The only force acting on a particle is conservative force

What is the spring constant of a spring that stores 25 J of elastic potential energy when compressed by 7.5 cm?

In Fig. 8-29, a single frictionless roller-coaster car of mass m ! 825 kg tops the first hill with speed v0 ! 17.0 m/s at height h ! 42.0 m. How much work does the gravitational force do on the car from that point to (a) point A, (b) point B, and (c) point C? If the gravitational potential energy of the carEarth system is taken to be zero at C, what is its value when the car is at (d) B and (e) A? (f) If mass m were doubled, would the change in the gravitational potential energy of the system between points A and B increase, decrease, or remain the same? First hill A B C h h h/2 v0 L r Ice flake Figure 8-32 Problems 5 and 11. Figure 8-29 Problems 2 and 9.

You drop a 2.00 kg book to a friend who stands on the ground at distance D ! 10.0 m below. If your friends outstretched hands are at distance d ! 1.50 m above the ground (Fig. 8-30), (a) how much work Wg does the gravitational force do on the book as it drops to her hands? (b) What is the change "U in the gravitational potential energy of the bookEarth system during the drop? If the gravitational potential energy U of that system is taken to be zero at ground level, what is U (c) when the book is released and (d) when it reaches her hands? Now take U to be 100 J at ground level and again find (e) Wg, (f) "U, (g) U at the release point, and (h) U at her hands. d D Figure 8-30 Problems 3 and 10

Figure 8-31 shows a ball with mass m ! 0.341 kg attached to the end of a thin rod with length L ! 0.452 m and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there. How much work is done on the ball by the gravitational force from the initial point to (a) the lowest point, (b) the highest point, and (c) the point on the right level with the initial point? If the gravitational potential energy of the ballEarth system is taken to be zero at the initial point, what is it when the ball reaches (d) the lowest point, (e) the highest point, and (f) the point on the right level with the initialpoint? (g) Suppose the rod were pushed harder so that the ball passed through the highest point with a nonzero speed.Would "Ug from the lowest point to the highest point then be greater than, less than, or the same as it was when the ball stopped at the highest point? L r Ice flake Figure 8-32 Problems 5 and 11. Figure 8-29 Problems 2 and 9. Figure 8-31 Problems 4 and 14.

In Fig. 8-32, a 2.00 g ice flake is released from the edge of a hemispherical bowl whose radius r is 22.0 cm. The flakebowl contact is frictionless. (a) How much work is done on the flake by the gravitational force during the flakes descent to the bottom of the bowl? (b) What is the change in the potential energy of the flakeEarth system during that descent? (c) If that potential energy is taken to be zero at the bottom of the bowl, what is its value when the flake is released? (d) If, instead, the potential energy is taken to be zero at the release point, what is its value when the flake reaches the bottom of the bowl? (e) If the mass of the flake were doubled, would themagnitudes of the answers to (a) through (d) increase, decrease, or remain the same? Ice flake Figure 8-32 Problems 5 and 11.

In Fig. 8-33, a small block of mass m ! 0.032 kg can slide along the frictionless loop-the-loop, with loop radius R ! 12 cm. The block is released from rest at point P, at height h ! 5.0R above the bottom of the loop. How much work does the gravitational force do on the block as the block travels from point P to (a) point Q and (b) the top of the loop? If the gravitational potential energy of the blockEarth system is taken to be zero at the bottom of the loop, what is that potential energy when the block is (c) at point P, (d) at point Q, and (e) at the top of the loop? (f) If, instead of merely being released, the block is given some initial speed downward along the track, do the answers to (a) through (e) increase, decrease, or remain the same? P R Q R Figure 8-33 Problems 6 and 17.

Figure 8-34 shows a thin rod, of length L ! 2.00 m and negligible mass, that can pivot about one end to rotate in a vertical circle. A ball of mass m ! 5.00 kg is attached to the other end. The rod is pulled aside to angle u0 ! 30.0# and released with initial velocity . As the ball descends to its lowest point, (a) how much work does the gravitational force do on it and (b) what is the change in the gravitational potential energy of

A 1.50 kg snowball is fired from a cliff 12.5 m high. The snowballs initial velocity is 14.0 m/s, directed 41.0# above the horizontal. (a) How much work is done on the snowball by the gravitational force during its flight to the flat ground below the cliff? (b) What is the change in the gravitational potential energy of the snowballEarth system during the flight? (c) If that gravitational potential energy is taken to be zero at the height of the cliff, what is its value when the snowball reaches the ground? 0 v0 m Figure 8-34 Problems 7, 18, and 21.

In Problem 2, what is the speed of the car at (a) point A, (b) point B, and (c) point C? (d) How high will the car go on the last hill, which is too high for it to cross? (e) If we substitute a second car with twice the mass, what then are the answers to (a) through (d)?

(a) In Problem 3, what is the speed of the book when it reaches the hands? (b) If we substituted a second book with twice the mass, what would its speed be? (c) If, instead, the book were thrown down, would the answer to (a) increase, decrease, or remain the same?

(a) In Problem 5, what is the speed of the flake when it reaches the bottom of the bowl? (b) If we substituted a second flake with twice the mass, what would its speed be? (c) If, instead, we gave the flake an initial downward speed along the bowl, would the answer to (a) increase, decrease, or remain the same?

(a) In Problem 8, using energy techniques rather than the techniques of Chapter 4, find the speed of the snowball as it reaches the ground below the cliff. What is that speed (b) if the launch angle is changed to 41.0# below the horizontal and (c) if the mass is changed to 2.50 kg?

A 5.0 g marble is fired vertically upward using a spring gun. The spring must be compressed 8.0 cm if the marble is to just reach a target 20 m above the marbles position on the compressed spring. (a) What is the change "Ug in the gravitational potential energy of the marbleEarth system during the 20 m ascent? (b) What is the change "Us in the elastic potential energy of the spring during its launch of the marble? (c) What is the spring constant of the spring?

(a) In Problem 4, what initial speed must be given the ball so that it reaches the vertically upward position with zero speed? What then is its speed at (b) the lowest point and (c) the point on the right at which the ball is level with the initial point? (d) If the balls mass were doubled, would the answers to (a) through (c) increase, decrease, or remain the same?

In Fig. 8-35, a runaway truck with failed brakes is moving downgrade at 130 km/h just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of u ! 15#.The trucks mass is 1.2 $ 104 kg. (a) What minimum length L must the ramp have if the truck is to stop (momentarily) along it? (Assume the truck is a particle, and justify that assumption.) Does the minimum length L increase, decrease, or remain the same if (b) the trucks mass is decreased and (c) its speed is decreased? L Figure 8-35 Problem 15.

A 700 g block is released from rest at height h0 above a vertical spring with spring constant k ! 400 N/m and negligible mass. The block sticks to the spring and momentarily stops after compressing the spring 19.0 cm. How much work is done (a) by the block on the spring and (b) by the spring on the block? (c) What is the value of h0? (d) If the block were released from height 2.00h0 above the spring, what would be the maximum compression of the spring?

In Problem 6, what are the magnitudes of (a) the horizontal component and (b) the vertical component of the net force acting on the block at point Q? (c) At what height h should the block be released from rest so that it is on the verge of losing contact with the track at the top of the loop? (On the verge of losing contact means that the normal force on the block from the track has just then become zero.) (d) Graph the magnitude of the normal force on the block at the top of the loop versus initial height h, for the range h ! 0 to h ! 6R.

(a) In Problem 7, what is the speed of the ball at the lowest point? (b) Does the speed increase, decrease, or remain the same if the mass is increased?

Figure 8-36 shows an 8.00 kg stone at rest on a spring. The spring is compressed 10.0 cm by the stone. (a) What is the spring constant? (b) The stone is pushed down an additional 30.0 cm and released. What is the elastic potential energy of the compressed spring just before that release? (c) What is the change in the gravitational potential energy of the stoneEarth system when the stone moves from the release point to its maximum height? (d) What is that maximum height, measured from the release point? k Figure 8-36 Problem 19.

A pendulum consists of a 2.0 kg stone swinging on a 4.0 m string of negligible mass. The stone has a speed of 8.0 m/s when it passes its lowest point. (a) What is the speed when the string is at 60# to the vertical? (b) What is the greatest angle with the vertical that the string will reach during the stones motion? (c) If the potential energy of the pendulumEarth system is taken to be zero at the stones lowest point, what is the total mechanical energy of the system?

Figure 8-34 shows a pendulum of length L ! 1.25 m. Its bob (which effectively has all the mass) has speed v0 when the cord makes an angle u0 ! 40.0# with the vertical. (a) What is the speed of the bob when it is in its lowest position if v0 ! 8.00 m/s? What is the least value that v0 can have if the pendulum is to swing down and then up (b) to a horizontal position, and (c) to a vertical position with the cord remaining straight? (d) Do the answers to (b) and (c) increase, decrease, or remain the same if u0 is increased by a few degrees?

A 60 kg skier starts from rest at height H ! 20 m above the end of a ski-jump ramp (Fig. 8-37) and leaves the ramp at angle u ! 28#. Neglect the effects of air resistance and assume the ramp is frictionless. (a) What is the maximum height h of his jump above the end of the ramp? (b) If he increased his weight by putting on a backpack, would h then be greater, less, or the same? H h End of ramp Figure 8-37 Problem 22.

The string in Fig. 8-38is L ! 120 cm long, has a ball attached to one end, and is fixed at its other end. The distance d from the fixed end to a fixed peg at point P is 75.0 cm. When the initially stationary ball is released with the string horizontal as shown, it will swing along the dashed arc. What is its speed when it reaches (a) its lowest point 31 A block with mass m ! 2.00 kg is placed against a spring and (b) its highest point after the string catches on the peg? r P L d Figure 8-38 Problems 23 and 70.

A block of mass m ! 2.0 kg is dropped from height h ! 40 cm onto a spring of spring constant k ! 1960 N/m (Fig. 8-39). Find the maximum distance the spring is compressed. h k m Figure 8-39 Problem 24

At t ! 0 a 1.0 kg ball is thrown from a tall tower with v . What is : ! (18 m/s)i & (24 m/s)j "U of the ballEarth system between t 0 and t 6.0 s (still free fall)?

A conservative force , where x is in meters, acts on a particle moving along an x axis. The potential energy U associated with this force is assigned a value of 27 J at x ! 0. (a) Write an expression for U as a function of x, with U in joules and x in meters. (b) What is the maximum positive potential energy? At what (c) negative value and (d) positive value of x is the potential energy equal to zero?

Tarzan, who weighs 688 N, swings from a cliff at the end of a vine 18 m long (Fig. 8-40). From the top of the cliff to the bottom of the swing, he descends by 3.2 m.The vine will break if the force on it exceeds 950 N. (a) Does the vine break? (b) If no, what is the greatest force on it during the swing? If yes, at what angle with the vertical does it break? Figure 8-40 Problem 27.

Figure 8-41a applies to the spring in a cork gun (Fig. 8-41b); it shows the spring force as a function of the stretch or compression of the spring. The spring is compressed by 5.5 cm and used to propel a 3.8 g cork from the gun. (a) What is the speed of the cork if it is released as the spring passes through its relaxed position? (b) Suppose, instead, that the cork sticks to the spring and stretches it 1.5 cm before separation occurs.What now is the speed of the cork at the time of release? x (cm) 24 2 4 0.4 0.2 (a) x 0 (b) Compressed spring Cork 0.2 0.4 Force (N) Figure 8-41 Problem 28.

In Fig. 8-42, a block of mass m ! 12 kg is released from rest on a frictionless incline of angle 30#. Below the block is a spring that can be compressed 2.0 cm by a force of 270 N. The block momentarily stops when it compresses the spring by 5.5 cm. (a) How far does the block move down the incline from its rest position to this stopping point? (b) What is the speed of the block just as it touches the spring? m Figure 8-42 Problems 29 and 35.

A 2.0 kg breadbox on a frictionless incline of angle u ! 40# is connected, by a cord that runs over a pulley, to a light spring of spring constant k ! 120 N/m, as shown in Fig. 8-43. The box is released from rest when the spring is unstretched. Assume that the pulley is massless and frictionless. (a) What is the speed of the box when it has moved 10 cm down the incline? (b) How far down the incline from its point of release does the box slide before momentarily stopping, and what are the (c) magnitude and (d) direction (up or down the incline) of the boxs acceleration at the instant the box momentarily stops? Figure 8-43 Problem 30.

A block with mass m ! 2.00 kg is placed against a spring on a frictionless incline with angle 30.0# (Fig. 8-44). (The block is not attached to the spring.) The spring, with spring constant k 19.6 N/cm, is compressed 20.0 cm and then released. (a) What is the elastic potential energy of the compressed spring? (b) What is the change in the gravitational potential energy of the blockEarth system as the block moves from the release point to its highest point on the incline? (c) How far along the incline is the highest point from the release point?! ' ! m k Figure 8-44 Problem 31

In Fig. 8-45, a chain is held on a frictionless table with onefourth of its length hanging over the edge. If the chain has length L ! 28 cm and mass m ! 0.012 kg, how much work is required to pull the hanging part back onto the table? Figure 8-45 Problem 32.

In Fig. 8-46, a spring with k ! 170 N/m is at the top of a frictionless incline of angle 37.0#. The lower end of the incline is distance D 1.00 m from the end of the spring, which is at its relaxed length. A 2.00 kg canister is pushed against the spring until the spring is compressed 0.200 m and released from rest. (a) What is the speed of the canister at the instant the spring returns to its relaxed length (which is when the canister loses contact with the spring)? (b) What is the speed of the canister when it reaches the lower end of the incline? D Figure 8-46 Problem 33.

A boy is initially seated on the top of a hemispherical ice mound of radius R 13.8 m. He begins to slide down the ice, with a negligible initial speed (Fig. 8-47). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice? R Figure 8-47 Problem 34

In Fig. 8-42, a block of mass m ! 3.20 kg slides from rest a distance d down a frictionless incline at angle u ! 30.0# where it runs into a spring of spring constant 431 N/m. When the block momentarily stops, it has compressed the spring by 21.0 cm. What are (a) distance d and (b) the distance between the point of the first blockspring contact and the point where the blocks speed is greatest?

Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a table. The target box is horizontal distance D ! 2.20 m from the edge of the table; see Fig. 8-48. Bobby compresses the spring 1.10 cm, but the center of the marble falls 27.0 cm short of the center of the box. How far should Rhoda compress the spring to score a direct hit? Assume that neither the spring nor the ball encounters friction in the gun. D Figure 8-48 Problem 36.

A uniform cord of length 25 cm and mass 15 g is initially stuck to a ceiling. Later, it hangs vertically from the ceiling with only one end still stuck. What is the change in the gravitational potential energy of the cord with this change in orientation? (Hint: Consider a differential slice of the cord and then use integral calculus.)

Figure 8-49 shows a plot of potential energy U versus position x of a 0.200 kg particle that can travel only along an x axis under the influence of a conservative force. The graph has these values: , and . The particle is released at the point where U forms a potential hill of height , with kinetic energy 4.00 J. What is the speed of the particle at (a) m and (b) m? What is the position of the turning point on (c) the right side and (d) the left side? x ! 3.5 x ! 6.5 UB ! 12.00 J UA ! 9.00 J, UC ! 20.00 J UD ! 24.00 J UA UB UC UD U (J) 0 1 2 3 4 5 6 7 8 9 x (m) Figure 8-49 Problem 38

Figure 8-50 shows a plot of potential energy U versus position x of a 0.90 kg particle that can travel only along an x axis. (Nonconservative forces are not involved.) Three values are and The particle is released at x 4.5 m with an initial speed of 7.0 m/s, headed in the negative x direction. (a) If the particle can reach x ! 1.0 m, what is its speed there, and if it cannot, what is its turning point? What are the (b) magnitude and (c) direction of the force on the particle as it begins to move to the left of x ! 4.0 m? Suppose, instead, the particle is headed in the positive x direction when it is released at x ! 4.5 m at speed 7.0 m/s. (d) If the particle can reach x ! 7.0 m, what is its speed there, and if it cannot, what is its turning point? What are the (e) magnitude and (f) direction of the force on the particle as it begins to move to the right of x ! 5.0 m? UC UB UA 2 46 x (m) U (J) Figure 8-50 Problem 39.

The potential energy of a diatomic molecule (a two-atom system like H2 or O2) is given by U ! A r12 % B r 6 ,where x is in meters. At x ! 5.0 m the particle has a kinetic energy of 2.0 J. (a) What is the mechanical energy of the system? (b) Make a plot of U(x) as a function of x for 0 ( x ( 10 m, and on the same graph draw the line that represents the mechanical energy of the system. Use part (b) to determine (c) the least value of x the particle can reach and (d) the greatest value of x the particle can reach. Use part (b) to determine (e) the maximum kinetic energy of the particle and (f) the value of x at which it occurs. (g) Determine an expression in newtons and meters for F(x) as a function of x. (h) For what (finite) value of x does F(x) ! 0?

A single conservative force F(x) acts on a 1.0 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is given by U(x) ! %4x e%x/4 J, where x is in meters. At x ! 5.0 m the particle has a kineticenergy of 2.0 J. (a) What is the mechanical energy of the system? (b) Make a plot of U(x) as a function of x for 0 ( x ( 10 m, and on the same graph draw the line that represents the mechanical energy of the system. Use part (b) to determine (c) the least value of x the particle can reach and (d) the greatest value of x the particle can reach. Use part (b) to determine (e) the maximum kinetic energy of the particle and (f) the value of x at which it occurs. (g) Determine an expression in newtons and meters for F(x) as a function of x. (h) For what (finite) value of x does F(x) ! 0?

A worker pushed a 27 kg block 9.2 m along a level floor at constant speed with a force directed 32# below the horizontal. If the coefficient of kinetic friction between block and floor was 0.20, what were (a) the work done by the workers force and (b) the increase in thermal energy of the blockfloor system?

A collie drags its bed box across a floor by applying a horizontal force of 8.0 N. The kinetic frictional force acting on the box has magnitude 5.0 N. As the box is dragged through 0.70 m along the way, what are (a) the work done by the collies applied force and (b) the increase in thermal energy of the bed and floor?

A horizontal force of magnitude 35.0 N pushes a block of mass 4.00 kg across a floor where the coefficient of kinetic friction is 0.600. (a) How much work is done by that applied force on the blockfloor system when the block slides through a displacement of 3.00 m across the floor? (b) During that displacement, the thermal energy of the block increases by 40.0 J. What is the increase in thermal energy of the floor? (c) What is the increase in the kinetic energy of the block?

A rope is used to pull a 3.57 kg block at constant speed 4.06 m along a horizontal floor. The force on the block from the rope is 7.68 N and directed 15.0# above the horizontal.What are (a) the work done by the ropes force, (b) the increase in thermal energy of the blockfloor system, and (c) the coefficient of kinetic friction between the block and floor?

An outfielder throws a baseball with an initial speed of 81.8 mi/h. Just before an infielder catches the ball at the same level, the balls speed is 110 ft/s. In foot-pounds, by how much is the mechanical energy of the ballEarth system reduced because of air drag? (The weight of a baseball is 9.0 oz.)

A 75 g Frisbee is thrown from a point 1.1 m above the ground with a speed of 12 m/s. When it has reached a height of 2.1 m, its speed is 10.5 m/s. What was the reduction in Emec of the FrisbeeEarth system because of air drag?

In Fig. 8-51, a block slides down an incline. As it moves from point A to point B, which are 5.0 m apart, force acts on the block, with magnitude 2.0 N and directed down the incline.The magnitude of the frictional force acting on the block is 10 N. If the kinetic energy of the block increases by 35 J between A and B, how much work is done on the block by the gravitational force as the block moves from A to B? A B Figure 8-51 Problems 48 and 71.

A 25 kg bear slides, from rest, 12 m down a lodgepole pine tree, moving with a speed of 5.6 m/s just before hitting the ground. (a) What change occurs in the gravitational potential energy of the bearEarth system during the slide? (b) What is the kinetic energy of the bear just before hitting the ground? (c) What is the average frictional force that acts on the sliding bear?

A 60 kg skier leaves the end of a ski-jump ramp with a velocity of 24 m/s directed 25# above the horizontal. Suppose that as a result of air drag the skier returns to the ground with a speed of 22 m/s, landing 14 m vertically below the end of the ramp. From the launch to the return to the ground, by how much is the mechanical energy of the skierEarth system reduced because of air drag?

During a rockslide, a 520 kg rock slides from rest down a hillside that is 500 m long and 300 m high. The coefficient of kinetic friction between the rock and the hill surface is 0.25. (a) If the gravitational potential energy U of the rockEarth system is zero at the bottom of the hill, what is the value of U just before the slide? (b) How much energy is transferred to thermal energy during the slide? (c) What is the kinetic energy of the rock as it reaches the bottom of the hill? (d) What is its speed then?

A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant k ! 400 N/m; the other end of the spring is fixed in place. The cookie has a kinetic energy of 20.0 J as it passes through the springs equilibrium position. As the cookie slides, a frictional force of magnitude 10.0 N acts on it. (a) How far will the cookie slide from the equilibrium position before coming momentarily to rest? (b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?

In Fig. 8-52, a 3.5 kg block is accelerated from rest by a compressed spring of spring constant 640 N/m. The block leaves the spring at the springs relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction mk ! 0.25. The frictional force stops the block in distance D ! 7.8 m. What are (a) the increase in the thermal energy of the blockfloor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring? No friction D ( k ) Figure 8-52 Problem 53.

A child whose weight is 267 N slides down a 6.1 m playground slide that makes an angle of 20# with the horizontal.The coefficient of kinetic friction between slide and child is 0.10. (a) How much energy is transferred to thermal energy? (b) If she starts at the top with a speed of 0.457 m/s, what is her speed at the bottom?

In Fig. 8-53, a block of mass m ! 2.5 kg slides head on into a spring of spring constant k ! 320 N/m. When the block ILW stops, it has compressed the spring by 7.5 cm. The coefficient of kinetic friction between block and floor is 0.25. While the block is in contact with the spring and being brought to rest, what are (a) the work done by the spring force and (b) the increase in thermal energy of the blockfloor system? (c) What is the blocks speed just as it reaches the spring? 0 Figure 8-53 Problem 55.

You push a 2.0 kg block against a horizontal spring, compressing the spring by 15 cm. Then you release the block, and the spring sends it sliding across a tabletop. It stops 75 cm from where you released it. The spring constant is 200 N/m. What is the blocktable coefficient of kinetic friction?

In Fig. 8-54, a block slides along a track from one level to a higher level after passing through an intermediate valley. The track is frictionless until the block reaches the higher level. There a frictional force stops the block in a distance d. The blocks initial speed v0 is 6.0 m/s, the height difference h is 1.1 m, and mk is 0.60. Find d. h k = 0 v0 d Figure 8-54 Problem 57.

A cookie jar is moving up a 40# incline. At a point 55 cm from the bottom of the incline (measured along the incline), the jar has a speed of 1.4 m/s. The coefficient of kinetic friction between jar and incline is 0.15. (a) How much farther up the incline will the jar move? (b) How fast will it be going when it has slid back to the bottom of the incline? (c) Do the answers to (a) and (b) increase, decrease, or remain the same if we decrease the coefficient of kinetic friction (but do not change the given speed or location)?

A stone with a weight of 5.29 N is launched vertically from ground level with an initial speed of 20.0 m/s, and the air drag on it is 0.265 N throughout the flight. What are (a) the maximum height reached by the stone and (b) its speed just before it hits the ground?

A 4.0 kg bundle starts up a 30# incline with 128 J of kinetic energy. How far will it slide up the incline if the coefficient of kinetic friction between bundle and incline is 0.30?

When a click beetle is upside down on its back, it jumps upward by suddenly arching its back, transferring energy stored in a muscle to mechanical energy.This launching mechanism produces an audible click, giving the beetle its name.Videotape of a certain clickbeetle jump shows that a beetle of mass m ! 4.0 $ 10%6 kg moved directly upward by 0.77 mm during the launch and then to a maximum height of h ! 0.30 m. During the launch, what are the average magnitudes of (a) the external force on the beetles back from the floor and (b) the acceleration of the beetle in terms of g?

In Fig. 8-55, a block slides along a path that is without friction until the block reaches the section of length L 0.75 m, which begins at height h ! 2.0 m on a ramp of angle u ! 30#. In that section, the coefficient of kinetic friction is 0.40. The block passes through point A with a speed of 8.0 m/s. If the block can reach point B (where the friction ends), what is its speed there, and if it cannot, what is its greatest height above A? ! A h L B Figure 8-55 Problem 62.

The cable of the 1800 kg elevator cab in Fig. 8-56 snaps when the cab is at rest at the first floor, where the cab bottom is a distance d ! 3.7 m above a spring of spring constant k ! 0.15 MN/m. A safety device clamps the cab against guide rails so that a constant frictional force of 4.4 kN opposes the cabs motion. (a) Find the speed of the cab just before it hits the spring. (b) Find the maximum distance x that the spring is compressed (the frictional force still acts during this compression). (c) Find the distance that the cab will bounce back up the shaft. (d) Using conservation of energy, find the approximate total distance that the cab will move before coming to rest. (Assume that the frictional force on the cab is negligible when the cab is stationary.) d k Figure 8-56 Problem 63.

In Fig. 8-57, a block is released from rest at height d ! 40 cm and slides down a frictionless ramp and onto a first plateau, which has length d and where the coefficient of kinetic friction is 0.50. If the block is still moving, it then slides down a second frictionless ramp through height d/2 and onto a lower plateau, which has length d/2 and where the coefficient of kinetic friction is again 0.50. If the block is still moving, it then slides up a frictionless ramp until it (momentarily) stops. Where does the block stop? If its final stop is on a plateau, state which one and give the distance L from the left edge of that plateau. If the block reaches the ramp, give the height H above the lower plateau where it momentarily stops. d d/2 d d/2 Figure 8-57 Problem 64

A particle can slide along a track with elevated ends and a flat central part, as shown in Fig. 8-58. The flat part has length L ! 40 cm. The curved portions of the track are frictionless, but for the flat part the coefficient of kinetic friction is mk ! 0.20.The particle is released from rest at point A, which is at height h ! L/2. How far from the left edge of the flat part does the particle finally stop? L h A Figure 8-58 Problem 65.

A 3.2 kg sloth hangs 3.0 m above the ground. (a) What is the gravitational potential energy of the slothEarth system if we take the reference point y 0 to be at the ground? If the sloth drops to the ground and air drag on it is assumed to be negligible, what are the (b) kinetic energy and (c) speed of the sloth just before it reaches the ground?

A spring (k 200 N/m) is fixed at the top of a frictionless plane inclined at angle 40# (Fig. 8-59). A 1.0 kg block is projected up the plane, from an initial position that is distance d 0.60 m from the end of the relaxed spring, with an initial kinetic energy of 16 J. (a) What is the kinetic energy of the block at the instant it has compressed the spring 0.20 m? (b) With what kinetic energy must the block be projected up the plane if it is to stop momentarily when it has compressed the spring by 0.40 m? d Figure 8-59 Problem 67.

From the edge of a cliff, a 0.55 kg projectile is launched with an initial kinetic energy of 1550 J. The projectiles maximum upward displacement from the launch point is &140 m. What are the (a) horizontal and (b) vertical components of its launch velocity? (c) At the instant the vertical component of its velocity is 65 m/s, what is its vertical displacement from the launch point?

In Fig. 8-60, the pulley has negligible mass, and both it and the inclined plane are frictionless. Block A has a mass of 1.0 kg, block B has a mass of 2.0 kg, and angle u is 30#. If the blocks are released from rest with the connecting cord taut, what is their total kinetic energy when block B has fallen 25 cm? A B Figure 8-60 Problem 69.

In Fig. 8-38, the string is L ! 120 cm long, has a ball attached to one end, and is fixed at its other end. A fixed peg is at point P. Released from rest, the ball swings down until the string catches on the peg; then the ball swings up, around the peg. If the ball is to swing completely around the peg, what value must distance d exceed? (Hint: The ball must still be moving at the top of its swing. Do you see why?)

In Fig. 8-51, a block is sent sliding down a frictionless ramp. Its speeds at points A and B are 2.00 m/s and 2.60 m/s, respectively. Next, it is again sent sliding down the ramp, but this time its speed at point A is 4.00 m/s. What then is its speed at point B?

Two snowy peaks are at heights H 850 m and h 750 m above the valley between them. A ski run extends between the peaks, with a total length of 3.2 km and an average slope of 30# (Fig. 8-61). (a) A skier starts from rest at the top of the higher peak. At what speed will he arrive at the top of the lower peak if he coasts without using ski poles? Ignore friction. (b) Approximately what coefficient of kinetic friction between snow and skis would make him stop just at the top of the lower peak? h H Figure 8-61 Problem 72.

The temperature of a plastic cube is monitored while the cube is pushed 3.0 m across a floor at constant speed by a horizontal force of 15 N. The thermal energy of the cube increases by 20 J. What is the increase in the thermal energy of the floor along which the cube slides?

A skier weighing 600 N goes over a frictionless circular hill of radius R 20 m (Fig. 8-62). Assume that the effects of air resistance on the skier are negligible. As she comes up the hill, her speed is 8.0 m/s at point B, at angle u ! 20#. (a) What is her speed at the hilltop (point A) if she coasts without using her poles? (b) What minimum speed can she have at B and still coast to the hilltop? (c) Do the answers to these two questions increase, decrease, or remain the same if the skier weighs 700 N instead of 600 N? R A B Figure 8-62 Problem 74.

To form a pendulum, a 0.092 kg ball is attached to one end of a rod of length 0.62 m and negligible mass, and the other end of the rod is mounted on a pivot. The rod is rotated until it is straight up, and then it is released from rest so that it swings down around the pivot. When the ball reaches its lowest point, what are (a) its speed and (b) the tension in the rod? Next, the rod is rotated until it is horizontal, and then it is again released from rest. (c) At what angle from the vertical does the tension in the rod equal the weight of the ball? (d) If the mass of the ball is increased, does the answer to (c) increase, decrease, or remain the same?

We move a particle along an x axis, first outward from x 1.0 m to x 4.0 m and then back to x 1.0 m, while an external force acts on it. That force is directed along the x axis, and its x component can have different values for the outward trip and for the return trip. Here are the values (in newtons) for four situations, where x is in meters: Outward Inward (a) &3.0 %3.0 (b) &5.0 &5.0 (c) &2.0x %2.0x (d) &3.0x2 &3.0x2 Find the net work done on the particle by the external force for the round trip for each of the four situations. (e) For which, if any, is the external force conservative?

A conservative force F(x) acts on a 2.0 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is graphed in Fig. 8-63. When the particle is at x ! 2.0 m, its velocity is %1.5 m/s. What are the (a) magnitude and (b) direction of F(x) at this position? Between what positions on the (c) left and (d) right does the particle move? (e) What is the particles speed at x ! 7.0 m? 0 5 10 15 20 U(x) ( J ) 5 x (m) 0 10 15 Figure 8-63 Problem 77.

At a certain factory, 300 kg crates are dropped vertically from a packing machine onto a conveyor belt moving at 1.20 m/s (Fig. 8-64). (A motor maintains the belts constant speed.) The coefficient of kinetic friction between the belt and each crate is 0.400. After a short time, slipping between the belt and the crate ceases, and the crate then moves along with the belt. For the period of time during which the crate is being brought to rest relative to the belt, calculate, for a coordinate system at rest in the factory, (a) the kinetic energy supplied to the crate, (b) the magnitude of the kinetic frictional force acting on the crate, and (c) the energy supplied by the motor. (d) Explain why answers (a) and (c) differ. Figure 8-64 Problem 78.

A 1500 kg car begins sliding down a 5.0# inclined road with a speed of 30 km/h. The engine is turned off, and the only forces acting on the car are a net frictional force from the road and the gravitational force. After the car has traveled 50 m along the road, its speed is 40 km/h. (a) How much is the mechanical energy of the car reduced because of the net frictional force? (b) What is the magnitude of that net frictional force?

In Fig. 8-65, a 1400 kg block of granite is pulled up an incline at a constant speed of 1.34 m/s by a cable and winch. The indicated distances are d1 40 m and d2 30 m. The coefficient of kinetic friction between the block and the incline is 0.40. What is the power due to the force applied to the block by the cable? ! ! SSM d2 d1 Figure 8-65 Problem 80

A particle can move along only an x axis, where conservative forces act on it (Fig. 8-66 and the following table). The particle is released at x 5.00 m with a kinetic energy of K 14.0 J and a potential energy of U 0. If its motion is in the negative direction of the x axis, what are its (a) K and (b) U at x 2.00 m and its (c) K and (d) U at x 0? If its motion is in the positive direction of the x axis, what are its (e) K and (f) U at x 11.0 m, its (g) K and (h) U at x 12.0 m, and its (i) K and ( j) U at x 13.0 m? (k) Plot U(x) versus x for the range x 0 to x !! 13.0 m. ! ! 0 2 4 6 8 10 12 F1 F2 F3 F4 x (m) Figure 8-66 Problems 81 and 82 Next, the particle is released from rest at x 0. What are (l) its kinetic energy at x 5.0 m and (m) the maximum positive position xmax it reaches? (n) What does the particle do after it reaches xmax? Range Force0 to 2.00 m2.00 m to 3.00 m3.00 m to 8.00 m F 08.00 m to 11.0 m11.0 m to 12.0 m12.0 m to 15.0 m F ! 0F:4 ! %(1.00 N)iF:3 ! %(4.00 N)i!F:2 ! &(5.00 N)iF:1 ! &(3.00 N)i

For the arrangement of forces in Problem 81, a 2.00 kg particle is released at x 5.00 m with an initial velocity of 3.45 m/s in the negative direction of the x axis. (a) If the particle can reach x 0 m, what is its speed there, and if it cannot, what is its turning point? Suppose, instead, the particle is headed in the positive x direction when it is released at x 5.00 m at speed 3.45 m/s. (b) If the particle can reach x 13.0 m, what is its speed there, and if it cannot, what is its turning point?

A 15 kg block is accelerated at 2.0 m/s2 along a horizontal frictionless surface, with the speed increasing from 10 m/s to 30 m/s. What are (a) the change in the blocks mechanical energy and (b) the average rate at which energy is transferred to the block? What is the instantaneous rate of that transfer when the blocks speed is (c) 10 m/s and (d) 30 m/s?

A certain spring is found not to conform to Hookes law. The force (in newtons) it exerts when stretched a distance x (in meters) is found to have magnitude 52.8x & 38.4x2 in the direction opposing the stretch. (a) Compute the work required to stretch the spring from x 0.500 m to x 1.00 m. (b) With one end of the spring fixed, a particle of mass 2.17 kg is attached to the other end of the spring when it is stretched by an amount x 1.00 m. If the particle is then released from rest, what is its speed at the instant the stretch in the spring is x 0.500 m? (c) Is the force exerted by the spring conservative or nonconservative? Explain.

Each second, 1200 m3 of water passes over a waterfall 100 m high. Three-fourths of the kinetic energy gained by the water in falling is transferred to electrical energy by a hydroelectric generator. At what rate does the generator produce electrical energy? (The mass of 1 m3 of water is 1000 kg.)

In Fig. 8-67, a small block is sent through point A with a speed of 7.0 m/s. Its path is without friction until it reaches the section of length L 12 m, where the coefficient of kinetic friction is 0.70. The indicated heights are h1 6.0 m and h2 2.0 m. What are the speeds of the block at (a) point B and (b) point C? (c) Does the block reach point D? If so, what is its speed there; if not, how far through the section of friction does it travel? ! ! ! 91 Two blocks, of masses M !2.0 kg and 2M, are connected to A B h1 C D h2 L Figure 8-67 Problem 86.

A massless rigid rod of length L has a ball of mass m attached to one end (Fig. 8-68). The other end is pivoted in such a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot. The system is launched downward from the horizontal position A with initial speed v0. The ball just barely reaches point D and then stops. (a) Derive an expression for v0 in terms of L, m, and g. (b) What is the tension in the rod when the ball passes through B? (c) A little grit is placed on the pivot to increase the friction there. Then the ball just barely reaches C when launched from A with the same speed as before. What is the decrease in the mechanical energy during this motion? (d) What is the decrease in the mechanical energy by the time the ball finally comes to rest at B after several oscillations? D B A L C Pivot point Rod Figure 8-68 Problem 87.

A 1.50 kg water balloon is shot straight up with an initial speed of 3.00 m/s. (a) What is the kinetic energy of the balloon just as it is launched? (b) How much work does the gravitational force do on the balloon during the balloons full ascent? (c) What is the change in the gravitational potential energy of the balloonEarth system during the full ascent? (d) If the gravitational potential energy is taken to be zero at the launch point, what is its value when the balloon reaches its maximum height? (e) If, instead, the gravitational potential energy is taken to be zero at the maximum height, what is its value at the launch point? (f) What is the maximum height?

A 2.50 kg beverage can is thrown directly downward from a height of 4.00 m, with an initial speed of 3.00 m/s. The air drag on the can is negligible. What is the kinetic energy of the can (a) as it reaches the ground at the end of its fall and (b) when it is halfway to the ground? What are (c) the kinetic energy of the can and (d) the gravitational potential energy of the canEarth system 0.200 s before the can reaches the ground? For the latter, take the reference point y ! 0 to be at the ground

A constant horizontal force moves a 50 kg trunk 6.0 m up a 30# incline at constant speed. The coefficient of kinetic friction is 0.20. What are (a) the work done by the applied force and (b) the increase in the thermal energy of the trunk and incline?

Two blocks, of masses M !2.0 kg and 2M, are connected to a spring of spring constant k ! 200 N/m that has one end fixed, as shown in Fig. 8-69. The horizontal surface and the pulley are frictionless, and the pulley hasnegligible mass. The blocks are released from rest with the spring relaxed. (a) What is the combined kinetic energy of the two blocks when the hanging block has fallen 0.090 m? (b) What is the kinetic energy of the hanging block when it has fallen that 0.090 m? (c) What maximum distance does the hanging block fall before momentarily stopping? 2M M Figure 8-69 Problem 91

A volcanic ash flow is moving across horizontal ground when it encounters a 10# upslope. The front of the flow then travels 920 m up the slope before stopping. Assume that the gases entrapped in the flow lift the flow and thus make the frictional force from the ground negligible; assume also that the mechanical energy of the front of the flow is conserved.What was the initial speed of the front of the flow?

A playground slide is in the form of an arc of a circle that has a radius of 12 m.The maximum height of the slide is h ! 4.0 m, and the ground is tangent to the circle (Fig. 8-70). A 25 kg child starts from rest at the top of the slide and has a speed of 6.2 m/s at the bottom. (a) What is the length of the slide? (b) What average frictional force acts on the child over this distance? If, instead of the ground, a vertical line through the top of the slide is tangent to the circle, what are (c) the length of the slide and (d) the average frictional force on the child? h Figure 8-70 Problem 93.

The luxury liner Queen Elizabeth 2 has a diesel-electric power plant with a maximum power of 92 MW at a cruising speed of 32.5 knots. What forward force is exerted on the ship at this speed? (1 knot ! 1.852 km/h.)

A factory worker accidentally releases a 180 kg crate that was being held at rest at the top of a ramp that is 3.7 m long and inclined at 39# to the horizontal.The coefficient of kinetic friction between the crate and the ramp, and between the crate and the horizontal factory floor, is 0.28. (a) How fast is the crate moving as it reaches the bottom of the ramp? (b) How far will it subsequently slide across the floor? (Assume that the crates kinetic energy does not change as it moves from the ramp onto the floor.) (c) Do the answers to (a) and (b) increase, decrease, or remain the same if we halve the mass of the crate?

If a 70 kg baseball player steals home by sliding into the plate with an initial speed of 10 m/s just as he hits the ground, (a) what is the decrease in the players kinetic energy and (b) what is the increase in the thermal energy of his body and the ground along which he slides?

A 0.50 kg banana is thrown directly upward with an initial speed of 4.00 m/s and reaches a maximum height of 0.80 m. What change does air drag cause in the mechanical energy of the bananaEarth system during the ascent?

A metal tool is sharpened by being held against the rim of a wheel on a grinding machine by a force of 180 N. The frictional forces between the rim and the tool grind off small pieces of the tool. The wheel has a radius of 20.0 cm and rotates at 2.50 rev/s. The coefficient of kinetic friction between the wheel and the tool is 0.320. At what rate is energy being transferred from the motor driving the wheel to the thermal energy of the wheel and tool and to the kinetic energy of the material thrown from the tool?

A swimmer moves through the water at an average speed of 0.22 m/s. The average drag force is 110 N. What average power is required of the swimmer?

An automobile with passengers has weight 16 400 N and is moving at 113 km/h when the driver brakes, sliding to a stop. The frictional force on the wheels from the road has a magnitude of 8230 N. Find the stopping distance.

A 0.63 kg ball thrown directly upward with an initial speed of 14 m/s reaches a maximum height of 8.1 m. What is the change in the mechanical energy of the ballEarth system during the ascent of the ball to that maximum height?

The summit of Mount Everest is 8850 m above sea level. (a) How much energy would a 90 kg climber expend against the gravitational force on him in climbing to the summit from sea level? (b) How many candy bars, at 1.25 MJ per bar, would supply an energy equivalent to this? Your answer should suggest that work done against the gravitational force is a very small part of the energy expended in climbing a mountain

A sprinter who weighs 670 N runs the first 7.0 m of a race in 1.6 s, starting from rest and accelerating uniformly. What are the sprinters (a) speed and (b) kinetic energy at the end of the 1.6 s? (c) What average power does the sprinter generate during the 1.6 s interval?

A 20 kg object is acted on by a conservative force given by F ! %3.0x % 5.0x2 , with F in newtons and x in meters. Take the potential energy associated with the force to be zero when the object is at x ! 0. (a) What is the potential energy of the system associated with the force when the object is at x ! 2.0 m? (b) If the object has a velocity of 4.0 m/s in the negative direction of the x axis when it is at x ! 5.0 m, what is its speed when it passes through the origin? (c) What are the answers to (a) and (b) if the potential energy of the system is taken to be %8.0 J when the object is at x ! 0?

A machine pulls a 40 kg trunk 2.0 m up a 40# ramp at constant velocity, with the machines force on the trunk directed parallel to the ramp. The coefficient of kinetic friction between the trunk and the ramp is 0.40. What are (a) the work done on the trunk by the machines force and (b) the increase in thermal energy of the trunk and the ramp?

The spring in the muzzle of a childs spring gun has a spring constant of 700 N/m. To shoot a ball from the gun, first the spring is compressed and then the ball is placed on it. The guns trigger then releases the spring, which pushes the ball through the muzzle. The ball leaves the spring just as it leaves the outer end of the muzzle. When the gun is inclined upward by 30# to the horizontal, a 57 g ball is shot to a maximum height of 1.83 m above the guns muzzle. Assume air drag on the ball is negligible. (a) At what speed does the spring launch the ball? (b) Assuming that friction on the ball within the gun can be neglected, find the springs initial compression distance.

The only force acting on a particle is conservative force . If the particle is at point A, the potential energy of the system associated with and the particle is 40 J. If the particle moves from point A to point B, the work done on the particle by is &25 J. What is the potential energy of the system with the particle at B?

In 1981, Daniel Goodwin climbed 443 m up the exterior of the Sears Building in Chicago using suction cups and metal clips. (a) Approximate his mass and then compute how much energy he had to transfer from biomechanical (internal) energy to the gravitational potential energy of the EarthGoodwin system to lift himself to that height. (b) How much energy would he have had to transfer if he had, instead, taken the stairs inside the building (to the same height)?

A 60.0 kg circus performer slides 4.00 m down a pole to the circus floor, starting from rest. What is the kinetic energy of the performer as she reaches the floor if the frictional force on her from the pole (a) is negligible (she will be hurt) and (b) has a magnitude of 500 N?

A 5.0 kg block is projected at 5.0 m/s up a plane that is inclined at 30# with the horizontal. How far up along the plane does the block go (a) if the plane is frictionless and (b) if the coefficient of kinetic friction between the block and the plane is 0.40? (c) In the latter case, what is the increase in thermal energy of block and plane during the blocks ascent? (d) If the block then slides back down against the frictional force, what is the blocks speed when it reaches the original projection point?

A 9.40 kg projectile is fired vertically upward. Air drag decreases the mechanical energy of the projectileEarth system by 68.0 kJ during the projectiles ascent. How much higher would the projectile have gone were air drag negligible?

A 70.0 kg man jumping from a window lands in an elevated fire rescue net 11.0 m below the window. He momentarily stops when he has stretched the net by 1.50 m. Assuming that mechanical energy is conserved during this process and that the net functions like an ideal spring, find the elastic potential energy of the net when it is stretched by 1.50 m.

A 30 g bullet moving a horizontal velocity of 500 m/s comes to a stop 12 cm within a solid wall. (a) What is the change in the bullets mechanical energy? (b) What is the magnitude of the average force from the wall stopping it?

A 1500 kg car starts from rest on a horizontal road and gains a speed of 72 km/h in 30 s. (a) What is its kinetic energy at the end of the 30 s? (b) What is the average power required of the car during the 30 s interval? (c) What is the instantaneous power at the end of the 30 s interval, assuming that the acceleration is constant?

A 1.50 kg snowball is shot upward at an angle of 34.0# to the horizontal with an initial speed of 20.0 m/s. (a) What is its initial kinetic energy? (b) By how much does the gravitational potential energy of the snowballEarth system change as the snowball moves from the launch point to the point of maximum height? (c) What is that maximum height?

A 68 kg sky diver falls at a constant terminal speed of 59 m/s. (a) At what rate is the gravitational potential energy of the Earthsky diver system being reduced? (b) At what rate is the systems mechanical energy being reduced?

A 20 kg block on a horizontal surface is attached to a horizontal spring of spring constant k ! 4.0 kN/m. The block is pulled to the right so that the spring is stretched 10 cm beyond its relaxed length, and the block is then released from rest.The frictional force between the sliding block and the surface has a magnitude of 80 N. (a) What is the kinetic energy of the block when it has moved 2.0 cm from its point of release? (b) What is the kinetic energy of the block when it first slides back through the point at which the spring is relaxed? (c) What is the maximum kinetic energy attained by the block as it slides from its point of release to the point at which the spring is relaxed?

Resistance to the motion of an automobile consists of road friction, which is almost independent of speed, and air drag, which is proportional to speed-squared. For a certain car with a weight of 12 000 N, the total resistant force F is given by F ! 300 & 1.8v2 , with F in newtons and v in meters per second. Calculate the power 212 CHAPTER 8 POTENTIAL ENERGY AND CONSERVATION OF ENERGY (in horsepower) required to accelerate the car at 0.92 m/s2 when the speed is 80 km/h.

A 50 g ball is thrown from a window with an initial velocity of 8.0 m/s at an angle of 30# above the horizontal. Using energy methods, determine (a) the kinetic energy of the ball at the top of its flight and (b) its speed when it is 3.0 m below the window. Does the answer to (b) depend on either (c) the mass of the ball or (d) the initial angle?

A spring with a spring constant of 3200 N/m is initially stretched until the elastic potential energy of the spring is 1.44 J. (U ! 0 for the relaxed spring.) What is "U if the initial stretch is changed to (a) a stretch of 2.0 cm, (b) a compression of 2.0 cm, and (c) a compression of 4.0 cm?

A locomotive with a power capability of 1.5 MW can accelerate a train from a speed of 10 m/s to 25 m/s in 6.0 min. (a) Calculate the mass of the train. Find (b) the speed of the train and (c) the force accelerating the train as functions of time (in seconds) during the 6.0 min interval. (d) Find the distance moved by the train during the interval

A 0.42 kg shuffleboard disk is initially at rest when a player uses a cue to increase its speed to 4.2 m/s at constant acceleration. The acceleration takes place over a 2.0 m distance, at the end of which the cue loses contact with the disk. Then the disk slides an additional 12 m before stopping. Assume that the shuffleboard court is level and that the force of friction on the disk is constant. What is the increase in the thermal energy of the diskcourt system (a) for that additional 12 m and (b) for the entire 14 m distance? (c) How much work is done on the disk by the cue?

A river descends 15 m through rapids. The speed of the water is 3.2 m/s upon entering the rapids and 13 m/s upon leaving. What percentage of the gravitational potential energy of the waterEarth system is transferred to kinetic energy during the descent? (Hint: Consider the descent of, say, 10 kg of water.)

The magnitude of the gravitational force between a particle of mass m1 and one of mass m2 is given by F(x) G m1m2 where G is a constant and x is the distance between the particles. (a) What is the corresponding potential energy function U(x)? Assume that U(x) 0 as x and that x is positive. (b) How much work is required to increase the separation of the particles from x ! x1 to x ! x1 & d? x2

Approximately 5.5 $ 106 kg of water falls 50 m over Niagara Falls each second. (a) What is the decrease in the gravitational potential energy of the waterEarth system each second? (b) If all this energy could be converted to electrical energy (it cannot be), at what rate would electrical energy be supplied? (The mass of 1 m3 of water is 1000 kg.) (c) If the electrical energy were sold at 1 cent/kW h, what would be the yearly income?

To make a pendulum, a 300 g ball is attached to one end of a string that has a length of 1.4 m and negligible mass. (The other end of the string is fixed.) The ball is pulled to one side until the string makes an angle of 30.0# with the vertical; then (with the string taut) the ball is released from rest. Find (a) the speed of the ball when the string makes an angle of 20.0# with the vertical and (b) the maximum speed of the ball. (c) What is the angle between the string and the vertical when the speed of the ball is one-third its maximum value?

In a circus act, a 60 kg clown is shot from a cannon with an initial velocity of 16 m/s at some unknown angle above the horizontal. A short time later the clown lands in a net that is 3.9 m vertically above the clowns initial position. Disregard air drag. What is the kinetic energy of the clown as he lands in the net?

A 70 kg firefighter slides, from rest, 4.3 m down a vertical pole. (a) If the firefighter holds onto the pole lightly, so that the frictional force of the pole on her is negligible, what is her speed just before reaching the ground floor? (b) If the firefighter grasps the pole more firmly as she slides, so that the average frictional force of the pole on her is 500 N upward, what is her speed just before reaching the ground floor?

The surface of the continental United States has an area of about 8 $ 106 km2 and an average elevation of about 500 m (above sea level). The average yearly rainfall is 75 cm. The fraction of this rainwater that returns to the atmosphere by evaporation is ; the rest eventually flows into the ocean. If the decrease in gravitational potential energy of the waterEarth system associated with that flow could be fully converted to electrical energy, what would be the average power? (The mass of 1 m3 of water is 1000 kg.)

A spring with spring constant k ! 200 N/m is suspended vertically with its upper end fixed to the ceiling and its lower end at position y ! 0. A block of weight 20 N is attached to the lower end, held still for a moment, and then released. What are (a) the kinetic energy K, (b) the change (from the initial value) in the gravitational potential energy "Ug, and (c) the change in the elastic potential energy "Ue of the springblock system when the block is at y ! %5.0 cm? What are (d) K, (e) "Ug, and (f) "Ue when y ! %10 cm, (g) K, (h) "Ug, and (i) "Ue when y ! %15 cm, and (j) K, (k) "Ug, and (l) "Ue when y ! %20 cm?

Fasten one end of a vertical spring to a ceiling, attach a cabbage to the other end, and then slowly lower the cabbage until the upward force on it from the spring balances the gravitational force on it. Show that the loss of gravitational potential energy of the cabbageEarth system equals twice the gain in the springs potential energy.

The maximum force you can exert on an object with one of your back teeth is about 750 N. Suppose that as you gradually bite on a clump of licorice, the licorice resists compression by one of your teeth by acting like a spring for which k ! 2.5 $ 105 N/m. Find (a) the distance the licorice is compressed by your tooth and (b) the work the tooth does on the licorice during the compression. (c) Plot the magnitude of your force versus the compression distance. (d) If there is a potential energy associated with this compression, plot it versus compression distance. In the 1990s the pelvis of a particular Triceratops dinosaur was found to have deep bite marks. The shape of the marks suggested that they were made by a Tyrannosaurus rex dinosaur. To test the idea, researchers made a replica of a T. rex tooth from bronze and aluminum and then used a hydraulic press to gradually drive the replica into cow bone to the depth seen in the Triceratops bone. A graph of the force required versus depth of penetration is given in Fig. 8-71 for one trial; the required force increased with depth because, as the nearly conical tooth penetrated the bone, more of the tooth came in contact with the bone. (e) How much work was done by the hydraulic pressand thus presumably by the T. rexin such a penetration? (f) Is there a potential energy associated with this penetration? (The large biting force and energy expenditure 0 2000 4000 6000 8000 1 2 3 4 5 6 Penetration depth (mm) 7 8 9 10 11 12 Force (N) Figure 8-71 Problem 132 attributed to the T. rex by this research suggest that the animal was a predator and not a scavenger.)

Conservative force F(x) acts on a particle that moves along an x axis. Figure 8-72 shows how the potential energy U(x) associated with force F(x) varies with the position of the particle, (a) Plot F(x) for the range 0 + x + 6 m. (b) The mechanical energy E of the system is 4.0 J. Plot the kinetic energy K(x) of the particle directly on Fig. 8-72. U(x) ( J ) 3 x (m) 10 4 5 6 4 3 2 1 2 Figure 8-72 Problem 133.

Figure 8-73a shows a molecule consisting of two atoms of masses m and M (with m M) and separation r. Figure 8-73b shows the potential energy U(r) of the molecule as a function of r. Describe the motion of the atoms (a) if the total mechanical energy E of the two-atom system is greater than zero (as is E1), and (b) if E is less than zero (as is E2). For E1 ! 1 $ 10%19 J and r ! 0.3 nm, find (c) the potential energy of the system, (d) the total kinetic energy of the atoms, and (e) the force (magnitude and direction) acting on each atom. For what values of r is the force (f) repulsive, (g) attractive, and (h) zero? U(r), E (1019 J) 3 2 1 0 1 2 3 0.2 r (nm) (b) E1 E2 r m M 0.10 0.3 0.4 (a) Figure 8-73 Problem 134

Repeat Problem 83, but now with the block accelerated up a frictionless plane inclined at 5.0# to the horizontal.

A spring with spring constant k ! 620 N/m is placed in a vertical orientation with its lower end supported by a horizontal surface. The upper end is depressed 25 cm, and a block with a weight of 50 N is placed (unattached) on the depressed spring. The system is then released from rest. Assume that the gravitational potential energy Ug of the block is zero at the release point (y ! 0) and calculate the kinetic energy K of the block for y equal to (a) 0, (b) 0.050 m, (c) 0.10 m, (d) 0.15 m, and (e) 0.20 m. Also, (f) how far above its point of release does the block rise?