Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 8

Conservation of Energy - all with Video Answers

Educators

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Chapter Questions

Problem 1

For each of the following systems and time intervals, write the appropriate expanded version of Equation 8.2, the conservation of energy equation. (a) the heating coils in your toaster during the first five seconds after you turn the toaster on (b) your automobile from just before you fill it with gasoline until you pull away from the gas station at speed v (c) your body while you sit quietly and eat a peanut butter and jelly sandwich for lunch (d) your home during five minutes of a sunny afternoon while the temperature in
the home remains fixed

Sheh Lit Chang

Sheh Lit Chang

University of Washington

Problem 2

A ball of mass m falls from a height h to the floor. (a) Write the appropriate version of Equation 8.2 for the system of the ball and the Earth and use it to calculate the speed of the ball just before it strikes the Earth. (b) Write the appropriate version of Equation 8.2 for the system of the ball and use it to calculate the speed of the ball just before it strikes the Earth.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 3

A block of mass 0.250 $\mathrm{kg}$ is placed on top of a light, vertical spring of force constant 5000 $\mathrm{N} / \mathrm{m}$ and pushed downward so that the spring is compressed by 0.100 $\mathrm{m}$ . After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise?

Vishal Gupta

Numerade Educator

Problem 4

At 11:00 a.m. on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the children’s bodies that was converted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 children of average mass 36.0 kg jumped
12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most produced high-frequency “microtremor” vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter
scale is given by $$ M=\frac{\log E-4.8}{1.5} $$ where $E$ is the seismic wave energy in joules. According to this model, what was the magnitude of the demonstration quake?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 5

A bead slides without friction around a loop-the-loop (Fig. P8.5). The bead is released from rest at a height
$h=3.50 R$ (a) What is its speed at point $@$ ? (b) How large is the normal force on the bead at point $@$ if its mass is 5.00 g?

Keshav Singh

Numerade Educator

Problem 6

A block of mass $m=5.00 \mathrm{kg}$ is released from point $\mathbb{Q}$ and slides on the frictionless track shown in Figure P8.6. Determine (a) the block's speed at points $\mathbb{B}$ and $\mathbb{C}$ and $(b)$ the net work done by the gravitational force on the block as it moves from point $@$ to point $\mathbb{C}$ .

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 7

Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass $m_{1}=5.00 \mathrm{kg}$ is released from rest at a height $h=4.00 \mathrm{m}$ above the table. Using the isolated system model, (a) determine the speed of the object of
mass $m_{2}=3.00 \mathrm{kg}$ just as the 5.00 -kg object hits the table and $(\mathrm{b})$ find the maximum height above the table to which the $3.00-\mathrm{kg}$ object rises.

Keshav Singh

Numerade Educator

Problem 8

Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass $m_{1}$ is released from rest at height $h$ above the table. Using the isolated system model, (a) determine the speed of $m_{2}$ just as $m_{1}$ hits the table and (b) find the maxi-
mum height above the table to which $m_{2}$ rises.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 9

A light, rigid rod is 77.0 $\mathrm{cm}$ long. Its top end is pivoted on a frictionless, horizontal axle. The rod hangs straight down at rest with a small, massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle?

Manish Kumar

Numerade Educator

Problem 10

A 20.0 -kg cannonball is fired from a cannon with muzzle speed of 1000 $\mathrm{m} / \mathrm{s}$ at an angle of $37.0^{\circ}$ with the horizontal. A second ball is fired at an angle of $90.0^{\circ} .$ Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ball- Earth system at the maximum height for each ball. Let $y=0$ at the cannon.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 11

The system shown in Figure P8.11 consists of a light, inextensible cord, light, frictionless pulleys, and
blocks of equal mass. Notice that block B is attached to one of the pulleys. The system is initially held
at rest so that the blocks are at the same height above the ground. The blocks are then released. Find the speed of block A at the moment the vertical separation of the blocks is $h .$

Keshav Singh

Numerade Educator

Problem 12

A sled of mass $m$ is given a kick on a frozen pond. The kick imparts to the sled an initial speed of 2.00 $\mathrm{m} / \mathrm{s}$ . The coefficient of kinetic friction between sled and ice is $0.100 .$ Use
energy considerations to find the distance the sled moves before it stops.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 13

A sled of mass $m$ is given a kick on a frozen pond. The kick imparts to the sled an initial speed of $v$ . The coefficient of kinetic friction between sled and ice is $\mu_{k} .$ Use energy considerations to find the distance the sled moves before it stops.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

A 40.0 -kg box initially at rest is pushed 5.00 $\mathrm{m}$ along a rough, horizontal floor with a constant applied horizontal force of 130 $\mathrm{N}$ . The coefficient of friction between box and floor is 0.300 . Find (a) the work done by the applied force, (b) the increase in internal energy in the box-floor system as a result of friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the
change in kinetic energy of the box, and (f) the final speed of the box.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 15

A block of mass $m=2.00 \mathrm{kg}$ is attached to a spring of force constant $k=500 \mathrm{N} / \mathrm{m}$ as shown in Figure $\mathrm{P} 8.15 .$ The block is pulled to a position $x_{i}=$
5.00 $\mathrm{cm}$ to the right of equilibrium and released from rest. Find the speed the block has as
it passes through equilibrium if $(a)$ the horizontal surface is frictionless and (b) the coef-
ficient of friction between block and surface is $\mu_{k}=0.350$ .

Manish Kumar

Numerade Educator

Problem 16

A crate of mass 10.0 $\mathrm{kg}$ is pulled up a rough incline with an initial speed of 1.50 $\mathrm{m} / \mathrm{s}$ . The pulling force is 100 $\mathrm{N}$ parallel to the incline, which makes an angle of $20.0^{\circ}$ with the horizontal. The coefficient of kinetic friction is 0.400 , and the crate is pulled $5.00 \mathrm{m} .$ (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crate-incline system owing to friction. (c) How much work is done by the $100-\mathrm{N}$ force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 $\mathrm{m}$ ?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 17

A smooth circular hoop with a radius of 0.500 m is placed flat on the floor. A 0.400-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.00 m/s. After one revolution, its speed has dropped to 6.00 m/s because of friction with the floor.
(a) Find the energy transformed from mechanical to internal in the particle–hoop–floor system as a result of friction in one revolution. (b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.

Keshav Singh

Numerade Educator

Problem 18

As shown in Figure P8.18, a green bead of mass 25 g slides along a straight wire. The length of the wire from point to point is 0.600 m, and point is 0.200 m higher than point . A constant friction force
of magnitude 0.025 0 N acts on the bead. (a) If the bead is released from rest at point
, what is its speed at point ? (b) A red bead of mass 25 g slides along a curved wire, subject to a friction force with the same constant magnitude as that on the green bead. If the green and red beads are released simultaneously from rest at point , which bead reaches point with a higher speed? Explain.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 19

A boy in a wheelchair (total mass 47.0 kg) has speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4 m long. At the bottom of the slope his speed is 6.20 m/s. Assume air resistance and rolling resistance can be modeled as a constant friction force of 41.0 N. Find the work he did in pushing forward on his wheels during the downhill ride.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

At time $t_{i},$ the kinetic energy of a particle is 30.0 $\mathrm{J}$ and the potential energy of the system to which it belongs is 10.0 $\mathrm{J}$ . At some later time $t_{f},$ the kinetic energy of the par-
ticle is 18.0 $\mathrm{J}$ . (a) If only conservative forces act on the particle, what are the potential energy and the the total energy of the system at time $t_{f} ?$ (b) If the potential energy of the system at time $t_{f}$ is 5.00 $\mathrm{J}$ , are any nonconservative forces acting on the particle? (c) Explain your answer to part $(\mathrm{b}) .$

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 21

A 200 -g block is pressed against a spring of force constant 1.40 $\mathrm{kN} / \mathrm{m}$ until the block compresses the spring $10.0 \mathrm{cm} .$ The spring rests at the bottom of a ramp inclined
at $60.0^{\circ}$ to the horizontal. Using energy considerations, determine how far up the incline the block moves from its initial position before it stops (a) if the ramp exerts no friction force on the block and (b) if the coefficient of kinetic friction is 0.400 .

Manish Kumar

Numerade Educator

Problem 22

The coefficient of friction between the block of mass $m_{1}=$ 3.00 $\mathrm{kg}$ and the surface in Figure $\mathrm{P} 8.22$ is $\mu_{k}=0.400$ . The system starts from rest. What is the speed of the ball of mass $m_{2}$ $=5.00 \mathrm{kg}$ when it has fallen a distance $h=1.50 \mathrm{m} ?$

Nicholas Mogoi

Numerade Educator

Problem 23

A $5.00-\mathrm{kg}$ block is set into motion up an inclined plane with an initial speed of $v_{i}=8.00 \mathrm{m} / \mathrm{s}$ (Fig. P8. 23 . The block comes to rest after traveling $d=3.00 \mathrm{m}$ along the plane, which is inclined at an angle of $\theta=30.0^{\circ}$ to the horizontal. For this motion,
determine (a) the change in the block's kinetic energy, (b) the change in the potential energy of the block-Earth system, and (c) the friction force exerted on the block (assumed to be constant). (d) What is
the coefficient of kinetic friction?

Manish Kumar

Numerade Educator

Problem 24

An 80.0-kg skydiver jumps out of a balloon at an altitude of 1 000 m and opens his parachute at an altitude of 200 m. (a) Assuming the total retarding force on the skydiver is constant at 50.0 N with the parachute closed and constant at 3 600 N with the parachute open, find the speed of the skydiver when he lands on the ground. (b) Do you think the skydiver will be injured? 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 m/s?
(d) How realistic is the assumption that the total retarding force is constant? Explain.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 25

A toy cannon uses a spring to project a 5.30 -g soft rubber ball. The spring is originally compressed by 5.00 $\mathrm{cm}$ and has a force constant of $8.00 \mathrm{N} / \mathrm{m} .$ When the cannon is
fired, the ball moves 15.0 $\mathrm{cm}$ through the horizontal barrel of the cannon, and the barrel exerts a constant friction force of 0.0320 $\mathrm{N}$ on the ball. (a) With what speed does the
projectile leave the barrel of the cannon? (b) At what point does the ball have maximum speed? (c) What is this maximum speed?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 26

A 1.50 -kg object is held 1.20 $\mathrm{m}$ above a relaxed massless, vertical spring with a force constant of $320 \mathrm{N} / \mathrm{m} .$ The object is dropped onto the spring. (a) How far does the object compress the spring? (b) What If? Repeat part (a), but this time assume a constant air-resistance force of 0.700 $\mathrm{N}$ acts on the object during its motion. (c) What If? How far does
the object compress the spring if the same experiment is performed on the Moon, where $g=1.63 \mathrm{m} / \mathrm{s}^{2}$ and air resistance is neglected?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 27

A child of mass $m$ starts from rest and slides without friction from a height $h$ along a slide next to a pool (Fig. P8.27). She is launched from a height $h / 5$ into the air over the pool. We wish to find the maximum height she reaches above the water in her projectile motion. (a) Is the child-Earth system isolated or nonisolated? Why? (b) Is there a nonconservative force acting within the system?
(c) Define the configuration of the system when the child is at the water level as having zero gravitational potential energy. Express the total energy of the system when the child is at the top of the water slide. (d) Express the total energy of the system when the child is at the launching point. (e) Express the total energy of the system when the child is at the highest point in her projectile motion.
(f) From parts (c) and (d), determine her initial speed $v_{i}$ at the launch point in terms of $g$ and $h .$ (g) From parts (d), $(\mathrm{e}),$ and $(\mathrm{f}),$ determine her maximum airborne height
$y_{\text { max in terms of } h \text { and the launch angle } \theta . \text { (h) Would your }}$ answers be the same if the waterslide were not frictionless? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

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 a $2.70-\mathrm{kW}$ pump to raise the same amount of water from the Earth's surface to the cloud's position?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 29

An $820-\mathrm{N}$ Marine in basic training climbs a $12.0-\mathrm{m}$ vertical rope at a constant speed in 8.00 $\mathrm{s}$ . What is his power output?

Manish Kumar

Numerade Educator

Problem 30

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 $\mathrm{g}$ . (a) Find the minimum power delivered to the train by electrical transmission from the metal rails during the acceleration. (b) Why is it the minimum power?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 31

Make an order-of-magnitude estimate of the power a car engine contributes to speeding the car up to highway speed. In your solution, state the physical quantities you take as data and the values you measure or estimate for them. The mass of a vehicle is often given in the owner’s manual.

Rashmi Sinha

Numerade Educator

Problem 32

Sewage at a certain pumping station is raised vertically by 5.49 m at the rate of 1 890 000 liters each day. The sewage, of density 1050 $\mathrm{kg} / \mathrm{m}^{3}$ , enters and leaves the pump at atmospheric pressure and through pipes of equal diameter.
(a) Find the output mechanical power of the lift station.
(b) Assume an electric motor continuously operating with average power 5.90 kW runs the pump. Find its efficiency.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 33

An energy-efficient lightbulb, taking in 28.0 $\mathrm{W}$ of power, can produce the same level of brightness as a conventional lightbulb operating at power 100 $\mathrm{W}$ . The lifetime of the energy-efficient bulb is 10000 $\mathrm{h}$ and its purchase price is $\$ 4.50,$ whereas the conventional bulb has a lifetime of 750 $\mathrm{h}$ and costs $\$ 0.42$ . Determine the total savings obtained by
using one energy-efficient bulb over its lifetime as opposed to using conventional bulbs over the same time interval. Assume an energy cost of $\$ 0.200$ per kilowatt-hour.

Manish Kumar

Numerade Educator

Problem 34

An electric scooter has a battery capable of supplying 120 $\mathrm{Wh}$ of energy. If friction forces and other losses account for 60.0$\%$ of the energy usage, what altitude change can a rider achieve when driving in hilly terrain if the rider and scooter have a combined weight of 890 $\mathrm{N} ?$

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 35

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

Manish Kumar

Numerade Educator

Problem 36

An older-model car accelerates from 0 to speed $v$ in a time interval of $\Delta t .$ A newer, more powerful sports car accelerates from 0 to 2$v$ in the same time period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 37

A 3.50 -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 75.0$\%$ efficient (so that 25.0$\%$ of the mechanical energy is transformed to other forms 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.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 38

A 650 -kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 $\mathrm{m} / \mathrm{s}$ . (a) What is the average power of the elevator motor during this time interval? (b) How does this power compare with the motor power when the elevator
moves at its cruising speed?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 39

A loaded ore car has a mass of 950 $\mathrm{kg}$ 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 winch motor provide?
(c) What total energy has transferred out of the motor by work by the time the car moves off the end of the track, which is of length 1250 $\mathrm{m}$ ?

Surjit Tewari

Numerade Educator

Problem 40

Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is one kilocalorie, defined as 1 kcal 5 4 186 J. Metabolizing 1 g of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. He plans to run up and down the stairs in a football stadium as fast as he can and as many times as necessary. To evaluate the program, suppose he runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For simplicity, ignore the energy he uses in coming down (which is
small). Assume a typical efficiency for human muscles is 20.0%. This statement means that when your body converts 100 J from metabolizing fat, 20 J goes into doing mechanical work (here, climbing stairs). The remainder goes into extra internal energy. Assume the student’s mass is 75.0 kg.
(a) How many times must the student run the flight of stairs to lose 1.00 kg of fat? (b) What is his average power output, in watts and in horsepower, as he runs up the stairs? (c) Is this activity in itself a practical way to lose weight?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 41

A small block of mass $m=200 \mathrm{g}$ is released from rest at point $@$ along the horizontal diameter on the inside of a frictionless, hemispherical bowl of radius $R=30.0 \mathrm{cm}$ (Fig. P8.41). Calculate (a) the gravitational potential energy of the block–Earth system when the block is at point relative to point , (b) the kinetic energy of the block at point , (c) its speed at point , and (d) its kinetic energy and the potential energy when the block is at point .

Ren Jie Tuieng

Numerade Educator

Problem 42

What If? The block of mass $m=200 \mathrm{g}$ described in Problem 41 (Fig. P8.41) is released from rest at point $\mathbb{Q}$ , and the surface of the bowl is rough. The block's speed at point is 1.50 m/s. (a) What is its kinetic energy at point ? (b) How much mechanical energy is transformed into
internal energy as the block moves from point to point ? (c) Is it possible to determine the coefficient of friction from these results in any simple manner? (d) Explain your answer to part (c).

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 43

A boy starts at rest and slides down a frictionless slide as in Figure P8.43. The bottom of the track is a
height h above the ground. The boy then leaves the track horizontally, striking the ground at a distance d as shown. Using energy methods, determine the initial height H of the boy above the ground in terms of h and d.

Vishal Gupta

Numerade Educator

Problem 44

Why is the following situation impossible? A softball pitcher has a strange technique: she begins with her hand at rest at the highest point she can reach and then quickly rotates her arm backward so that the ball moves through a half-circle path. She releases the ball when her hand reaches the bot- tom of the path. The pitcher maintains a component of force on the 0.180-kg ball of constant magnitude 12.0 N in
the direction of motion around the complete path. As the ball arrives at the bottom of the path, it leaves her hand with a speed of 25.0 m/s.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 45

A skateboarder with his board can be modeled as a particle of mass 76.0 kg, located at his center of mass (which we will study in Chapter 9). As shown in Figure P8.45, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point ). The half-pipe is one half of a cylinder of radius 6.80 m with its axis horizontal. On his descent, the skateboarder moves without friction so that his center of mass moves through one quarter of a circle of radius 6.30 m. (a) Find his speed at the bottom of the half-pipe (point ). (b) Immediately after passing point , he stands up and raises his arms, lifting his center of mass from 0.500 m to 0.950 m above the concrete (point). Next, the skateboarder glides upward with his center of mass moving in a quarter circle of radius 5.85 m. His body is horizontal when he passes point , the far lip of the half-pipe. As he passes through point , the speed of the skateboarder is 5.14 m/s. How much chemical potential energy in the body of the skateboarder was converted to mechanical energy in the skateboarder–Earth system when he stood up at point ? (c) How high above point does he rise? Caution: Do not try this stunt yourself without the
required skill and protective equipment.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 46

As shown in Figure P8.46, a light string that does not stretch changes from horizontal to vertical as it passes over the edge of a table. The string connects $m_{1},$ a 3.50 -kg block originally
at reston the horizontal table at a height $h=1.20 \mathrm{m}$ above the floor, to $m_{2},$ a hanging
$1.90-\mathrm{kg}$ block originally a distance $d=0.900 \mathrm{m}$ above the floor. Neither the surface of the table nor its edge exerts a force of kinetic friction. The blocks start to move from rest. The sliding block $m_{1}$ is projected horizontally after reaching the edge of the table. The hanging block $m_{2}$ stops without bouncing when it strikes the floor. Consider the two blocks plus the Earth as the system. (a) Find the speed at which $m_{1}$ leaves the edge of the table. (b) Find the impact speed of m1 on the floor. (c) What is the shortest length of the string so that it does not go taut
while $m_{1}$ is in flight? (d) Is the energy of the system when it is released from rest equal to the energy of the system just before $m_{1}$ strikes the ground? (e) Why or why not?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 47

A 4.00 -kg particle moves along the $x$ axis. Its position varies with time according to $x=t+2.0 t^{3},$ where $x$ is in meters and $t$ is in seconds. Find $(a)$ the kinetic energy of the particle at any time $t,$ (b) the acceleration of the particle and the force acting on it at time $t,(c)$ the power being delivered to the particle at time $t,$ and $(\mathrm{d})$ the work done on the particle in the interval $t=0$ to $t=2.00 \mathrm{s}$ .

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 48

Heedless of danger, a child leaps onto a pile of old mattresses to use them as a trampoline. His motion
between two particular points is described by the energy conservation equation $\begin{aligned} \frac{1}{2}(46.0 \mathrm{kg})(2.40 \mathrm{m} / \mathrm{s})^{2}+(46.0 \mathrm{kg})\left(9.80 \mathrm{m} / \mathrm{s}^{2}\right)(2.80 \mathrm{m}+x) & \\=\frac{1}{2}\left(1.94 \times 10^{4} \mathrm{N} / \mathrm{m}\right) x^{2} \end{aligned}$
(a) Solve the equation for $x .$ (b) Compose the statement of a problem, including data, for which this equation gives the solution. (c) Add the two values of $x$ obtained in part
(a) and divide by $2 .$ (d) What is the significance of the resulting value in part (c)?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 49

Jonathan is riding a bicycle and encounters a hill of height 7.30 m. At the base of the hill, he is traveling at 6.00 m/s. When he reaches the top of the hill, he is traveling at 1.00 m/s. Jonathan and his bicycle together have a mass of 85.0 kg. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathan’s body during this process?
(c) How much work does Jonathan do on the bicycle pedals within the Jonathan–bicycle–Earth system during this process?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 50

Jonathan is riding a bicycle and encounters a hill of height h. At the base of the hill, he is traveling at a speed $v_{i}$ . When he reaches the top of the hill, he is traveling at a speed $v_{f}$. Jonathan and his bicycle together have a mass m. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle
between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathan’s body during this process? (c) How much work does Jonathan do on the bicycle pedals within the Jonathan–bicycle–Earth system during this process?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 51

As the driver steps on the gas pedal, a car of mass 1 160 kg accelerates from rest. During the first few seconds of motion, the car’s acceleration increases with time according to the expression $$
a=1.16 t-0.210 t^{2}+0.240 t^{3} $$ where $t$ is in seconds and $a$ is in $\mathrm{m} / \mathrm{s}^{2}$ . (a) What is the change in kinetic energy of the car during the interval from $t=0$ to $t=2.50 \mathrm{s}^{2}$ (b) What is the minimum average power out- put of the engine over this time interval? (c) Why is the value in part (b) described as the minimum value?

Keshav Singh

Numerade Educator

Problem 52

Why is the following situation impossible? A new high- speed roller coaster is claimed to be so safe that the passengers do not need to wear seat belts or any other restraining device. The coaster is designed with a vertical circular section over which the coaster travels on the inside of the circle so that the passengers are upside down for a short time interval. The radius of the circular section is 12.0 m,
and the coaster enters the bottom of the circular section at a speed of 22.0 m/s. Assume the coaster moves without friction on the track and model the coaster as a particle.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 53

A horizontal spring attached to a wall has a force constant of $k=850 \mathrm{N} / \mathrm{m} .$ A block of mass $m=1.00 \mathrm{kg}$ is attached to the spring and rests on a frictionless, horizontal
surface as in Figure $\mathrm{P} 8.53$ . (a) The block is pulled to a position $x_{i}=6.00 \mathrm{cm}$ from equilibrium and released. Find the elastic potential energy stored in the spring when the block
is 6.00 $\mathrm{cm}$ from equilibrium and when the block passes through equilibrium. (b) Find the speed of the block as it passes through the equilibrium point. (c) What is the speed of the block when it is at a position $x_{i} / 2=3.00 \mathrm{cm} ?(\mathrm{d})$ Why isn't the answer to part (c) half the answer to part (b)?

Keshav Singh

Numerade Educator

Problem 54

As it plows a parking lot, a snowplow pushes an ever-growing pile of snow in front of it. Suppose a car moving through the air is similarly modeled as a cylinder of area A pushing a growing disk of air in front of it. The originally stationary air is set into motion at the constant speed v of the cylinder as shown in Figure
P8.54. In a time interval $\Delta t$ a new disk of air of mass $\Delta m$ must be moved a distance $v \Delta t$ and hence must be given a kinetic energy $\frac{1}{2}(\Delta m) v^{2} .$ Using this model, show that the car's power loss owing to air resistance is $\frac{1}{2} \rho A v^{3}$ and that the resistive force acting on the car is $\frac{1}{2} \rho A v^{2},$ where $\rho$ is the density of air. Compare this
result with the empirical expression $\frac{1}{2} D \rho A v^{2}$ for the resistive force.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 55

A wind turbine on a wind farm turns in response to a force of high-speed air resistance, $R=\frac{1}{2} D \rho A v^{2} .$ The power available is $P=R v=\frac{1}{2} D \rho \pi r^{2} v^{3},$ where $v$ is the wind speed and we have assumed a circular face for the wind turbine of radius $r$ . Take the drag coefficient as $D=1.00$ and the density of air from the front endpaper. For a wind turbine having $r=1.50 \mathrm{m},$ calculate the power available with (a) $v=$ 8.00 $\mathrm{m} / \mathrm{s}$ and $(\mathrm{b}) v=24.0 \mathrm{m} / \mathrm{s} .$ The power delivered to the generator is limited by the efficiency of the system, about $25 \% .$ For comparison, a large American home uses about 2 $\mathrm{kW}$ of electric power.

Manish Kumar

Numerade Educator

Problem 56

Consider the popgun in Example 8.3. Suppose the projectile mass, compression distance, and spring constant remain the same as given or calculated in the example. Suppose, however, there is a friction force of magnitude 2.00 N acting on the projectile as it rubs against the interior of the barrel. The vertical length from point to the end of the barrel is 0.600 m. (a) After the spring is compressed and the popgun fired, to what height does the projectile rise above point ? (b) Draw four energy bar charts for this situation, analogous to those in Figures 8.6c-d.

Ren Jie Tuieng

Numerade Educator

Problem 57

Consider the block–spring–surface system in part (B) of Example 8.6. (a) Using an energy approach, find the position x of the block at which its speed is a maximum. (b) In the What If? section of this example, we explored the effects of an increased friction force of 10.0 N. At what position of the block does its maximum speed occur in this situation?

Ren Jie Tuieng

Numerade Educator

Problem 58

More than 2 300 years ago, the Greek teacher Aristotle wrote the first book called Physics. Put into more precise terminology, this passage is from the end of its Section
Eta:
Let $P$ be the power of an agent causing motion; $w,$ the load moved; $d$ , the distance covered; and $\Delta t,$ the time interval required. Then $(1)$ a power equal to $P$ will in an interval of time equal to $\Delta t$ move $w / 2$ a distance $2 d ;$ or $(2)$ it will move $w / 2$ the given distance
$d$ in the time interval $\Delta t / 2 .$ Also, if $(3)$ the given power $P$ moves the given load $w$ a distance $d / 2$ in time interval $\Delta t / 2,$ then $(4) P / 2$ will move $w / 2$ the given distance $d$ in the given time interval $\Delta t$
(a) Show that Aristotle's proportions are included in the equation $P \Delta t=b w d,$ where $b$ is a proportionality constant. (b) Show that our theory of motion includes this part of Aristotle’s theory as one special case. In particular, describe a situation in which it is true, derive the equation
representing Aristotle’s proportions, and determine the proportionality constant.

Ren Jie Tuieng

Numerade Educator

Problem 59

The mass of a car is 1500 $\mathrm{kg}$ . The shape of the car's body is such that its aerodynamic drag coefficient is $D=$ 0.330 and its frontal area is $2.50 \mathrm{m}^{2} .$ Assuming the drag force is proportional to $v^{2}$ and ignoring other sources of friction, calculate the power required to maintain a speed of 100 $\mathrm{km} / \mathrm{h}$ as the car climbs a long hill sloping at $3.20^{\circ} .$

Manish Kumar

Numerade Educator

Problem 60

Review. As a prank, someone has balanced a pumpkin at the highest point of a grain silo. The silo is topped with a hemispherical cap that is frictionless when wet. The line from the center of curvature of the cap to the pumpkin makes an angle $\theta_{i}=0^{\circ}$ with the vertical. While you happen to be standing nearby in the middle of a rainy night, a breath of wind makes the pumpkin start sliding downward
from rest. It loses contact with the cap when the line from the center of the hemisphere to the pumpkin makes a certain angle with the vertical. What is this angle?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 61

A child's pogo stick (Fig. $\mathrm{P} 8.61$ ) stores energy in a spring with a force constant of $2.50 \times 10^{4} \mathrm{~N} / \mathrm{m} .$
At position $A\left(Xa=-0.100 \mathrm{~m}\right)$ the spring compression is a maximum and the child is momentarily at rest. At position
$B (Xb=0)$, 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. The combined mass of
child and pogo stick is 25.0 kg. Although the boy must lean for- ward to remain balanced, the angle is small, so let’s assume the pogo stick is vertical. Also assume the boy does not bend his legs during the motion.
(a) Calculate the total energy of the child-stick-Earth system, taking both gravitational and elastic potential energies as zero for $x=0 .$
(b) Determine $Xc$
(c) Calculate the speed of the child at x 5 0.
(d) Determine the value of x for which the kinetic energy of the system is a maximum
(e) Calculate the child’s maximum up ward speed.

Ren Jie Tuieng

Numerade Educator

Problem 62

A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P8.62a). The object has a speed of vi 5 3.00 m/s when it makes contact with a light spring (Fig. P8.62b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P8.62c). The object is then forced toward the left by the spring (Fig. P8.62d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P8.62e).
Find (a) the distance of compression d, (b) the speed v at the unstretched position when the object is moving to the left (Fig. P8.62d), and (c) the distance D where the object comes to rest.

Ren Jie Tuieng

Numerade Educator

Problem 63

A 10.0-kg block is released from rest at point in Figure P8.63. The track is frictionless except for the portion between points and , which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between points and

Keshav Singh

Numerade Educator

Problem 64

A block of mass $m_{1}=20.0 \mathrm{kg}$ $m_{2}=30.0 \mathrm{kg}$ by a massless string that passes over a light, frictionless pulley. The $30.0-\mathrm{kg}$ block is connected to a spring that has negli-
gible mass and a force constant of $k=250 \mathrm{N} / \mathrm{m}$ as shown in Figure P8.64. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0 -kg block is pulled a distance $h=20.0 \mathrm{cm}$ down the incline of angle $\theta=40.0^{\circ}$ and released from rest. Find the speed of each block when the spring is again unstretched.

Ren Jie Tuieng

Numerade Educator

Problem 65

Q C A block of mass $0.500 \mathrm{~kg}$ is pushed against a horizontal spring of negligible mass until the spring is compressed a distance $x$ (Fig. $\mathrm{P} 8.65$ ). The force constant of the spring is $450 \mathrm{~N} / \mathrm{m} .$ When it is released, the block travels along a frictionless, horizontal surface to point $@$, the bottom of a vertical circular track of radius $R=1.00 \mathrm{~m},$ and continues to move up the track. The block's speed at the bottom of the track is $v_{@}=12.0 \mathrm{~m} / \mathrm{s},$ and the block experiences an average friction force of $7.00 \mathrm{~N}$ while sliding up the track.
(a) What is $x$ ? (b) If the block were to reach the top of the track, what would be its speed at that point? (c) Does the block actually reach the top of the track, or does it fall off before reaching the top?

Ren Jie Tuieng

Numerade Educator

Problem 66

A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force $3 m g,$ so the spring compression is $4 m g / k .$ In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of $m,$ what is the greatest possible
value for $M ?$

Ren Jie Tuieng

Numerade Educator

Problem 67

A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane.
The string hits a peg located a distance d below the point of suspension (Fig. P8.67). (a) Show that if
the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position ($\theta= 90^{\circ} )$) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3$L / 5$.

Ren Jie Tuieng

Numerade Educator

Problem 68

Why is the following situation impossible? An athlete tests her hand strength by having an assistant hang weights from her belt as she hangs onto a horizontal bar with her hands. When the weights hanging on her belt have increased to 80% of her body weight, her hands can no longer support her and she drops to the floor. Frustrated at not meeting her hand-strength goal, she decides to swing on a trapeze. The trapeze consists of a bar suspended by two parallel ropes, each of length $\ell$ allowing performers to swing in a vertical circular arc (Fig. P8. 68 ). The athlete holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle $\theta_{i}=$ $60.0^{\circ}$ with respect to the vertical. As she swings several times back and forth in a circular arc, she forgets her frustration related to the hand-strength test. Assume the size of the performer's body is small compared to the length $\ell$ and air resistance is negligible.

Ren Jie Tuieng

Numerade Educator

Problem 69

A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assuming the total energy of the ball–Earth system remains constant, show that the tension in the string at the bottom is greater than the tension at the top by six times the ball’s weight.

Manish Kumar

Numerade Educator

Problem 70

An airplane of mass $1.50 \times 10^{4} \mathrm{kg}$ is in level flight, initially moving at 60.0 $\mathrm{m} / \mathrm{s}$ . The resistive force exerted by air on the airplane has a magnitude of $4.0 \times 10^{4} \mathrm{N} .$ By Newton's third law, if the engines exert a force on the exhaust
gases to expel them out of the back of the engine, the exhaust gases exert a force on the engines in the direction of the airplane's travel. This force is called thrust, and the value of the thrust in this situation is $7.50 \times 10^{4} \mathrm{N}$ . (a) Is the work done by the exhaust gases on the airplane during some time interval equal to the change in the airplane's kinetic energy? Explain. (b) Find the speed of the airplane after it has traveled $5.0 \times 10^{2} \mathrm{m} .$

Ren Jie Tuieng

Numerade Educator

Problem 71

Consider the block–spring collision discussed in Example 8.8. (a) For the situation in part (B), in which the surface exerts a friction force on the block, show that the block never arrives back at x 5 0. (b) What is the maximum value of the coefficient of friction that would allow the block to return to $x=0 ?$

Ren Jie Tuieng

Numerade Educator

Problem 72

A roller-coaster car shown in Figure P8.72 is released from rest from a height h and then moves freely with negligible friction. The roller-coaster track includes a circular loop of radius R in a vertical plane. (a) First suppose the car barely makes it around the loop; at the top of the loop, the riders are upside down and feel weightless. Find the required height h of the release point above the bottom of
the loop in terms of R. (b) Now assume the release point is at or above the minimum required height. Show that the normal force on the car at the bottom of the loop exceeds the normal force at the top of the loop by six times the car’s weight. The normal force on each rider follows the same rule. Such a large normal force is dangerous and very uncomfortable for the riders. Roller coasters are therefore
not built with circular loops in vertical planes. Figure P6.19 (page 159) shows an actual design.

Ren Jie Tuieng

Numerade Educator

Problem 73

In 1887 in Bridgeport, Connecticut, C. J. Belknap built the water slide shown in Figure P8.73 (page 232).
A rider on a small sled, of total mass 80.0 kg, pushed off to start at the top of the slide (point ) with a speed of 2.50 m/s. The chute was 9.76 m high at the top and 54.3 m long. Along its length, 725 small wheels made friction negligible. Upon leaving the chute horizontally at its bottom end (point ), the rider skimmed across the water of Long Island Sound for as much as 50 m, “skipping along like a flat pebble,” before at last coming to rest and swimming ashore, pulling his sled after him. (a) Find the speed of the sled and rider at point . (b) Model the force of water friction as a constant retarding force acting on a particle. Find the magnitude of the friction force the water exerts on the sled. (c) Find the magnitude of the force the chute exerts on the sled at point . (d) At point , the chute is horizontal but curving in the vertical plane. Assume its radius of curvature is 20.0 m. Find the force the chute exerts on the sled at point.

Ren Jie Tuieng

Numerade Educator

Problem 74

Starting from rest, a 64.0-kg person bungee jumps from a tethered hot-air balloon 65.0 m above the ground. The bungee cord has negligible mass and unstretched length 25.8 m. One end is tied to the basket of the balloon and the other end to a harness around the person’s body. The cord is modeled as a spring that obeys Hooke’s law with a spring constant of 81.0 N/m, and the person’s body is modeled as a particle. The hot-air balloon does not move. (a) Express the gravitational potential energy of the
person–Earth system as a function of the person’s variable height y above the ground. (b) Express the elastic potential energy of the cord as a function of y. (c) Express the total potential energy of the person–cord–Earth system as a function of y. (d) Plot a graph of the gravitational, elastic, and total potential energies as functions of y. (e) Assume air resistance is negligible. Determine the minimum height of the person above the ground during his plunge. (f) Does the potential energy graph show any equilibrium position or positions? If so, at what elevations? Are they stable or unstable? (g) Determine the jumper’s maximum speed.

Ren Jie Tuieng

Numerade Educator

Problem 75

A uniform board of length L is sliding along a smooth, frictionless, horizontal plane as shown in Figure P8.75a. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface is $\mu_{k^{*}}$. (a) Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary. (b) The board stops at the moment its back end reaches the boundary as shown in Figure P8.75b. Find the initial speed v of the board.

Manish Kumar

Numerade Educator

Problem 76

A uniform chain of length 8.00 m initially lies stretched out on a horizontal table. (a) Assuming the coefficient of static friction between chain and table is 0.600, show that the chain will begin to slide off the table if at least 3.00 m of it hangs over the edge of the table. (b) Determine the speed of the chain as its last link leaves the table, given that the coefficient of kinetic friction between the chain and the table is 0.400.

Vipender Yadav

Numerade Educator

Problem 77

Jane, whose mass is 50.0 $\mathrm{kg}$ , needs to swing across a river (having width $D$ ) filled with person-eating crocodiles to save Tarzan from danger. She must swing into a wind exerting constant horizontal force $\mathbf{F},$ on a vine having length $L$ and initially making an angle $\theta$ with the vertical (Fig. P8.77). Take $D=50.0 \mathrm{m}, F=110 \mathrm{N}, L=40.0 \mathrm{m},$ and $\theta=50.0^{\circ} .$ (a) With what minimum speed must Jane begin her swing to just make it to the other side? (b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing? Assume Tarzan has a mass of 80.0 $\mathrm{kg}$ .

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

A ball of mass $m=300 \mathrm{g}$ is connected by a strong string of length $L=80.0 \mathrm{cm}$ to a pivot and held in place with the string vertical. A wind exerts constant force $F$ to the right on the ball as shown in Figure $\mathrm{P} 8.78$ . The ball is released from rest. The wind makes it swing up to attain maximum height $H$ above its starting point before it swings down again. (a) Find $H$ as a function of $F$ . Evaluate $H$ for $(b) F=$ 1.00 $\mathrm{N}$ and $(\mathrm{c}) F=10.0 \mathrm{N}$ . How does $H$ behave (d) as $F$ approaches zero and (e) as $F$ approaches infinity? (f) Now consider the equilibrium height of the ball with the wind blowing. Determine it as a function of $F$ . Evaluate the equilibrium height for $(g) F=10 \mathrm{N}$ and $(\mathrm{h})$ Fgoing to infinity.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 79

A daredevil plans to bungee jump from a balloon 65.0 m above the ground. He will use a uniform elastic cord, tied to a harness around his body, to stop his fall at a point 10.0 m above the ground. Model his body as a particle and the cord as having negligible mass and obeying Hooke’s law. In a preliminary test he finds that when hanging at rest from a 5.00-m length of the cord, his body weight stretches it by 1.50 m. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon. (a) What length of cord should he use? (b) What maximum acceleration will he experience?

Ren Jie Tuieng

Numerade Educator

Problem 80

Consider the roller coaster described in Problem 52. Because of some friction between the coaster
and the track, the coaster enters the circular section at a speed of 15.0 m/s rather than the 22.0 m/s in Problem 52. Is this situation more or less dangerous for the passengers than that in Problem 52? Assume the circular section is still frictionless.

Ren Jie Tuieng

Numerade Educator