Engineering Dynamics · Work & Energy · Open Problem Companion

Chapter Fourteen
Kinetics of a Particle: Work and Energy

P Preliminary Problems 3 problems
Concept Check
  1. P14-2Determine the kinetic energy of the 10-kg block. figure — add your own
  2. P14-3Determine the potential energy of the block that has a weight of \(100\,\mathrm{N}\). figure — add your own
  3. P14-4Determine the potential energy in the spring that has an unstretched length of \(4\,\mathrm{m}\). figure — add your own
F Fundamental Problems 18 problems
Fundamental
  1. F14-1The spring is placed between the wall and the \(10\,\mathrm{kg}\) block. If the block is subjected to a force of \(F=500\,\mathrm{N}\) determine its velocity when \(s=0.5\,\mathrm{m}\). When \(s=0\) the block is at rest and the spring is uncompressed. The contact surface is smooth. figure — add your own
  2. F14-2If the motor exerts a constant force of \(300\,\mathrm{N}\) on the cable, determine the speed of the \(20\,\mathrm{kg}\) crate when it travels \(s=10\,\mathrm{m}\) up the plane, starting from rest. The coefficient of kinetic friction between the crate and the plane is \(\mu_k=0.3\). figure — add your own
  3. F14-3If the motor exerts a force of \(F=(600+2s^2)\,\mathrm{N}\) on the cable, determine the speed of the \(100\,\mathrm{kg}\) crate when it rises to \(s=15\,\mathrm{m}\). The crate is initially at rest on the ground. figure — add your own
  4. F14-4The \(1.8\,\mathrm{Mg}\) dragster is traveling at \(125\,\mathrm{m/s}\) 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 \(400\,\mathrm{m}\). figure — add your own
  5. F14-5When \(s=0.6\,\mathrm{m}\), the spring is unstretched and the \(10\,\mathrm{kg}\) block has a speed of \(5\,\mathrm{m/s}\) down the smooth plane. Determine the distance \(s\) when the block stops. figure — add your own
  6. F14-6The \(20\,\mathrm{kg}\) crate is subjected to a force having a constant direction and a magnitude \(F=100\,\mathrm{N}\). When \(s=15\,\mathrm{m}\) the crate is moving to the right with a speed of \(8\,\mathrm{m/s}\). Determine its speed when \(s=25\,\mathrm{m}\). The coefficient of kinetic friction between the crate and the ground is \(\mu_k=0.25\). figure — add your own
  7. F14-7If the contact surface between the \(20\,\mathrm{kg}\) block and the ground is smooth, determine the power of force \(\mathbf{F}\) when \(t=4\,\mathrm{s}\). Initially, the block is at rest. figure — add your own
  8. F14-8If \(F=(10s)\,\mathrm{N}\), where \(s\) is in meters, and the contact surface between the block and the ground is smooth, determine the power of force \(\mathbf{F}\) when \(s=5\,\mathrm{m}\). When \(s=0\) the \(20\,\mathrm{kg}\) block is moving at \(v=1\,\mathrm{m/s}\). figure — add your own
  9. F14-9If the motor winds in the cable with a constant speed of \(v=3\,\mathrm{m/s}\), determine the power supplied to the motor. The load weighs \(100\,\mathrm{N}\) and the efficiency of the motor is \(\varepsilon=0.8\). Neglect the mass of the pulleys. figure — add your own
  10. F14-10The coefficient of kinetic friction between the \(20\,\mathrm{kg}\) block and the inclined plane is \(\mu_k=0.2\). If the block is traveling up the inclined plane with a constant velocity \(v=5\,\mathrm{m/s}\), determine the power of force \(\mathbf{F}\). figure — add your own
  11. F14-11If the \(50\,\mathrm{kg}\) load \(A\) is hoisted by motor \(M\) so that the load has a constant velocity of \(1.5\,\mathrm{m/s}\), determine the power input to the motor, which operates at an efficiency \(\varepsilon=0.8\). figure — add your own
  12. F14-12At the instant shown, point \(P\) on the cable has a velocity \(v_P=12\,\mathrm{m/s}\), which is increasing at a rate of \(a_P=6\,\mathrm{m/s^2}\). 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 \(50\,\mathrm{kg}\). figure — add your own
  13. F14-13The 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, \(B\). figure — add your own
  14. F14-14The 2-kg package leaves the conveyor belt at \(A\) with a speed of \(v_A=1\,\mathrm{m/s}\) 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 \(\rho_B=2\,\mathrm{m}\). figure — add your own
  15. F14-15The 2-kg collar is given a downward velocity of \(4\,\mathrm{m/s}\) when it is at \(A\). If the spring has an unstretched length of \(1\,\mathrm{m}\) and a stiffness of \(k=30\,\mathrm{N/m}\), determine the velocity of the collar at \(s=1\,\mathrm{m}\). figure — add your own
  16. F14-16The 5-kg 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 \(0.5\,\mathrm{m}\). figure — add your own
  17. F14-17The 35-kg block is released from rest \(1.5\,\mathrm{m}\) 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. figure — add your own
  18. F14-18The 4-kg collar \(C\) has a velocity of \(v_A=2\,\mathrm{m/s}\) 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 \(l_0=0.2\,\mathrm{m}\). figure — add your own
14 Practice Problems 91 problems
Work and Energy — Basic
  1. 14-1The \(20\,\mathrm{kg}\) crate is subjected to a force having a constant direction and a magnitude \(F=100\,\mathrm{N}\). When \(s=15\,\mathrm{m}\) the crate is moving to the right with a speed of \(8\,\mathrm{m/s}\). Determine its speed when \(s=25\,\mathrm{m}\). The coefficient of kinetic friction between the crate and the ground is \(\mu_k=0.25\). figure — add your own
  2. 14-2The crate, which has a mass of \(100\,\mathrm{kg}\), 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 \(6\,\mathrm{m/s}\). The coefficient of kinetic friction between the crate and the surface is \(\mu_k=0.2\). figure — add your own
  3. 14-3The \(100\,\mathrm{kg}\) 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 \(v=8\,\mathrm{m/s}\). The coefficient of kinetic friction between the crate and the surface is \(\mu_k=0.2\). figure — add your own
  4. 14-4Determine 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 \(100\,\mathrm{km/h}\) 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 \(4mg=(39.24m)\,\mathrm{N}\)? Neglect the size of the car and passenger. figure — add your own
  5. 14-5For protection, the barrel barrier is placed in front of the bridge pier. If the relation between the force and deflection of the barrier is \(F=[800(10^3)x^{1/2}]\,\mathrm{N}\), where \(x\) is in m, determine the car's maximum penetration in the barrier. The car has a mass of \(2\,\mathrm{Mg}\) and it is traveling with a speed of \(20\,\mathrm{m/s}\) just before it hits the barrier. figure — add your own
  6. 14-6The force of \(F=50\,\mathrm{N}\) is applied to the cord when \(s=2\,\mathrm{m}\). If the 6-kg collar is originally at rest, determine its velocity at \(s=0\). Neglect friction. figure — add your own
  7. 14-7As 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 \(6\,\mathrm{N}\). If observer \(A\) is in a fixed frame \(x\), determine the final speed of the block if it has an initial speed of \(5\,\mathrm{m/s}\) and travels \(10\,\mathrm{m}\), 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 \(2\,\mathrm{m/s}\) 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. figure — add your own
  8. 14-8When the driver applies the brakes of a light truck traveling \(40\,\mathrm{km/h}\), it skids \(3\,\mathrm{m}\) before stopping. How far will the truck skid if it is traveling at \(80\,\mathrm{km/h}\) when the brakes are applied? figure — add your own
  9. 14-9The “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 \(20\,\mathrm{kg}\) and is suspended a height \(d=0.4\,\mathrm{m}\) 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. figure — add your own
  10. 14-10The 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, find 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 \(70\,\mathrm{kg}\). figure — add your own
Springs and Friction
  1. 14-11The force \(\mathbf{F}\), acting in a constant direction on the \(20\,\mathrm{kg}\) block, has a magnitude which varies with the position \(s\) of the block. Determine how far the block must slide before its velocity becomes \(15\,\mathrm{m/s}\). When \(s=0\) the block is moving to the right at \(v=6\,\mathrm{m/s}\). The coefficient of kinetic friction between the block and surface is \(\mu_k=0.3\). figure — add your own
  2. 14-12Design considerations for the bumper \(B\) on the \(5\,\mathrm{Mg}\) 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 \(0.2\,\mathrm{m}\) when the car, traveling at \(4\,\mathrm{m/s}\) strikes the rigid stop. Neglect the mass of the car wheels. figure — add your own
  3. 14-13The 8-kg cylinder \(A\) and 3-kg cylinder \(B\) are released from rest. Determine the speed of \(A\) after it has moved \(2\,\mathrm{m}\) starting from rest. Neglect the mass of the cord and pulleys. figure — add your own
  4. 14-14Cylinder \(A\) has a mass of \(3\,\mathrm{kg}\) and cylinder \(B\) has a mass of \(8\,\mathrm{kg}\). Determine the speed of \(A\) after it has moved \(2\,\mathrm{m}\) starting from rest. Neglect the mass of the cord and pulleys. figure — add your own
  5. 14-15The collar has a mass of \(20\,\mathrm{kg}\) and is supported on the smooth rod. The attached springs are undeformed when \(d=0.5\,\mathrm{m}\). Determine the speed of the collar after the applied force \(F=100\,\mathrm{N}\) causes it to be displaced so that \(d=0.3\,\mathrm{m}\). When \(d=0.5\,\mathrm{m}\) the collar is at rest. figure — add your own
  6. 14-16A 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. figure — add your own
  7. 14-17If the cord is subjected to a constant force of \(F=300\,\mathrm{N}\) and the \(15\,\mathrm{kg}\) smooth collar starts from rest at \(A\), determine the velocity of the collar when it reaches point \(B\). Neglect the size of the pulley. figure — add your own
  8. 14-18If the force exerted by the motor \(M\) on the cable is \(250\,\mathrm{N}\), determine the speed of the \(100\,\mathrm{kg}\) crate when it is hoisted to \(s=3\,\mathrm{m}\). The crate is at rest when \(s=0\). figure — add your own
  9. 14-19When a \(7\,\mathrm{kg}\) projectile is fired from a cannon barrel that has a length of \(2\,\mathrm{m}\), 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. figure — add your own
  10. 14-20The steel ingot has a mass of \(1800\,\mathrm{kg}\). It travels along the conveyor at a speed \(v=0.5\,\mathrm{m/s}\) when it collides with the “nested” spring assembly. If the stiffness of the outer spring is \(k_A=5\,\mathrm{kN/m}\), determine the required stiffness \(k_B\) of the inner spring so that the motion of the ingot is stopped at the moment the front \(C\) of the ingot is \(0.3\,\mathrm{m}\) from the wall. figure — add your own
Nonlinear Springs & Impact
  1. 14-21The \(1.5\,\mathrm{kg}\) block slides along a smooth plane and strikes a nonlinear spring with a speed of \(v=4\,\mathrm{m/s}\). The spring is termed “nonlinear” because it has a resistance of \(F_s=ks^2\), where \(k=900\,\mathrm{N/m^2}\). Determine the speed of the block after it has compressed the spring \(s=0.2\,\mathrm{m}\). figure — add your own
  2. 14-22A car is equipped with a bumper \(B\) designed to absorb collisions. The bumper is mounted to the car using pieces of flexible tubing \(T\). Upon collision with a rigid barrier at \(A\), a constant horizontal force \(\mathbf{F}\) is developed which causes a car deceleration of \(3g=29.43\,\mathrm{m/s^2}\) (the highest safe deceleration for a passenger without a seatbelt). If the car and passenger have a total mass of \(1.5\,\mathrm{Mg}\) and the car is initially coasting with a speed of \(1.5\,\mathrm{m/s}\), determine the magnitude of \(\mathbf{F}\) needed to stop the car and the deformation \(x\) of the bumper tubing. figure — add your own
  3. 14-23The catapulting mechanism is used to propel the \(10\,\mathrm{kg}\) 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 \(F=20\,\mathrm{kN}\) to rod \(BC\) such that it moves it \(0.2\,\mathrm{m}\), determine the speed attained by the slider if it was originally at rest. Neglect the mass of the pulleys, cable, piston, and rod \(BC\). figure — add your own
  4. 14-24The \(12\,\mathrm{kg}\) block has an initial speed of \(v_0=4\,\mathrm{m/s}\) 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. figure — add your own
  5. 14-25The \(8\,\mathrm{kg}\) block is moving with an initial speed of \(5\,\mathrm{m/s}\). 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. figure — add your own
  6. 14-26Marbles having a mass of \(5\,\mathrm{g}\) 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. figure — add your own
  7. 14-27The collar has a mass of \(20\,\mathrm{kg}\) 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 \(1\,\mathrm{m}\) and the collar has a speed of \(2\,\mathrm{m/s}\) when \(s=0\), determine the maximum compression of each spring due to the back-and-forth (oscillating) motion of the collar. figure — add your own
  8. 14-28The train car has a mass of \(10\,\mathrm{Mg}\) and is traveling at \(5\,\mathrm{m/s}\) 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. figure — add your own
  9. 14-29The \(0.5\,\mathrm{kg}\) ball is fired up the smooth vertical circular track using the spring plunger. The plunger keeps the spring compressed \(0.08\,\mathrm{m}\) when \(s=0\). Determine how far \(s\) it must be pulled back and released so that the ball will begin to leave the track when \(\theta=135^\circ\). figure — add your own
  10. 14-30The “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, \(v_t=3\,\mathrm{m/s}\). 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 \(250\,\mathrm{kg}\) and the center of mass of the car and rider moves along a circular path having a radius of \(8\,\mathrm{m}\). figure — add your own
Pulleys, Cords & Inclines
  1. 14-31The man at the 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 \(R\). figure — add your own
  2. 14-32The conveyor belt delivers each 12-kg crate to the ramp at \(A\) such that the crate's velocity is \(v_A=2.5\,\mathrm{m/s}\) directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_k=0.3\), determine the speed at which each crate slides off the ramp at \(B\). Assume that no tipping occurs. figure — add your own
  3. 14-33The cyclist travels to point \(A\) pedaling until he reaches a speed \(v_A=4\,\mathrm{m/s}\). He then coasts freely up the curved surface. Determine how high he reaches up the surface before he comes to a stop. 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 \(75\,\mathrm{kg}\). Neglect friction, the mass of the wheels, and the size of the bicycle. figure — add your own
  4. 14-34The block has a mass of \(0.8\,\mathrm{kg}\) 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 \(v_B=2.5\,\mathrm{m/s}\) when it reaches \(B\); \(s_B=0.15\,\mathrm{m}\). Neglect the size and mass of the pulley. Hint: The work of \(\mathbf{F}\) can be determined by finding the difference \(\Delta l\) in cord lengths \(AC\) and \(BC\) and using \(U_F=F\Delta l\). figure — add your own
  5. 14-35If the \(60\,\mathrm{kg}\) skier passes point \(A\) with a speed of \(5\,\mathrm{m/s}\), determine his speed when he reaches point \(B\). Also find the normal force exerted on him by the slope at this point. Neglect friction. figure — add your own
  6. 14-36The spring in the toy gun has an unstretched length of \(100\,\mathrm{mm}\). It is compressed and locked in the position shown. When the trigger is pulled, the spring unstretches \(12.5\,\mathrm{mm}\), and the \(20\,\mathrm{g}\) ball moves along the barrel. Determine the speed of the ball when it leaves the gun. Neglect friction. figure — add your own
  7. 14-37If 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. figure — add your own
  8. 14-38The 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, find 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 \(75\,\mathrm{kg}\). figure — add your own
  9. 14-39If the \(75\,\mathrm{kg}\) crate starts from rest at \(A\), determine its speed when it reaches point \(B\). The cable is subjected to a constant force of \(F=300\,\mathrm{N}\). Neglect friction and the size of the pulley. figure — add your own
  10. 14-40If the \(75\,\mathrm{kg}\) crate starts from rest at \(A\), and its speed is \(6\,\mathrm{m/s}\) when it passes point \(B\), determine the constant force \(\mathbf{F}\) exerted on the cable. Neglect friction and the size of the pulley. figure — add your own
  11. 14-41A spring having a stiffness of \(5\,\mathrm{kN/m}\) is compressed \(400\,\mathrm{mm}\). The stored energy in the spring is used to drive a machine which requires \(90\,\mathrm{W}\) of power. Determine how long the spring can supply energy at the required rate.
Power and Efficiency
  1. 14-42To dramatize the loss of energy in an automobile, consider a car having a weight of \(25000\,\mathrm{N}\) that is traveling at \(50\,\mathrm{km/h}\). If the car is brought to a stop, determine how long a \(100\,\mathrm{W}\) light bulb must burn to expend the same amount of energy.
  2. 14-43If the engine of a \(1.5\,\mathrm{Mg}\) car generates a constant power of \(15\,\mathrm{kW}\), determine the speed of the car after it has traveled a distance of \(200\,\mathrm{m}\) on a level road starting from rest. Neglect friction.
  3. 14-44If the engine of a \(1.5\,\mathrm{Mg}\) car generates a constant power of \(15\,\mathrm{kW}\), determine the speed of the car after it has traveled a distance of \(200\,\mathrm{m}\) on a level road starting from rest. Neglect friction.
  4. 14-45The \(2\,\mathrm{Mg}\) car increases its speed uniformly from rest to \(25\,\mathrm{m/s}\) in \(30\,\mathrm{s}\) 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. figure — add your own
  5. 14-46A car has a mass \(m\) and accelerates along a horizontal straight road from rest such that the power is always a constant amount \(P\). Determine how far it must travel to reach a speed of \(v\).
  6. 14-47An automobile having a mass of \(2\,\mathrm{Mg}\) travels up a \(7^\circ\) slope at a constant speed of \(v=100\,\mathrm{km/h}\). If mechanical friction and wind resistance are neglected, determine the power developed by the engine if the automobile has an efficiency \(\varepsilon=0.65\). figure — add your own
  7. 14-48A rocket having a total mass of \(8\,\mathrm{Mg}\) is fired vertically from rest. If the engines provide a constant thrust of \(T=300\,\mathrm{kN}\), determine the power output of the engines as a function of time. Neglect the effect of drag resistance and the loss of fuel mass and weight.
  8. 14-49The sports car has a mass of \(2.3\,\mathrm{Mg}\) and while it is traveling at \(28\,\mathrm{m/s}\) the driver causes it to accelerate at \(5\,\mathrm{m/s^2}\). If the drag resistance on the car due to the wind is \(F_D=(0.3v^2)\,\mathrm{N}\), where \(v\) is the velocity in \(\mathrm{m/s}\), determine the power supplied to the engine at this instant. The engine has a running efficiency of \(\varepsilon=0.68\). figure — add your own
  9. 14-50The sports car has a mass of \(2.3\,\mathrm{Mg}\) and accelerates at \(6\,\mathrm{m/s^2}\), starting from rest. If the drag resistance on the car due to the wind is \(F_D=(10v)\,\mathrm{N}\), where \(v\) is the velocity in \(\mathrm{m/s}\), determine the power supplied to the engine when \(t=5\,\mathrm{s}\). The engine has a running efficiency of \(\varepsilon=0.68\). figure — add your own
  10. 14-51A motor hoists a \(60\,\mathrm{kg}\) crate at a constant velocity to a height of \(h=5\,\mathrm{m}\) in \(2\,\mathrm{s}\). If the indicated power of the motor is \(3.2\,\mathrm{kW}\), determine the motor's efficiency. figure — add your own
  11. 14-52The \(50\,\mathrm{kg}\) crate is hoisted up the \(30^\circ\) incline by the pulley system and motor \(M\). If the crate starts from rest and, by constant acceleration, attains a speed of \(4\,\mathrm{m/s}\) after traveling \(8\,\mathrm{m}\) along the plane, determine the power that must be supplied to the motor at the instant. Neglect friction along the plane. The motor has an efficiency of \(\varepsilon=0.74\). figure — add your own
  12. 14-53The \(500\,\mathrm{kg}\) elevator starts from rest and travels upward with a constant acceleration \(a_c=2\,\mathrm{m/s^2}\). Determine the power output of the motor \(M\) when \(t=3\,\mathrm{s}\). Neglect the mass of the pulleys and cable. figure — add your own
  13. 14-54The escalator steps move with a constant speed of \(0.6\,\mathrm{m/s}\). If the steps are \(125\,\mathrm{mm}\) high and \(250\,\mathrm{mm}\) in length, determine the power of a motor needed to lift an average mass of \(150\,\mathrm{kg}\) per step. There are 32 steps. figure — add your own
Mixed Applications
  1. 14-55If the escalator in Prob. 14-54 is not moving, determine the constant speed at which a man having a mass of \(80\,\mathrm{kg}\) must walk up the steps to generate \(100\,\mathrm{W}\) of power — the same amount that is needed to power a standard light bulb. figure — add your own
  2. 14-56The elevator \(E\) and its freight have a total mass of \(400\,\mathrm{kg}\). Hoisting is provided by the motor \(M\) and the \(60\,\mathrm{kg}\) 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 \(v_E=4\,\mathrm{m/s}\). figure — add your own
  3. 14-57The crate has a mass of \(150\,\mathrm{kg}\) and rests on a surface for which the coefficients of static and kinetic friction are \(\mu_s=0.3\) and \(\mu_k=0.2\), respectively. If the motor \(M\) supplies a cable force of \(F=(8t^2+20)\,\mathrm{N}\), where \(t\) is in seconds, determine the power output developed by the motor when \(t=5\,\mathrm{s}\). figure — add your own
  4. 14-58The material hoist and the load have a total mass of \(800\,\mathrm{kg}\) and the counterweight \(C\) has a mass of \(150\,\mathrm{kg}\). At a given instant, the hoist has an upward velocity of \(2\,\mathrm{m/s}\) and an acceleration of \(1.5\,\mathrm{m/s^2}\). Determine the power generated by the motor \(M\) at this instant if it operates with an efficiency of \(\varepsilon=0.8\). figure — add your own
  5. 14-59The material hoist and the load have a total mass of \(800\,\mathrm{kg}\) and the counterweight \(C\) has a mass of \(150\,\mathrm{kg}\). If the upward speed of the hoist increases uniformly from \(0.5\,\mathrm{m/s}\) to \(1.5\,\mathrm{m/s}\) in \(1.5\,\mathrm{s}\), determine the average power generated by the motor \(M\) during this time. The motor operates with an efficiency of \(\varepsilon=0.8\). figure — add your own
  6. 14-60An 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 \(t=0.3\,\mathrm{s}\). figure — add your own
  7. 14-61An 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. figure — add your own
  8. 14-62If the jet on the dragster supplies a constant thrust of \(T=20\,\mathrm{kN}\), 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 \(1\,\mathrm{Mg}\) and starts from rest. figure — add your own
  9. 14-63The rocket sled has a mass of \(4\,\mathrm{Mg}\) and travels from rest along the horizontal track for which the coefficient of kinetic friction is \(\mu_k=0.20\). If the engine provides a constant thrust \(T=150\,\mathrm{kN}\), determine the power output of the engine as a function of time. Neglect the loss of fuel mass and air resistance. figure — add your own
  10. 14-64The block has a mass of \(150\,\mathrm{kg}\) 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 \(F=(60t^2)\,\mathrm{N}\), where \(t\) is in seconds, is applied to the cable, determine the power developed by the force when \(t=5\,\mathrm{s}\). Hint: First determine the time needed for the force to cause motion. figure — add your own
  11. 14-65The assembly consists of two blocks \(A\) and \(B\) which have a mass of \(20\,\mathrm{kg}\) and \(30\,\mathrm{kg}\), respectively. Determine the distance \(B\) must descend in order for \(A\) to achieve a speed of \(3\,\mathrm{m/s}\) starting from rest. figure — add your own
Conservation of Energy — Pulleys & Blocks
  1. 14-66The assembly consists of two blocks \(A\) and \(B\) which have a mass of \(20\,\mathrm{kg}\) and \(30\,\mathrm{kg}\), respectively. Determine the speed of each block when \(B\) descends \(1.5\,\mathrm{m}\). The blocks are released from rest. Neglect the mass of the pulleys and cords. figure — add your own
  2. 14-67The girl has a mass of \(40\,\mathrm{kg}\) 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. figure — add your own
  3. 14-68Each of the two elastic rubber bands of the slingshot has an unstretched length of \(180\,\mathrm{mm}\). If they are pulled back to the position shown and released from rest, determine the maximum height the \(30\,\mathrm{g}\) 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 \(k=80\,\mathrm{N/m}\). figure — add your own
  4. 14-69Two 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 \(s=0.5\,\mathrm{m}\) above the top of the springs from an at-rest position, and the maximum compression of the springs is to be \(0.2\,\mathrm{m}\), determine the required stiffness of the inner spring, \(k_B\), if the outer spring has a stiffness \(k_A=400\,\mathrm{N/m}\). figure — add your own
  5. 14-70The 5-kg collar has a velocity of \(5\,\mathrm{m/s}\) 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 \(100\,\mathrm{mm}\) and \(B\) is located just before the end of the curved portion of the rod. figure — add your own
  6. 14-71The 5-kg collar has a velocity of \(5\,\mathrm{m/s}\) 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 \(100\,\mathrm{mm}\) and \(B\) is located just before the end of the curved portion of the rod. figure — add your own
  7. 14-72The roller coaster car has a mass of \(700\,\mathrm{kg}\) including its passenger. If it starts from the top of the hill \(A\) with a speed \(v_A=3\,\mathrm{m/s}\), 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 \(\rho_B=7.5\,\mathrm{m}\) and \(\rho_C=5\,\mathrm{m}\). figure — add your own
  8. 14-73The roller coaster car has a mass of \(700\,\mathrm{kg}\) 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 \(\rho_B=7.5\,\mathrm{m}\) and \(\rho_C=5\,\mathrm{m}\). figure — add your own
  9. 14-74The 2-kg ball of negligible size is fired from point \(A\) with an initial velocity of \(10\,\mathrm{m/s}\) 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? figure — add your own
  10. 14-75The 4-kg smooth collar has a speed of \(3\,\mathrm{m/s}\) when it is at \(s=0\). Determine the maximum distance \(s\) it travels before it stops momentarily. The spring has an unstretched length of \(1\,\mathrm{m}\). figure — add your own
  11. 14-76The spring has a stiffness \(k=200\,\mathrm{N/m}\) and an unstretched length of \(0.5\,\mathrm{m}\). 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. figure — add your own
  12. 14-77The 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. figure — add your own
Conservation of Energy — Advanced
  1. 14-78A \(750\,\mathrm{mm}\)-long spring is compressed and confined by the plate \(P\), which can slide freely along the vertical \(60\,\mathrm{mm}\)-long rods. The \(40\,\mathrm{kg}\) block is given a speed of \(v=5\,\mathrm{m/s}\) when it is \(h=2\,\mathrm{m}\) above the plate. Determine how far the plate moves downwards when the block momentarily stops after striking it. Neglect the mass of the plate. figure — add your own
  2. 14-79The spring has a stiffness \(k=50\,\mathrm{N/m}\) and an unstretched length of \(0.3\,\mathrm{m}\). If it is attached to the \(2\,\mathrm{kg}\) 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. figure — add your own
  3. 14-80If 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 \(r\to\infty\). Also, prove that \(F\) is a conservative force.
  4. 14-81A 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. figure — add your own
  5. 14-82When \(s=0\), the spring on the firing mechanism is unstretched. If the arm is pulled back such that \(s=100\,\mathrm{mm}\) and released, determine the speed of the \(0.3\,\mathrm{kg}\) 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. figure — add your own
  6. 14-83When \(s=0\), the spring on the firing mechanism is unstretched. If the arm is pulled back such that \(s=100\,\mathrm{mm}\) 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. figure — add your own
  7. 14-84When the 5-kg box reaches point \(A\) it has a speed \(v_A=10\,\mathrm{m/s}\). Determine the normal force the box exerts on the surface when it reaches point \(B\). Neglect friction and the size of the box. figure — add your own
  8. 14-85When the 5-kg box reaches point \(A\) it has a speed \(v_A=10\,\mathrm{m/s}\). 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. figure — add your own
  9. 14-86When the 6-kg box reaches point \(A\) it has a speed of \(v_A=2\,\mathrm{m/s}\). 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. figure — add your own
  10. 14-87The 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 \(70\,\mathrm{kg}\). figure — add your own
  11. 14-88A 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 \(v_A=40\,\mathrm{Mm/h}\). What is the speed of the satellite when it reaches point \(B\) where \(r_B=80\,\mathrm{Mm}\)? Hint: See Prob. 14-80, where \(M_e=5.976(10^{24})\,\mathrm{kg}\) and \(G=66.73(10^{-12})\,\mathrm{m^3/(kg\cdot s^2)}\). figure — add your own
  12. 14-89The block has a mass of \(20\,\mathrm{kg}\) and is released from rest when \(s=0.5\,\mathrm{m}\). If the mass of the bumpers \(A\) and \(B\) can be neglected, determine the maximum deformation of each spring due to the collision. figure — add your own
Conservation of Energy — Final Problems
  1. 14-90The \(0.75\,\mathrm{kg}\) bob of a pendulum is fired from rest at position \(A\) by a spring which has a stiffness \(k=6\,\mathrm{kN/m}\) and is compressed \(125\,\mathrm{mm}\). Determine the speed of the bob and the tension in the cord when the bob is at positions \(B\) and \(C\). Point \(B\) is located on the path where the radius of curvature is still \(0.6\,\mathrm{m}\), i.e., just before the cord becomes horizontal. figure — add your own
  2. 14-91The Raptor is an outside loop roller coaster in which riders are belted into seats resembling ski-lift chairs. If the cars travel at \(v_0=4\,\mathrm{m/s}\) 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 \(70\,\mathrm{kg}\) passenger on his seat at this instant. The car has a mass of \(50\,\mathrm{kg}\). Take \(h=12\,\mathrm{m}\), \(\rho=5\,\mathrm{m}\). Neglect friction and the size of the car and passenger. figure — add your own
  3. 14-92If 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 \(h=0.5\,\mathrm{m}\). Each spring has an unstretched length of \(1\,\mathrm{m}\). figure — add your own
  4. 14-93The cylinder has a mass of \(20\,\mathrm{kg}\) and is released from rest when \(h=0\). Determine its speed when \(h=3\,\mathrm{m}\). Each spring has a stiffness \(k=40\,\mathrm{N/m}\) and an unstretched length of \(2\,\mathrm{m}\). figure — add your own
  5. 14-94A 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. figure — add your own
  6. 14-95The 10-kg sphere \(C\) is released from rest when \(\theta=0^\circ\) and the tension in the spring is \(100\,\mathrm{N}\). Determine the speed of the sphere at the instant \(\theta=90^\circ\). Neglect the mass of rod \(AB\) and the size of the sphere. figure — add your own
  7. 14-96A pan of negligible mass is attached to two identical springs of stiffness \(k=250\,\mathrm{N/m}\). If a 10-kg box is dropped from a height of \(0.5\,\mathrm{m}\) above the pan, determine the maximum vertical displacement \(d\). Initially each spring has a tension of \(50\,\mathrm{N}\). figure — add your own
  8. 14-97If a \(50\,\mathrm{kg}\) crate is released from rest at \(A\), determine its speed after it slides \(10\,\mathrm{m}\) down the plane. The coefficient of kinetic friction between the crate and plane is \(\mu_k=0.3\). figure — add your own
R Review Problems 8 problems
Chapter Review
  1. R14-1If a \(50\,\mathrm{kg}\) crate is released from rest at \(A\), determine its speed after it slides \(10\,\mathrm{m}\) down the plane. The coefficient of kinetic friction between the crate and plane is \(\mu_k=0.3\). figure — add your own
  2. R14-2The small 1-kg collar starting from rest at \(A\) slides down along the smooth rod. During the motion, the collar is acted upon by a force \(\mathbf{F}=[50\mathbf{i}+30\mathbf{j}+10z\mathbf{k}]\,\mathrm{N}\), where \(x,y,z\) are in meters. Determine the collar's speed when it strikes the wall at \(B\). figure — add your own
  3. R14-3The block has a mass of \(0.75\,\mathrm{kg}\) and slides along the smooth chute \(AB\). It is released from rest at \(A\), which has coordinates of \(A(2.5\,\mathrm{m},0.5\,\mathrm{m})\). Determine the speed at which it slides off at \(B\), which has coordinates of \(B(0,4\,\mathrm{m},0)\). figure — add your own
  4. R14-4The block has a mass of \(0.5\,\mathrm{kg}\) 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 \(v_B=2.5\,\mathrm{m/s}\) when it reaches \(B\); \(s_B=0.15\,\mathrm{m}\). Neglect the mass of the cord and pulley. figure — add your own
  5. R14-5The crate, having a mass of \(25\,\mathrm{kg}\), is hoisted by the pulley system and motor \(M\). If the crate starts from rest and, by constant acceleration, attains a speed of \(6\,\mathrm{m/s}\) after rising \(3\,\mathrm{m}\), determine the power that must be supplied to the motor at the instant \(s=3\,\mathrm{m}\). The motor has an efficiency \(\varepsilon=0.74\). figure — add your own
  6. R14-6The \(25\,\mathrm{kg}\) load is hoisted by the pulley system and motor \(M\). If the motor exerts a constant force of \(150\,\mathrm{N}\) on the cable, determine the power that must be supplied to the motor if the load has been hoisted \(s=3\,\mathrm{m}\) starting from rest. The motor has an efficiency of \(\varepsilon=0.76\). figure — add your own
  7. R14-7The collar of negligible size has a mass of \(0.25\,\mathrm{kg}\) and is attached to a spring having an unstretched length of \(100\,\mathrm{mm}\). If the collar is released from rest at \(A\) and travels along the smooth guide, determine its speed just before it strikes \(B\). figure — add your own
  8. R14-8The blocks \(A\) and \(B\) have the mass of \(5\,\mathrm{kg}\) and \(15\,\mathrm{kg}\), 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 \(2\,\mathrm{m}\) up along the plane. The blocks are released from rest. figure — add your own