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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 5

Energy

Educators


Problem 1

A weight lifter lifts a $350-\mathrm{N}$ set of weights from ground level to a position over his head, a vertical distance of 2.00 $\mathrm{m} .$ How much work does the weight lifter do, assuming he moves the weights at constant speed?

Averell H.
Carnegie Mellon University

Problem 2

In 1990 walter Arfeuille of Belgium lifted a 281.5 -kg object through a distance of 17.1 $\mathrm{cm}$ using only his teeth. (a) How much work did Arfeuille do on the object? (b) What magnitude force did he exert on the object during the lift, assuming the force was constant?

Averell H.
Carnegie Mellon University

Problem 3

A cable exerts a constant upward tension of magnitude $1.25 \times 10^{4} \mathrm{N}$ on a $1.00 \times 10^{3}$ -kg elevator as it rises through a vertical distance of 2.00 $\mathrm{m} .$ (a) Find the work done by the tension force on the elevator. (b) Find the work done by the force of gravity on the elevator.

Averell H.
Carnegie Mellon University

Problem 4

A shopper in a supermarket pushes a cart with a force of 35 $\mathrm{N}$ directed at an angle of $25^{\circ}$ below the horizontal. The force is just sufficient to overcome various frictional forces, so the cart moves at constant speed. (a) Find the work done by the shopper as she moves down a $50.0-\mathrm{m}$ length aisle. (b) What is the net work done on the cart? Why? (c) The shopper goes down the next aisle, pushing horizontally and maintaining the same speed as before. If the work done by frictional forces doesn't change, would the shopper's applied force be larger, smaller, or the same? What about the work done on the cart by the shopper?

Averell H.
Carnegie Mellon University

Problem 5

Starting from rest, a $5.00-\mathrm{kg}$ block slides 2.50 $\mathrm{m}$ down a rough $30.0^{\circ}$ incline. The coefficient of kinetic friction between the block and the incline is $\mu_{k}=0.436$ . Determine (a) the work done by the force of gravity, (b) the work done by the friction force between block and incline, and (c) the work done by the normal force. (d) Qualitatively, how would the answers change if a shorter ramp at a steeper angle were used to span the same vertical height?

Averell H.
Carnegie Mellon University

Problem 6

A horizontal force of 150 $\mathrm{N}$ is used to push a $40.0-\mathrm{kg}$ packing crate a distance of 6.00 $\mathrm{m}$ on a rough horizontal surface. If the crate moves at constant speed, find (a) the work done by the 150 -N force and (b) the coefficient of kinetic friction between the crate and surface.

Averell H.
Carnegie Mellon University

Problem 7

A tension force of 175 $\mathrm{N}$ inclined at $20.0^{\circ}$ above the horizontal is used to pull a 40.0 -kg packing crate a distance of 6.00 $\mathrm{m}$ on a rough surface. If the crate moves at a constant speed, find (a) the work done by the tension force and (b) the coefficient of kinetic friction between the crate and surface.

Yaqub K.
Numerade Educator

Problem 8

A block of mass $m=$ 2.50 $\mathrm{kg}$ is pushed a distance $d=2.20 \mathrm{m}$ along a frictionless horizontal table by a constant applied force of magnitude $F=16.0 \mathrm{N}$ directed at an angle $\theta=25.0^{\circ}$ below the horizontal as shown in Figure P5.8.Determine the work done by (a) the applied force, (b) the normal force exerted by the table, (c) the force of gravity, and (d) the net force on the block.

Keshav S.
Numerade Educator

Problem 9

A mechanic pushes a $2.50 \times 10^{3}$ -kg car from rest to a speed of $v$ doing $5.00 \times 10^{3} \mathrm{J}$ of work in the process. During this time, the car moves 25.0 $\mathrm{m}$ . Neglecting friction between car and road, find (a) $v$ and (b) the horizontal force exerted on the car.

Averell H.
Carnegie Mellon University

Problem 10

A $7.00-\mathrm{kg}$ bowling ball moves at 3.00 $\mathrm{m} / \mathrm{s}$ . How fast must a $2.45-\mathrm{g}$ Ping-Pong ball move so that the two balls have the same kinetic energy?

Averell H.
Carnegie Mellon University

Problem 11

A 65.0 -kg runner has a speed of 5.20 $\mathrm{m} / \mathrm{s}$ at one instant during a long-distance event. (a) What is the runner's kinetic energy at this instant? (b) How much net work is required to double his speed?

Averell H.
Carnegie Mellon University

Problem 12

A worker pushing a 35.0 -kg wooden crate at a constant speed for 12.0 $\mathrm{m}$ along a wood floor does 350 $\mathrm{J}$ of work by applying a constant horizontal force of magnitude $F_{0}$ on the crate. (a) Determine the value of $F_{0}$ . (b) If the worker now applies a force greater than $F_{0},$ describe the subsequent motion of the crate. (c) Describe what would happen to the crate if the applied force is less than $F_{0}$ .

Averell H.
Carnegie Mellon University

Problem 13

A 70 -kg base runner begins his slide into second base when he is moving at a speed of 4.0 $\mathrm{m} / \mathrm{s}$ . The coefficient of friction between his clothes and Earth is $0.70 .$ He slides so that his speed is zero just as he reaches the base. (a) How much mechanical energy is lost due to friction acting on the runner? (b) How far does he slide?

Averell H.
Carnegie Mellon University

Problem 14

A 62.0 -kg cheetah accelerates from rest to its top speed of 32.0 $\mathrm{m} / \mathrm{s}$ . (a) How much net work is required for the cheetah to reach its top speed? (b) One food Calorie equals 4186 $\mathrm{J}$ . How many Calories of net work are required for the cheetah to reach its top speed? Note: Due to inefficiencies in converting chemical energy to mechanical energy, the amount calculated here is only a fraction of the power that must be produced by the cheetah’s body.

Averell H.
Carnegie Mellon University

Problem 15

A 7.80 -g bullet moving at 575 $\mathrm{m} / \mathrm{s}$ penetrates a tree trunk to a depth of 5.50 $\mathrm{cm} .$ (a) Use work and energy considerations to find the average frictional force that stops the bullet. (b) Assuming the frictional force is constant, determine how much time elapses between the moment the bullet enters the tree and the moment it stops moving.

Averell H.
Carnegie Mellon University

Problem 16

A 0.60 -kg particle has a speed of 2.0 $\mathrm{m} / \mathrm{s}$ at point $A$ and a kinetic energy of 7.5 $\mathrm{J}$ at point $B$ . What is (a) its kinetic energy at $A^{2}$ (b) Its speed at point $B ?(c)$ The total work done on the particle as it moves from $A$ to $B ?$

Averell H.
Carnegie Mellon University

Problem 17

A large cruise ship of mass $6.50 \times 10^{7} \mathrm{kg}$ has a speed of 12.0 $\mathrm{m} / \mathrm{s}$ at some instant. (a) What is the ship's kinetic energy at this time? (b) How much work is required to stop it? (c) What is the magnitude of the constant force required to stop it as it undergoes a displacement of 2.50 $\mathrm{km}$ ?

Averell H.
Carnegie Mellon University

Problem 18

A man pushing a crate of mass $m=92.0 \mathrm{~kg}$ at a speed of $v=0.850 \mathrm{~m} / \mathrm{~s}$ encounters a rough horizontal surface of length
$\ell=0.65 \mathrm{~m}$ as in Figure $\mathrm{P} 5.18 .$ If the coefficient of kinetic friction between the crate and rough surface is 0.358 and
he exerts a constant horizontal force of $275 \mathrm{N}$ on the crate, find
(a) the magnitude and direction of the net force on the crate
while it is on the rough surface,
(b) the net work done on the
crate while it is on the rough surface, and
(c) the speed of the
crate when it reaches the end of the rough surface.

Averell H.
Carnegie Mellon University

Problem 19

A $0.20-\mathrm{kg}$ stone is held 1.9 $\mathrm{m}$ above the top edge of a water well and then dropped into it. The well has a depth of 5.0 $\mathrm{m} .$ Taking $y=0$ at the top edge of the well, what is the gravitational potential energy of the stone-Earth system (a) before the stone is released and (b) when it reaches the bottom of the well. (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well?

Guilherme B.
Numerade Educator

Problem 20

When a 2.50 -kg object is hung vertically on a certain light spring described by Hooke's law, the spring stretches 2.76 $\mathrm{cm} .$ (a) What is the force constant of the spring? (b) If the $2.50-\mathrm{kg}$ . object is removed, how far will the spring stretch if a $1.25-\mathrm{kg}$ block is hung on it? (c) How much work must an external agent do to stretch the same spring 8.00 $\mathrm{cm}$ from its unstretched position?

Averell H.
Carnegie Mellon University

Problem 21

A block of mass 3.00 $\mathrm{kg}$ is placed against a horizontal spring of constant $k=875 \mathrm{N} / \mathrm{m}$ and pushed so the spring compresses by 0.0700 $\mathrm{m} .$ (a) What is the elastic potential energy of the block-spring system? (b) If the block is now released and the surface is frictionless, calculate the block's speed after leaving the spring.

Keshav S.
Numerade Educator

Problem 22

A 60.0 -kg athlete leaps straight up into the air from a trampoline with an initial speed of 9.0 $\mathrm{m} / \mathrm{s}$ . The goal of this problem is to find the maximum height she attains and her speed at half maximum height. (a) What are the interacting objects and how do they interact? (b) Select the height at which the athlete's speed is 9.0 $\mathrm{m} / \mathrm{s}$ as $y=0 .$ What is her kinetic energy at this point? What is the gravitational potential energy associated with the athlete? (d) Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer. (e) Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.

Averell H.
Carnegie Mellon University

Problem 23

A $2.10 \times 10^{3}-$ kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5.00 $\mathrm{m}$ before coming into contact with the top of the beam, and it drives the beam 12.0 $\mathrm{cm}$ farther into the ground as it comes to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

Averell H.
Carnegie Mellon University

Problem 24

Two blocks are connected by a light string that passes over two frictionless pulleys as in Figure P5.24. The block of mass $m_{2}$ is attached to a spring of force constant $k$ and $m_{1}>m_{2}$ If the system is released from rest, and the spring is initially not stretched or compressed, find an expres- sion for the maximum displacement $d$ of $m_{2}$

Averell H.
Carnegie Mellon University

Problem 25

A daredevil on a motorcycle leaves the end of a ramp with a speed of 35.0 $\mathrm{m} / \mathrm{s}$ as in Figure $\mathrm{P} 5.25 .$ If his speed is 33.0 $\mathrm{m} / \mathrm{s}$ when he reaches the peak of the path, what is the maximum height that he reaches? Ignore friction and air resistance.

Averell H.
Carnegie Mellon University

Problem 26

Truck suspensions often have "helper springs" that engage at high loads. One such arrangement is a leaf spring with a helper coil spring mounted on the axle, as shown in Figure P5.26. When the main leaf spring is compressed by distance $y_{0},$ the helper spring engages and then helps to support any additional load. Suppose the leaf spring constant is $5.25 \times 10^{5} \mathrm{N} / \mathrm{m}$ , the helper spring constant is $3.60 \times 10^{5} \mathrm{N} / \mathrm{m},$ and $y_{0}=0.500 \mathrm{m} .$ (a) What is the compression of the leaf spring for a load of $5.00 \times 10^{5} \mathrm{N}$ ? (b) How much work is done in compressing the springs?

Averell H.
Carnegie Mellon University

Problem 27

The chin-up is one exercise that can be used to strengthen the biceps muscle. This muscle can exert a force of approximately $8.00 \times 10^{2} \mathrm{N}$ as it contracts a distance of 7.5 $\mathrm{cm}$ in a $75-\mathrm{kg}$ male. $^{3}$ . How much work can the biceps muscles (one in each arm) perform in a single contraction? (b) Compare this amount of work with the energy required to lift a 75 -kg person $40 . \mathrm{cm}$ in performing a chin-up. (c) Do you think the biceps muscle is the only muscle involved in performing a chin-up?

Averell H.
Carnegie Mellon University

Problem 28

A flea is able to jump about 0.5 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100-story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 1 000, the cross section of its muscle would increase by $1000^{2}$ and the length of contraction would increase by 1000 . How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)

Averell H.
Carnegie Mellon University

Problem 29

A 50.0 -kg projectile is fired at an angle of $30.0^{\circ}$ above the horizontal with an initial speed of $1.20 \times 10^{2} \mathrm{m} / \mathrm{s}$ from the top of a cliff 142 $\mathrm{m}$ above level ground, where the ground is taken to be $y=0 .$ (a) What is the initial total mechanical energy of the projectile? (b) Suppose the projectile is traveling 85.0 $\mathrm{m} / \mathrm{s}$ at its maximum height of $y=427 \mathrm{m}$ . How much work has been done on the projectile by air friction? (c) What is the speed of the projectile immediately before it hits the ground if air friction does one and a half times as much work on the projectile when it is going down as it did when it was going up?

Averell H.
Carnegie Mellon University

Problem 30

A projectile of mass $m$ is fired horizontally with an initial speed of $\tau_{0}$ from a height of $h$ above a flat, desert surface. Neglecting air friction, at the instant before the projectile hits the ground, find the following in terms of $m, v_{0}, h,$ and $g :(\text { a) the }$ work done by the force of gravity on the projectile, (b) the change in kinetic energy of the projectile since it was fired, and $(\mathrm{c})$ the final kinetic energy of the projectile. (d) Are any of the answers changed if the initial angle is changed?

Averell H.
Carnegie Mellon University

Problem 31

A horizontal spring attached to a wall has a force constant of 850 $\mathrm{N} / \mathrm{m}$ . A block of mass 1.00 $\mathrm{kg}$ is attached to the spring and oscillates freely on a horizontal, frictionless surface as in Figure $5.22 .$ The initial goal of this problem is to find the velocity at the equilibrium point after the block is released. (a) What objects constitute the system, and through what forces do they interact? (b) What are the two points of interest? (c) Find the energy stored in the spring when the mass is stretched 6.00 $\mathrm{cm}$ from equilibrium and again when the mass passes through equilibrium after being released from rest. (d) Write the conservation of energy equation for this situation and solve it for the speed of the mass as it passes equilibrium. Substitute to obtain a numerical value. (c) What is the speed at the halfway point? Why isn't it half the speed at equilibrium?

Averell H.
Carnegie Mellon University

Problem 32

A $50 .$ -kg pole vaulter running at $10 . \mathrm{m} / \mathrm{s}$ vaults over the bar. Her speed when she is above the bar is 1.0 $\mathrm{m} / \mathrm{s}$ . Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she crosses the bar.

Averell H.
Carnegie Mellon University

Problem 33

A child and a sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.00 $\mathrm{m} / \mathrm{s}$ at the bottom, what is the height of the hill?

Averell H.
Carnegie Mellon University

Problem 34

A 35.0 -cm long spring is hung vertically from a ceiling and stretches to 41.5 $\mathrm{cm}$ when a $7.50-\mathrm{kg}$ weight is hung from its free end. (a) Find the spring constant. (b) Find the length of the spring if the $7.50-\mathrm{kg}$ weight is replaced with a $195-\mathrm{N}$ weight.

Averell H.
Carnegie Mellon University

Problem 35

A 0.250 -kg block along a horizontal track has a speed of 1.50 $\mathrm{m} / \mathrm{s}$ immediately before colliding with a light spring of force constant 4.60 $\mathrm{N} / \mathrm{m}$ located at the end of the track. (a) What is the spring's maximum compression if the track. is frictionless? (b) If the track is not frictionless, would the spring's maximum compression be greater than, less than, or equal to the value obtained in part (a)?

Averell H.
Carnegie Mellon University

Problem 36

A block of mass $m=5.00 \mathrm{kg}$ is released from rest from point $@$ and slides on the frictionless track shown in Figure P5.36. Determine (a) the block’s speed at points B and C and (b) the net work done by the gravitational force on the block as it moves from point from A to C.

Averell H.
Carnegie Mellon University

Problem 37

Tarzan swings on a 30.0 -m-long vine initially inclined at an angle of $37.0^{\circ}$ with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) If he pushes off with a speed of 4.00 $\mathrm{m} / \mathrm{s}$ ?

Averell H.
Carnegie Mellon University

Problem 38

Two blocks are connected by a light string that passes over a frictionless pulley as in Figure P5.38. The system is released from rest while $m_{2}$ is on the floor and $m_{1}$ is a distance $h$ above the floor. (a) Assuming $m_{1}>m_{2},$ find an expression for the speed of $m_{1}$ just as it reaches the floor. (b) Taking $m_{1}=6.5 \quad \mathrm{kg}, m_{2}=4.2 \mathrm{kg},$ and $h=3.2 \mathrm{m},$ evaluate your answer to part $(\mathrm{a}),$ and $(\mathrm{c})$ find the speed of each block when $m_{1}$ has fallen a distance of 1.6 $\mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 39

The launching mechanism of a toy gun consists of a spring of unknown spring constant, as shown in Figure P5.39a. If the spring is compressed a distance of 0.120 $\mathrm{m}$ and the the gun fired vertically as shown, the gurcan launch a 20.0 -g projectile from rest to a maximum height of 20.0 $\mathrm{m}$ above the starting point of the projectile. Neglecting all resistive forces, (a) describe the mechanical energy transformations that occur from the time the gun is fired until the projectile reaches its maximum height, (b) determine the spring constant, and (c) find the speed of the projectile as it moves through the equilibrium position of the spring $(\text { where } x=0),$ as shown in Figure P5. 39 $\mathrm{b}$ .

Averell H.
Carnegie Mellon University

Problem 40

(a) A block with a mass $m$ is pulled along a horizontal surface for a distance $x$ by a constant force $\overrightarrow{\mathbf{F}}$ at an angle $\theta$ with respect to the horizontal. The coefficient of kinetic friction between block and table is $\mu_{k} .$ Is the force exerted by friction equal to $\mu_{k} m g^{\circ}$ If not, what is the force exerted by friction? (b) How much work is done by the friction force and by $\overrightarrow{\mathbf{F}}$ ? (Don't forget the signs.) (c) Identify all the forces that do no work on the block. (d) Let $m=2.00 \mathrm{kg},$ $x=4.00 \mathrm{m}, \theta=37.0^{\circ}, F=15.0 \mathrm{N},$ and $\mu_{k}=0.400,$ and find the answers to parts (a) and (b).

Averell H.
Carnegie Mellon University

Problem 41

(a) A child slides down a water slide at an amusement park from an initial height $h$. The slide can be considered frictionless because of the water flowing down it. Can the equation for conservation of mechanical energy be used on the child? (b) Is the mass of the child a factor in determining his speed at the bottom of the slide? (c) The child drops straight down rather than following the curved ramp of the slide. In which case will he be traveling faster at ground level? (d) If friction is present, how would the conservation-of-energy equation be modified? (e) Find the maximum speed of the child when the slide is frictionless if the initial height of the slide is $12.0 \mathrm{m}$.

Averell H.
Carnegie Mellon University

Problem 42

An airplane of mass $1.50 \times 10^{4} \mathrm{kg}$ is moving at 60.0 $\mathrm{m} / \mathrm{s}$. The pilot then increases the engine's thrust to $7.50 \times 10^{4} \mathrm{N}$ . The resistive force exerted by air on the airplane has a magnitude of $4.00 \times 10^{4} \mathrm{N}$ . (a) Is the work done by the engine on the airplane equal to the change in the airplane's kinetic energy after it travels through some distance through the air? Is mechanical energy conserved? Explain. (b) Find the speed of the airplane after it has traveled $5.00 \times 10^{2} \mathrm{m}$ . Assume the airplane is in level flight throughout the motion.

Averell H.
Carnegie Mellon University

Problem 43

The system shown in Figure P5.43 is used to lift an object of mass $m=$ 76.0 $\mathrm{kg} .$ A constant downward force of magnitude $F$ is applied to the loose end of the rope such that the hanging object moves upward at constant speed. Neglecting the masses of the rope and pulleys, find (a) the required value of $F,$ (b) the tensions $T_{1}, T_{2}$ and $T_{3},$ and $(\mathrm{c})$ the work done by the applied force in raising the object a distance of 1.80 $\mathrm{m}$ .

Averell H.
Carnegie Mellon University

Problem 44

A 25.0 -kg child on a 2.00 -m-long swing is released from rest when the ropes of the swing make an angle of $30.0^{\circ}$ with the vertical. (a) Neglecting friction, find the child's speed at the lowest position. (b) If the actual speed of the child at the lowest position is $2.00 \mathrm{m} / \mathrm{s},$ what is the mechanical energy lost due to friction?

Averell H.
Carnegie Mellon University

Problem 45

A $2.1 \times 10^{3}-\mathrm{kg}$ car starts from rest at the top of a $5.0-\mathrm{m}$ -long driveway that is inclined at $20.0^{\circ}$ with the horizontal. If an average friction force of $4.0 \times 10^{3} \mathrm{N}$ impedes the motion, find the speed of the car at the bottom of the driveway.

Averell H.
Carnegie Mellon University

Problem 46

A child of mass $m$ starts from rest and slides without friction from a height $h$ along a curved waterslide (Fig. P5.46). She is launched from a height $h / 5$ into the pool. (a) Is mechanical energy conserved? Why? (b) Give the gravitational potential energy associated with the child and her kinetic energy in terms of mgh at the following positions: the top of the waterslide, the launching point, and the point where she lands in the pool. (c) Determine her initial speed $v_{0}$ at the launch point in terms of $g$ and $h .$ (d) Determine her maximum airborne height $y_{\max }$ in terms of $h, g,$ and the horizontal speed at that height, $v_{0 x}$ (e) Use the $x$ -component of the answer to part (c) to eliminate $v_{0}$ from the answer to part (d), giving the height $y_{\text { max }}$ in terms of $g, h,$ and the launch angle $\theta$ . (f) Would your answers be the same if the waterslide were not frictionless? Explain.

Averell H.
Carnegie Mellon University

Problem 47

A skier starts from rest at the top of a hill that is inclined $10.5^{\circ}$ with respect to the horizontal. The hillside is $2.00 \times 10^{2} \mathrm{m}$ long, and the coefficient of friction between snow and skis is 0.0750 . At the bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier glide along the horizontal portion of the snow before coming to rest?

Averell H.
Carnegie Mellon University

Problem 48

In a circus performance, a monkey is strapped to a sled and both are given an initial speed of 4.0 $\mathrm{m} / \mathrm{s}$ up a $20.0^{\circ}$ inclined track. The combined mass of monkey and sled is $20 . \mathrm{kg},$ and the coefficient of kinetic friction between sled and incline is $0.20 .$ How far up the incline do the monkey and sled move?

Averell H.
Carnegie Mellon University

Problem 49

An 80.0 -kg skydiver jumps out of a balloon at an altitude of $1.00 \times 10^{3} \mathrm{m}$ and opens the parachute at an altitude of 200.0 $\mathrm{m}$ . (a) Assuming that the total retarding force on the diver is constant at 50.0 $\mathrm{N}$ with the parachute closed and constant at $3.60 \times 10^{9} \mathrm{N}$ with the parachute open, what is the speed of the diver when he lands on the ground? (b) Do you think the skydiver will get hurt? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 $\mathrm{m} / \mathrm{s}$ ? (d) How realistic is the assumption that the total retarding force is constant? Explain.

Averell H.
Carnegie Mellon University

Problem 50

A skier of mass 70.0 $\mathrm{kg}$ is pulled up a slope by a motordriven cable. (a) How much work is required to pull him 60.0 $\mathrm{m}$ up a $30.0^{\circ}$ slope (assumed frictionless) at a constant speed of 2.00 $\mathrm{m} / \mathrm{s}$ ? (b) What power (expressed in hp) must a motor have to perform this task?

Averell H.
Carnegie Mellon University

Problem 51

What average mechanical power must a 70.0 -kg mountain climber generate to climb to the summit of a hill of height 325 $\mathrm{m}$ in 45.0 $\mathrm{min}$ ? Note: Due to inefficiencies in converting chemical energy to mechanical energy, the amount calculated here is only a fraction of the power that must be produced by the climber's body.

Averell H.
Carnegie Mellon University

Problem 52

While running, a person dissipates about 0.60 $\mathrm{J}$ of mechanical energy per step per kilogram of body mass. If a 60 -kg person develops a power of $70 .$ W during a race, how fast is the person running? (Assume a running step is 1.5 $\mathrm{m}$ long. $)$

Averell H.
Carnegie Mellon University

Problem 53

The electric motor of a model train accelerates the train from rest to 0.620 $\mathrm{m} / \mathrm{s}$ in 21.0 $\mathrm{ms}$ . The total mass of the train is 875 g. Find the average power delivered to the train during its acceleration.

Averell H.
Carnegie Mellon University

Problem 54

When an automobile moves with constant speed down a high-way, most of the power developed by the engine is used to compensate for the mechanical energy loss due to frictional forces exerted on the car by the air and the road. If the power developed by an engine is 175 hp, estimate the total frictional force acting on the car when it is moving at a speed of 29 $\mathrm{m} / \mathrm{s}$ . One horsepower equals 746 $\mathrm{W}$ .

Averell H.
Carnegie Mellon University

Problem 55

Under normal conditions the human heart converts about 13.0 $\mathrm{J}$ of chemical energy per second into 1.30 $\mathrm{W}$ of mechanical power as it pumps blood throughout the body. (a) Determine the number of Calories required to power the heart for one day, given that 1 Calorie equals 4186 $\mathrm{J}$ . (b) Metabolizing 1.00 $\mathrm{kg}$ of fat can release about $9.00 \times 10^{3}$ Calories of energy. What mass of metabolized fat would power the heart for one day?

Averell H.
Carnegie Mellon University

Problem 56

A certain rain cloud at an altitude of 1.75 $\mathrm{km}$ contains $3.20 \times 10^{7} \mathrm{kg}$ of water vapor. How long would it take for a 2.70 -kW pump to raise the same amount of water from Earth's surface to the cloud's position?

Averell H.
Carnegie Mellon University

Problem 57

A $1.50 \times 10^{3}-$ kg car starts from rest and accelerates uniformly to 18.0 $\mathrm{m} / \mathrm{s}$ in 12.0 $\mathrm{s}$ . Assume that air resistance remains constant at $4.00 \times 10^{2} \mathrm{N}$ during this time. Find (a) the average power developed by the engine and (b) the instantaneous power output of the engine at $t=12.0 \mathrm{s}$ , just before the car stops accelerating.

Averell H.
Carnegie Mellon University

Problem 58

A $6.50 \times 10^{2}$ -kg elevator starts from rest and moves upward for 3.00 s with constant acceleration until it reaches its cruising speed, 1.75 $\mathrm{m} / \mathrm{s}$ . (a) What is the average power of the elevator motor during this period? (b) How does this amount of power compare with its power during an upward trip with constant speed?

Averell H.
Carnegie Mellon University

Problem 59

The force acting on a particle varies as in Figure P5.59. Find the work done by the force as the particle moves (a) from $x=0$ to $x=8.00 \mathrm{m},(\mathrm{b})$ from $x=8.00 \mathrm{m}$ to $x=10.0 \mathrm{m},$ and $(\mathrm{c})$ from $x=0$ to $x=10.0 \mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 60

An object of mass 3.00 $\mathrm{kg}$ is subject to a force $F_{x}$ that varies with position as in Figure $\mathrm{P} 5.60 .$ Find the work done by the force on the object as it moves (a) from $x=0$ to $x=5.00 \mathrm{m}$ , (b) from $x=5.00 \mathrm{m}$ to $x=10.0 \mathrm{m},$ and $(\mathrm{c})$ from $x=10.0 \mathrm{m}$ to $x=15.0 \mathrm{m} .(\mathrm{d})$ If the object has a speed of 0.500 $\mathrm{m} / \mathrm{s}$ at $x=0,$ find its speed at $x=5.00 \mathrm{m}$ and its speed at $x=15.0 \mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 61

The force acting on an object is given by $F_{x}=$ $(8 x-16) \mathrm{N},$ where $x$ is in meters. (a) Make a plot of this force vs. $x$ from $x=0$ to $x=3.00 \mathrm{m}$ . (b) From your graph, find the net work done by the force as the object moves from $x=0$ to $x=3.00 \mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 62

An outfielder throws a 0.150 -kg baseball at a speed of 40.0 $\mathrm{m} / \mathrm{s}$ and an initial angle of $30.0^{\circ} .$ What is the kinetic energy of the ball at the highest point of its motion?

Averell H.
Carnegie Mellon University

Problem 63

A roller-coaster car of mass $1.50 \times 10^{3} \mathrm{kg}$ is initially at the top of a rise at point @. It then moves 35.0 $\mathrm{m}$ at an angle of $50.0^{\circ}$ below the horizontal to a lower point B. (a) Find both the potential energy of the system when the car is at points A and B and the change in potential energy as the car moves from point A to point B, assuming $y=0$ at point at point B. (b) Repeat
part (a), this time choosing $y=0$ at point C, which is another 15.0 m down the same slope from point B.

Averell H.
Carnegie Mellon University

Problem 64

A ball of mass $m=1.80 \mathrm{kg}$ is released from rest at a height $h=65.0 \mathrm{cm}$ above a light vertical spring of force constant $k$ as in Figure $\mathrm{P} 5.64 \mathrm{a}$ . The ball strikes the top of the spring and compresses it a distance $d=9.00 \mathrm{cm}$ as in Figure $\mathrm{P} 5.64 \mathrm{b}$ . Neglecting any energy losses during the collision, find (a) the speed of the ball just as it touches the spring and (b) the force constant of the spring.

Averell H.
Carnegie Mellon University

Problem 65

An archer pulls her bowstring back 0.400 $\mathrm{m}$ by exerting a force that increases uniformly from $\mathrm{zero}$ to 230 $\mathrm{N}$ (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do in pulling the bow?

Averell H.
Carnegie Mellon University

Problem 66

A block of mass 12.0 $\mathrm{kg}$ slides from rest down a frictionless $35.0^{\circ}$ incline and is stopped by a strong spring with $k=3.00 \times 10^{4} \mathrm{N} / \mathrm{m} .$ The block slides 3.00 $\mathrm{m}$ from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?

Averell H.
Carnegie Mellon University

Problem 67

(a) A 75 -kg man steps out a window and falls (from rest) 1.0 $\mathrm{m}$ to a sidewalk. What is his speed just before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for his fall is an approximately $0.50-\mathrm{cm}$ give in the pads of his feet. Calculate the average force exerted on him by the ground during this 0.50 $\mathrm{cm}$ of travel. This aver- age force is sufficient to cause damage to cartilage in the joints or to break bones.

Averell H.
Carnegie Mellon University

Problem 68

A toy gun uses a spring to project a 5.3 -g soft rubber sphere horizontally. The spring constant is 8.0 $\mathrm{N} / \mathrm{m}$ , the barrel of the gun is 15 $\mathrm{cm}$ long, and a constant frictional force of 0.032 $\mathrm{N}$ exists between barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed 5.0 $\mathrm{cm}$ for this launch?

Averell H.
Carnegie Mellon University

Problem 69

Two objects $\left(m_{1}=5.00 \mathrm{kg} \text { and }\right.$ $m_{2}=3.00 \mathrm{kg}$ ) are connected $\mathrm{by}$ a light string passing over a light, frictionless pulley as in Figure P5.69. The $5.00-\mathrm{kg}$ object is released from rest at a point $h=4.00 \mathrm{m}$ above the table. (a) Determine the speed of each other. (b) Determine the speed of each object at the moment the 5.00-kg object hits the table. (c) How much higher does the $3.00-\mathrm{kg}$ object travel after the 5.00 -kg object hits the table?

Averell H.
Carnegie Mellon University

Problem 70

A $3.50-\mathrm{kN}$ piano is lifted by three workers at constant speed to an apartment 25.0 $\mathrm{m}$ above the street using a pulley system fastened to the roof of the building. Each worker is able to deliver 165 $\mathrm{W}$ of power, and the pulley system is 75$\%$ efficient (so that 25$\%$ of the mechanical energy is lost due to friction in the pulley). Neglecting the mass of the pulley, find the time required to lift the piano from the street to the apartment.

Averell H.
Carnegie Mellon University

Problem 71

A $2.00 \times 10^{2}$ -g particle is released from rest at point $A$ on the inside of a smooth hemispherical bowl of radius $R=30.0 \mathrm{cm}$ (Fig. P5.71). Calculate (a) its gravitational potential energy at A relative to $B,(\mathrm{b})$ its kinetic energy at $B,(\mathrm{c})$ its speed at $B$ , (d) its potential energy at $C$ relative to $B,$ and $(c)$ its kinetic energy at $C .$

Averell H.
Carnegie Mellon University

Problem 72

The particle described in Problem 71 (Fig. P5.71) is released from point $A$ at rest. Its speed at $B$ is 1.50 $\mathrm{m} / \mathrm{s}$ . (a) What is its kinetic energy at $B ?$ (b) How much mechanical energy is lost as a result of friction as the particle goes from $A$ to $B ?(\mathrm{c})$ Is it possible to determine $\mu$ from these results in a simple manner? Explain.

Averell H.
Carnegie Mellon University

Problem 73

In terms of saving energy, bicycling and walking are far more efficient means of transportation than is travel by automobile. For example, when riding at 10.0 $\mathrm{mi} / \mathrm{h}$ , a cyclist uses food energy at a rate of about 400 $\mathrm{kcal} / \mathrm{h}$ above what he would use if he were merely sitting still. (In exercise physiology, power is often measured in kcal/h rather than in watts. Here, $1 \mathrm{kcal}=1$ nutritionist's Calorie $=4186 \mathrm{J.}$ .) Walking at 3.00 $\mathrm{mi} / \mathrm{h}$ requires about 220 $\mathrm{kcal} / \mathrm{h}$ . It is interesting to compare these values with the energy consumption required for travel by car. Gasoline yields about $1.30 \times 10^{8} \mathrm{J} / \mathrm{gal}$ . Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.

Averell H.
Carnegie Mellon University

Problem 74

A 50.0 -kg student evaluates a weight loss program by calculating the number of times she would need to climb a $12.0-\mathrm{m}$ high flight of steps in order to lose one pound $(0.45 \mathrm{kg})$ of fat. Metabolizing 1.00 $\mathrm{kg}$ of fat can release $3.77 \times 10^{7} \mathrm{J}$ of chemical energy and the body can convert about 20.0$\%$ of this into mechanical energy. (The rest goes into internal energy.) (a) How much mechanical energy can the body produce from 0.450 $\mathrm{kg}$ of fat? (b) How many trips up the flight of steps are required for the student to lose 0.450 $\mathrm{kg}$ of fat? Ignore the relatively small amount of energy required to return down the stairs.

Averell H.
Carnegie Mellon University

Problem 75

A ski jumper starts from rest 50.0 $\mathrm{m}$ above the ground on a frictionless track and flies off the track at an angle of $45.0^{\circ}$ above the horizontal and at a height of 10.0 $\mathrm{m}$ above the level ground. Neglect air resistance. (a) What is her speed when she leaves the track? (b) What is the maximum altitude she attains after leaving the track? (c) Where does she land relative to the end of the track?

Averell H.
Carnegie Mellon University

Problem 76

A 5.0 -kg block is pushed 3.0 $\mathrm{m}$ up a vertical wall with constant speed by a constant force of magnitude $F$ applied at an angle of $\theta=30^{\circ}$ with the horizontal, as shown in Figure $P 5.76$ . If the coefficient of kinetic friction between block and wall is $0.30,$ determine the work done by (a) $\overrightarrow{\mathbf{F}},(\mathrm{b})$ the force of gravity, and (c) the normal force between block and wall. (d) By how much does the gravitational potential energy increase during the block's motion?

Averell H.
Carnegie Mellon University

Problem 77

A child's pogo stick (Fig. P5.77) stores energy in a spring $(k=$ $2.50 \times 10^{4} \mathrm{N} / \mathrm{m}$ ). At position (A) $\left(x_{1}=-0.100 \mathrm{m}\right),$ the spring compression is a maximum and the child is momentarily at rest. At position {B} $(x=0)$ , the spring is relaxed and the child is moving upward. At position C, the child is again momentarily at rest at the top of the jump. Assuming that the combined mass of child and pogo stick is 25.0 $\mathrm{kg}$ (a) calculate the total energy of the system if both potential energies are zero at $x=0$ , (b) determine $x_{2},(\mathrm{c})$ calculate the speed of the child at $x=0$ , (d) determine the value of $x$ for which the kinetic energy of the system is a maximum, and (e) obtain the child's maximum upward speed.

Averell H.
Carnegie Mellon University

Problem 78

A hummingbird hovers by exerting a downward force on the air equal, on average, to its weight. By Newton’s third law, the air exerts an upward force of the same magnitude on the bird's wings. Find the average mechanical power delivered by a 3.00 -g hummingbird while hovering if its wings beat 80.0 times per second through a stroke 3.50 $\mathrm{cm}$ long.

Averell H.
Carnegie Mellon University

Problem 79

In the dangerous “sport” of bungee jumping, a daring student jumps from a hot air balloon with a specially designed elastic cord attached to his waist. The unstretched length of the cord is $25.0 \mathrm{m},$ the student weighs $7.00 \times 10^{2} \mathrm{N},$ and the balloon is 36.0 $\mathrm{m}$ above the surface of a river below. Calculate the required force constant of the cord if the student is to stop safely 4.00 $\mathrm{m}$ above the river.

Averell H.
Carnegie Mellon University

Problem 80

Apollo 14 astronaut Alan Shepard famously took two golf shots on the Moon where it's been estimated that an expertly hit shot could travel for 70.0 s through the Moon's reduced gravity, airless environment to a maximum range of 4.00 $\mathrm{km}$ (about 2.5 miles). Assuming such an expert shot has a launch angle of $45.0^{\circ},$ determine the golf ball's (a) kinetic energy as it leaves the club, and (b) maximum altitude in $\mathrm{km}$ above the lunar surface. Take the mass of a golf ball to be 0.0450 $\mathrm{kg}$ and the Moon's gravitational acceleration to be $g_{\text { moon }}=1.63 \mathrm{m} / \mathrm{s}^{2}$

Averell H.
Carnegie Mellon University

Problem 81

A truck travels uphill with constant velocity on a highway with a $7.0^{\circ}$ slope. $A 50$ -kg package sits on the floor of the back of the truck and does not slide, due to a static frictional force. During an interval in which the truck travels $340 \mathrm{m},(\mathrm{a})$ what is the net work done on the package? What is the work done on the package by (b) the force of gravity, (c) the normal force, and (d) the friction force?

Averell H.
Carnegie Mellon University

Problem 82

As a 75.0 -kg man steps onto a bathroom scale, the spring inside the scale compresses by 0.650 $\mathrm{mm}$ . Excited to see that he has lost 2.50 $\mathrm{kg}$ since his previous weigh-in, the man jumps 0.300 $\mathrm{m}$ straight up into the air and lands directly on the scale. (a) What is the spring's maximum compression? (b) If the scale reads in kilograms, what reading does it give when the spring is at its maximum compression?

Averell H.
Carnegie Mellon University

Problem 83

A loaded ore car has a mass of $9.50 \times 10^{2} \mathrm{kg}$ and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at $30.0^{\circ}$ above the horizontal. The car accelerates uniformly to a speed of 2.20 $\mathrm{m} / \mathrm{s}$ in 12.0 $\mathrm{s}$ and then continues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed? (b) What maximum power must the motor provide? (c) What total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length 1250 $\mathrm{m} ?$

Averell H.
Carnegie Mellon University

Problem 84

A cat plays with a toy mouse suspended from a light string of length $1.25 \mathrm{m},$ rapidly batting the mouse so that it acquires a speed of 2.75 $\mathrm{m} / \mathrm{s}$ while the string is still vertical. Use energy conservation to find the mouse's maximum height above its original position. (Assume the string always remains taut.)

Averell H.
Carnegie Mellon University

Problem 85

Three objects with masses $m_{1}=5.00 \mathrm{kg}, m_{2}=10.0 \mathrm{kg},$ and $m_{3}=15.0 \mathrm{kg},$ respectively, are attached by strings over frictionless pulleys as indicated in Figure P5.85. The horizontal surface exerts a force of friction of 30.0 $\mathrm{N}$ on $m_{2} .$ If the system is released from rest, use energy concepts to find the speed of $m_{3}$ after it moves down 4.00 $\mathrm{m} .$

Averell H.
Carnegie Mellon University

Problem 86

Two blocks, $A$ and $B$ (with mass 50.0 $\mathrm{kg}$ and $1.00 \times 10^{2} \mathrm{kg}$ , respectively), are connected by a string, as shown in Figure $\mathrm{P} 5.86 .$ The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between block $A$ and the incline is $\mu_{k}=0.250 .$ Determine the change in the kinetic energy of block $A$ as it moves from C to D, a distance of 20.0 m up the incline (and block $B$ drops downward a distance of 20.0 m) if the system starts from rest.

Averell H.
Carnegie Mellon University