kinetic coefficient of friction between the incline and block A is \(\mu_{k} = 0.2\), determine the speed of A after it moves 3 ft down the plane starting from rest. Neglect the mass of the cord and pulleys.

A small box of mass m is given a speed of \(v = \sqrt{\frac{1}{4}gr}\) at the top of the smooth half cylinder. Determine the angle \(\theta\) at which the box leaves the cylinder.

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.

When the 12-lb block A is released from rest it lifts the two 15-lb weights B and C. Determine the maximum distance A will fall before its motion is momentarily stopped. Neglect the weight of the cord and the size of the pulleys.

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.

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.

Determine the velocity of the 60-lb block A if the two blocks are 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 \(\mu_{k} = 0.10\).

The 25-lb block has an initial speed of \(v_{0} = 10 \ ft/s\) 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 \(\mu_{k} = 0.4\), determine the total distance traveled by the block before it comes to rest.

The 8-kg block is moving with an initial speed of 5 m/s. If the coefficient of kinetic friction between the block and plane is \(\mu_{k} = 0.25\), determine the compression in the spring when the block momentarily stops.

At a given instant the 10-lb block A is moving downward with a speed of 6 ft/s. Determine its speed 2 s later. Block B has a weight of 4 lb, and the coefficient of kinetic friction between it and the horizontal plane is \(\mu_{k} = 0.2\). Neglect the mass of the cord and pulleys.

The 5-lb cylinder is falling from A with a speed \(v_{A} = 10 \ ft/s\) onto the platform. Determine the maximum displacement of the platform, caused by the collision. The spring has an unstretched length of 1.75 ft and is originally kept in compression by the 1-ft long cables attached to the platform. Neglect the mass of the platform and spring and any energy lost during the collision.

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.

The “flying car” is a ride at an amusement park which consists of a car having wheels that roll along a track mounted inside a rotating drum. By design the car cannot fall off the track, however motion of the car is developed by applying the car’s brake, thereby gripping the car to the track and allowing it to move with a constant speed of the track, \(v_{t} = 3 \ m/s\). If the rider applies the brake when going from B to A and then releases it at the top of the drum, A, so that the car coasts freely down along the track to B (\(\theta = \pi \ rad\)), determine the speed of the car at B and the normal reaction which the drum exerts on the car at B. Neglect friction during the motion from A to B. The rider and car have a total mass of 250 kg and the center of mass of the car and rider moves along a circular path having a radius of 8 m.

The 10-lb box falls off the conveyor belt at 5-ft/s. If the coefficient of kinetic friction along AB is \(\mu_{k} = 0.2\), determine the distance x when the box falls into the cart.

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.

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 from 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 \(\mu_{k} = 0.6\).

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 block has a mass of 0.8 kg and moves within the smooth vertical slot. If it starts from rest when the attached spring is in the unstretched position at A, determine the constant vertical force F which must be applied to the cord so that the block attains a speed \(v_{B} = 2.5 \ m/s\) when it reaches B; \(s_{B} = 0.15 \ m\). Neglect the size and mass of the pulley. Hint: The work of F can be determined by finding the difference \(\Delta l\) in cord lengths AC and BC and using \(U_{F} = F \ \Delta l\).

The 10-lb block is pressed against the spring so as to compress it 2 ft when it is at A. If the plane is smooth, determine the distance d, measured from the wall, to where the block strikes the ground. Neglect the size of the block.

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 lb/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.

When the 150-lb skier is at point A he has a speed of 5 ft/s. Determine his speed when he reaches point B on the smooth slope. For this distance the slope follows the cosine curve shown. Also, what is the normal force on his skis at B and his rate of increase in speed? Neglect friction and air resistance.

The spring has a stiffness k = 50 lb/ft and an unstretched length of 2 ft. As shown, it is confined by the plate and wall using cables so that its length is 1.5 ft. A 4-lb block is given a speed \(v_{A}\) when it is at A, and it slides down the incline having a coefficient of kinetic friction \(\mu_{k} = 0.2\). If it strikes the plate and pushes it forward 0.25 ft before stopping, determine its speed at A. Neglect the mass of the plate and spring.

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 \(h_{A}\) and \(h_{C}\) so that this does not occur. The roller coaster starts from rest at position A. Neglect friction.

If the 60-kg skier passes point A with a speed of 5 m/s, determine his speed when he reaches point B. Also find the normal force exerted on him by the slope at this point. Neglect friction.

If the 75-kg crate starts from rest at A, determine its speed when it reaches point B. The cable is subjected to a constant force of F = 300 N. Neglect friction and the size of the pulley.

If the 75-kg crate starts from rest at A, and its speed is 6 m/s when it passes point B, determine the constant force F exerted on the cable. Neglect friction and the size of the pulley.

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 \(A (\theta = 0^{\circ})\), determine the unstretched length of the cord so the block begins to leave the semicylinder at the instant \(\theta = 45^{\circ}\). Neglect the size of the block.

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 \(\theta\) of the largest incline the jeep can climb at a constant speed v = 30 ft/s.

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 \(\varepsilon = 0.65\).

An automobile having a mass of 2 Mg travels up a 7° slope at a constant speed of v = 100 km/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 lb. If the engine develops a constant thrust of 5200 lb, determine the power output of the plane when it is just ready to take off with a speed of 600 mi/h.

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.)

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.

The man having the weight of 150 lb is able to run up a 15-ft-high flight of stairs in 4 s. Determine the power generated. How long would a 100-W light bulb have to burn to expend the same amount of energy? Conclusion: Please turn off the lights when they are not in use!

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 \(\varepsilon = 0.8\). Also, find the average power supplied by the engine.

Determine the power output of the draw-works motor M necessary to lift the 600-lb drill pipe upward with a constant speed of 4 ft/s. The cable is tied to the top of the oil rig, wraps around the lower pulley, then around the top pulley, and then to the motor.

The 1000-lb elevator is hoisted by the pulley system and motor M. If the motor exerts a constant force of 500 lb on the cable, determine the power that must be supplied to the motor at the instant the load has been hoisted s = 15 ft starting from rest. The motor has an efficiency of \(\varepsilon = 0.65\).

The 50-lb crate is given a speed of 10 ft/s in t = 4 s starting from rest. If the acceleration is constant, determine the power that must be supplied to the motor when t = 2 s. The motor has an efficiency \(\varepsilon = 0.65\). Neglect the mass of the pulley and cable.

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 \(5 \ m/s^{2}\). If the drag resistance on the car due to the wind is \(F_{D} = (0.3v^{2}) \ N\), 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 \(\varepsilon = 0.68\).

The sports car has a mass of 2.3 Mg and accelerates at \(6 \ m/s^{2}\), starting from rest. If the drag resistance on the car due to the wind is \(F_{D} = (10v) \ N\), where v is the velocity in m/s, determine the power supplied to the engine when t = 5 s. The engine has a running efficiency of \(\varepsilon = 0.68\).

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 \(\varepsilon = 0.6\), determine the power that must be supplied to the motor when the elevator is hoisted upward at a constant speed of \(v_{E} = 4 \ m/s\).

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 \(\theta = 60^{\circ}\).

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.

The 50-lb block rests on the rough surface for which the coefficient of kinetic friction is \(\mu_{k} = 0.2\). A force \(F = (40 + s^{2}) \ lb\), 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.

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.

If the escalator in Prob. 14–46 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 power—the same amount that is needed to power a standard light bulb.

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.

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 athlete’s 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 t = 0.3 s.

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 athlete’s 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.

The block has a weight of 80 lb and rests on the floor for which \(\mu_{k} = 0.4\). If the motor draws in the cable at a constant rate of 6 ft/s, determine the output of the motor at the instant \(\theta = 30^{\circ}\). Neglect the mass of the cable and pulleys.

The block has a mass of 150 kg and rests on a surface for which the coefficients of static and kinetic friction are \(\mu_{s} = 0.5\) and \(\mu_{k} = 0.4\), respectively. If a force \(F = (60t^{2}) \ N\), where t is in seconds, is applied to the cable, determine the power developed by the force when t = 5 s. Hint: First determine the time needed for the force to cause motion.

The girl has a mass of 40 kg and center of mass at G. If she is swinging to a maximum height defined by \(\theta = 60^{\circ}\), determine the force developed along each of the four supporting posts such as AB at the instant \(\theta = 0^{\circ}\). The swing is centrally located between the posts.

The 30-lb block A is placed on top of two nested springs B and C and then pushed down to the position shown. If it is then released, determine the maximum height h to which it will rise.

The 5-kg collar has a velocity of 5 m/s to the right when it is at A. It then travels down along the smooth guide. Determine the speed of the collar when it reaches point B, which is located just before the end of the curved portion of the rod. The spring has an unstretched length of 100 mm and B is located just before the end of the curved portion of the rod.

The 5-kg collar has a velocity of 5 m/s to the right when it is at A. It then travels along the smooth guide. Determine its speed when its center reaches point B and the normal force it exerts on the rod at this point. The spring has an unstretched length of 100 mm and B is located just before the end of the curved portion of the rod.

The ball has a weight of 15 lb and is fixed to a rod having a negligible mass. If it is released from rest when \(\theta = 0^{\circ}\), determine the angle \(\theta\) at which the compressive force in the rod becomes zero.

The car C and its contents have a weight of 600 lb, whereas block B has a weight of 200 lb. If the car is released from rest, determine its speed when it travels 30 ft down the \(20^{\circ}\) incline. Suggestion: To measure the gravitational potential energy, establish separate datums at the initial elevations of B and C.

The roller coaster car has a mass of 700 kg, including its passenger. If it starts from the top of the hill A with a speed \(v_{A} = 3 \ m/s\), determine the minimum height h of the hill crest so that the car travels around the inside loops without leaving the track. Neglect friction, the mass of the wheels, and the size of the car. What is the normal reaction on the car when the car is at B and when it is at C? Take \(\rho_{B} = 7.5 \ m\) and \(\rho_{C} = 5 \ m\).

The roller coaster car has a mass of 700 kg, including its passenger. If it is released from rest at the top of the hill A, determine the minimum height h of the hill crest so that the car travels around both inside the loops without leaving the track. Neglect friction, the mass of the wheels, and the size of the car. What is the normal reaction on the car when the car is at B and when it is at C? Take \(\rho_{B} = 7.5 \ m\) and \(\rho_{C} = 5 \ m\).

The assembly consists of two blocks A and B which have a mass of 20 kg and 30 kg, respectively. Determine the speed of each block when B descends 1.5 m. The blocks are released from rest. Neglect the mass of the pulleys and cords.

The assembly consists of two blocks A and B, which have a mass of 20 kg and 30 kg, respectively. Determine the distance B must descend in order for A to achieve a speed of 3 m/s starting from rest.

The spring has a stiffness k = 50 N/m and an unstretched length of 0.3 m. If it is attached to the 2-kg smooth collar and the collar is released from rest at \(A \ (\theta = 0^{\circ})\), determine the speed of the collar when \(\theta = 60^{\circ}\). The motion occurs in the horizontal plane. Neglect the size of the collar.

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 h. Also, find the speed of the car when it reaches point C. Neglect friction.

The spring has a stiffness k = 200 N/m and an unstretched length of 0.5 m. If it is attached to the 3-kg smooth collar and the collar is released from rest at A, determine the speed of the collar when it reaches B. Neglect the size of the collar.

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

When s = 0, the spring on the firing mechanism is unstretched. If the arm is pulled back such that s = 100 mm and released, determine the speed of the 0.3-kg ball and the normal reaction of the circular track on the ball when \(\theta = 60^{\circ}\). Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

When s = 0, the spring on the firing mechanism is unstretched. If the arm is pulled back such that s = 100 mm and released, determine the maximum angle \(\theta\) the ball will travel without leaving the circular track. Assume all surfaces of contact to be smooth. Neglect the mass of the spring and the size of the ball.

If the mass of the earth is \(M_{e}\), show that the gravitational potential energy of a body of mass m located a distance r from the center of the earth is \(V_{g} = -GM_{e}m/r\). Recall that the gravitational force acting between the earth and the body is \(F = G(M_{e}m/r^{2})\), Eq. 13–1. For the calculation, locate the datum at \(r \rightarrow \infty\). Also, prove that F is a conservative force.

A rocket of mass m is fired vertically from the surface of the earth, i.e., at \(r = r_{1}\). Assuming that no mass is lost as it travels upward, determine the work it must do against gravity to reach a distance \(r_{2}\). The force of gravity is \(F = GM_{e}m/r^{2}\) (Eq. 13–1), where \(M_{e}\) is the mass of the earth and r the distance between the rocket and the center of the earth.

The 4-kg smooth collar has a speed of 3 m/s when it is at s = 0. Determine the maximum distance s it travels before it stops momentarily. The spring has an unstretched length of 1 m.

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

The skier starts from rest at A and travels down the ramp. If friction and air resistance can be neglected, determine his speed \(v_{B}\) when he reaches B. Also, compute the distance s to where he strikes the ground at C, if he makes the jump traveling horizontally at B. Neglect the skier’s size. He has a mass of 70 kg.

The block has a mass of 20 kg and is released from rest when s = 0.5 m. If the mass of the bumpers A and B can be neglected, determine the maximum deformation of each spring due to the collision.

The 2-lb collar has a speed of 5 ft/s at A. The attached spring has an unstretched length of 2 ft and a stiffness of k = 10 lb/ft. If the collar moves over the smooth rod, determine its speed when it reaches point B, the normal force of the rod on the collar, and the rate of decrease in its speed.

When the 6-kg box reaches point A it has a speed of \(v_{A} = 2 \ m/s\). Determine the angle \(\theta\) at which it leaves the smooth circular ramp and the distance s to where it falls into the cart. Neglect friction.

When the 5-kg box reaches point A it has a speed \(v_{A} = 10 \ m/s\). Determine the normal force the box exerts on the surface when it reaches point B. Neglect friction and the size of the box.

When the 5-kg box reaches point A it has a speed \(v_{A} = 10 \ m/s\). Determine how high the box reaches up the surface before it comes to a stop. Also, what is the resultant normal force on the surface at this point and the acceleration? Neglect friction and the size of the box.

The roller coaster car has a speed of 15 ft/s when it is at the crest of a vertical parabolic track. Determine the car’s velocity and the normal force it exerts on the track when it reaches point B. Neglect friction and the mass of the wheels. The total weight of the car and the passengers is 350 lb.

The 10-kg sphere C is released from rest when \(\theta = 0^{\circ}\) and the tension in the spring is 100 N. Determine the speed of the sphere at the instant \(\theta = 90^{\circ}\). Neglect the mass of rod AB and the size of the sphere.

A quarter-circular tube AB of mean radius r contains a smooth chain that has a mass per unit length of \(m_{0}\). If the chain is released from rest from the position shown, determine its speed when it emerges completely from the tube.

The cylinder has a mass of 20 kg and is released from rest when h = 0. Determine its speed when h = 3 m. Each spring has a stiffness k = 40 N/m and an unstretched length of 2 m.

If the 20-kg cylinder is released from rest at h = 0, determine the required stiffness k of each spring so that its motion is arrested or stops when h = 0.5 m. Each spring has an unstretched length of 1 m.

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 roller coaster is momentarily at rest at A. Determine the approximate normal force it exerts on the track at B. Also determine its approximate acceleration at this point. Use numerical data, and take scaled measurements from the photo with a known height at A.

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 are 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 seat on a passenger when this speed is reached. Use numerical values to explain your answer.

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 unstretched length and stiffness of each cord is known.

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.

If a 150-lb crate is released from rest at A, determine its speed after it slides 30 ft down the plane. The coefficient of kinetic friction between the crate and plane is \(\mu_{k} = 0.3\).

The small 2-lb collar starting from rest at A slides down along the smooth rod. During the motion, the collar is acted upon by a force \(\mathbf{F} = \{10 \mathbf{i} + 6y \mathbf{j} + 2z \mathbf{k}\} \ lb\), where x, y, z are in feet. Determine the collar’s speed when it strikes the wall at B.

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, 0, 10 ft). Determine the speed at which it slides off at B, which has coordinates of B(0, 8 ft, 0).

The block has a mass of 0.5 kg and moves within the smooth vertical slot. If the block starts from rest when the attached spring is in the unstretched position at A, determine the constant vertical force F which must be applied to the cord so that the block attains a speed \(v_{B} = 2.5 \ m/s\) when it reaches B; \(s_{B} = 0.15 \ m\). Neglect the mass of the cord and pulley.

The crate, having a weight of 50 lb, is hoisted by the pulley system and motor M. If the crate starts from rest and, by constant acceleration, attains a speed of 12 ft/s after rising 10 ft, determine the power that must be supplied to the motor at the instant s = 10 ft. The motor has an efficiency \(\varepsilon = 0.74\).

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 \(\varepsilon = 0.76\).

The collar of negligible size has a mass of 0.25 kg and is attached to a spring having an unstretched length of 100 mm. If the collar is released from rest at A and travels along the smooth guide, determine its speed just before it strikes B.

The blocks A and B weigh 10 and 30 lb, respectively. They are connected together by a light cord and ride in the frictionless grooves. Determine the speed of each block after block A moves 6 ft up along the plane. The blocks are released from rest.