• Home
  • Textbooks
  • Physics for Scientists and Engineers with Modern Physics
  • Conservation of Energy

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

Chapter Questions

01:51

Problem 1

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.

Massimo Antonelli
Massimo Antonelli
Numerade Educator
06:26

Problem 2

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 ballEarth system at the maximum height for each ball. Let $y=0$ at the cannon.

Jacob Schulze
Jacob Schulze
Numerade Educator
03:56

Problem 3

A block of mass $m=5.00 \mathrm{~kg}$ is released from point $@$ and slides on the frictionless track shown in Figure $\mathrm{P} 8.3 .$ Determine (a) the block's speed at points (B) and @ and (b) the net work done by the gravitational force on the block as it moves from point $(\mathrm{A}$ to point $\mathrm{C}$ ).

Jacob Schulze
Jacob Schulze
Numerade Educator
03:28

Problem 4

At $11: 00 \mathrm{a} \cdot \mathrm{m}$. 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 $\begin{array}{llll}1 & 050 & 000 \text { children of average mass } & 36.0 \mathrm{~kg} \text { jumped } 12\end{array}$
times each, raising their centers of mass by $25.0 \mathrm{~cm}$ 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?

Jacob Schulze
Jacob Schulze
Numerade Educator
01:28

Problem 5

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
Manish Kumar
Numerade Educator
03:39

Problem 6

The system shown in Figure P8.6 consists of a light, inextensible cord, light, frictionless pulleys, and blocks of equal mass. Notice that block $\mathrm{B}$ 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 $\mathrm{A}$ at the moment the vertical separation of the blocks is $h$.

Supratim Pal
Supratim Pal
Numerade Educator
06:58

Problem 7

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 crateincline 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 $5.00 \mathrm{~m} ?$

Jacob Schulze
Jacob Schulze
Numerade Educator
04:54

Problem 8

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.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:00

Problem 9

A smooth circular hoop with a radius of $0.500 \mathrm{~m}$ 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 $8.00 \mathrm{~m} / \mathrm{s}$. After one revolution, its speed has dropped to $6.00 \mathrm{~m} / \mathrm{s}$ 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.

Jacob Schulze
Jacob Schulze
Numerade Educator
03:38

Problem 10

As shown in Figure $\mathrm{P} 8.10$, a green bead of mass $25 \mathrm{~g}$ slides along a straight wire. The length of the wire from point $@$ to point
(B) is $0.600 \mathrm{~m},$ and point $\mathbb{Q}$ is $0.200 \mathrm{~m}$ higher than point
(B). A constant friction force of magnitude 0.0250 N acts on the bead. (a) If the bead is released from rest at point $@$, what is its speed at point (B)? (b) A red bead of mass $25 \mathrm{~g}$ 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 $(\bar{\theta}),$ which bead reaches point $@$ with a higher speed? Explain.

Jacob Schulze
Jacob Schulze
Numerade Educator
01:57

Problem 11

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 particle is $18.0 \mathrm{~J}$. (a) If only conservative forces act on the particle, what are the potential energy and 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 (b).

Jacob Schulze
Jacob Schulze
Numerade Educator
09:05

Problem 12

A $1.50-\mathrm{kg}$ 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
08:34

Problem 13

A child of mass $m$ starts from rest and slides without friction from a height $h$ along a slide next to a pool (Fig. $\mathrm{P} 8.13$ ). 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 waterslide.
(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

Jacob Schulze
Jacob Schulze
Numerade Educator
03:33

Problem 14

An 80.0 -kg skydiver jumps out of a balloon at an altitude of $1000 \mathrm{~m}$ and opens his parachute at an altitude of $200 \mathrm{~m}$
(a) Assuming the total retarding force on the skydiver is constant at $50.0 \mathrm{~N}$ with the parachute closed and constant at $3600 \mathrm{~N}$ 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 $5.00 \mathrm{~m} / \mathrm{s}$ ? (d) How realistic is the assumption that the total retarding force is constant? Explain.

Surjit Tewari
Surjit Tewari
Numerade Educator
03:23

Problem 15

You have spent a long day skiing and are tired. You are standing at the top of a hill, looking at the lodge at the bottom of the hill. You are so tired that you want to simply start from rest and coast down the slope, without pushing with your poles or doing anything else to change your motion. You want to let gravity do all the work! You have a choice of two trails to reach the lodge. Both trails have the same coefficient of friction $\mu_{k}$. In addition, both trails represent the same horizontal separation between the initial and final points. Trail A has a short, steep downslope and then a long, flat coast to the lodge. Trail $\mathrm{B}$ has a long, gentle downslope and then a short remaining flat coast to the lodge. Which trail will result in your arriving at the lodge with the highest final speed?

Jacob Schulze
Jacob Schulze
Numerade Educator
02:09

Problem 16

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. (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
03:16

Problem 17

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
Manish Kumar
Numerade Educator
01:37

Problem 18

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
01:39

Problem 19

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.

Surjit Tewari
Surjit Tewari
Numerade Educator
02:54

Problem 20

There is a $5 \mathrm{~K}$ event coming up in your town. While talking to your grandmother, who uses an electric scooter for mobility, she says that she would like to accompany you on her scooter while you walk the $5.00-\mathrm{km}$ distance. The manual

Jacob Schulze
Jacob Schulze
Numerade Educator
03:00

Problem 21

For 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 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.

Surjit Tewari
Surjit Tewari
Numerade Educator
03:34

Problem 22

Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is one kilocalorie, defined as $1 \mathrm{kcal}=4186 \mathrm{~J} .$ Metabolizing $1 \mathrm{~g}$ 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 $0.150 \mathrm{~m}$ high, in $65.0 \mathrm{~s}$. 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 $100 \mathrm{~J}$ from metabolizing fat, $20 \mathrm{~J}$ goes into doing mechanical work (here, climbing stairs). The remainder goes into extra internal energy. Assume the student's mass is $75.0 \mathrm{~kg}$. (a) How many times must the student run the flight of stairs to lose $1.00 \mathrm{~kg}$ 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?

Surjit Tewari
Surjit Tewari
Numerade Educator
03:51

Problem 23

A block of mass $m=200 \mathrm{~g}$ is released from rest at point $@$ along the horizontal diameter on the inside of hemispherical bowl of radius $R=30.0 \mathrm{~cm},$ and the surface of the bowl is rough (Fig. P8.23). The block's speed at point (B) is $1.50 \mathrm{~m} / \mathrm{s}$.(a) What is its kinetic energy at point (B)? (b) How much mechanical energy is transformed into internal energy as the block moves from point $@$ to point $(\mathrm{B})$ ? determine the coefficient of friction from these results in any simple manner?
(d) Explain your answer to part (c).

Jacob Schulze
Jacob Schulze
Numerade Educator
01:32

Problem 24

Make an order-of-magnitude estimate of your power output as you climb stairs. In your solution, state the physical quantities you take as data and the values you measure or estimate for them. Do you consider your peak power or your sustainable power?

Surjit Tewari
Surjit Tewari
Numerade Educator
08:32

Problem 25

You are working with a team that is designing a new roller coaster-type amusement park ride for a major theme park. You are present for the testing of the ride, in which an empty $250-\mathrm{kg}$ car is sent along the entire ride. Near the end of the ride, the car is at near rest at the top of a $110-\mathrm{m}$ tall track. It then enters a final section, rolling down an undulating hill to ground level. The total length of track for this final section from the top to the ground is $250 \mathrm{~m}$. For the first $230 \mathrm{~m},$ a constant friction force of $50.0 \mathrm{~N}$ acts from computer-controlled brakes. For the last $20 \mathrm{~m},$ which is horizontal at ground level, the computer increases the friction force to a value required for the speed to be reduced to zero just as the car arrives at the point on the track at which the passengers exit. (a) Determine the required constant friction force for the last $20 \mathrm{~m}$ for the empty test car.
(b) Find the highest speed reached by the car during the final section of track length $250 \mathrm{~m}$.
(c) You are asked by your team supervisor to determine the answers to parts (a) and (b) for a fully loaded car with an upper limit of $450 \mathrm{~kg}$ of passenger mass. Find these new values.
(d) The required friction force in part (c) is well within design limits. The fastest speed, however, is well below that of current leading rides, so you would like to increase the maximum speed. You can't make the tower taller above ground, so you decide to include a feature where part of the track goes underground. Determine the depth to which the underground part of the ride must go to increase the maximum speed to $55.0 \mathrm{~m} / \mathrm{s}$. Assume the overall length of the first part of the track remains at $230 \mathrm{~m}$ and the length of track from the top to the lowest underground point is $150 \mathrm{~m}$. The same 50.0 - $\mathrm{N}$ friction force acts on the entire $230-\mathrm{m}$ section of track. (e) Is the construction in part
(d) feasible?

Jacob Schulze
Jacob Schulze
Numerade Educator
08:02

Problem 26

Review. As shown in Figure $\mathrm{P8} .26, \mathrm{a}$ 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 rest on the horizontal
table at a height $h=1.20 \mathrm{~m}$ above the floor, to $m_{2},$ a hanging 1.90 -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 $m_{1}$ on the floor. (c) What is the shortest length of the string so thatit 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?

Jacob Schulze
Jacob Schulze
Numerade Educator
06:59

Problem 27

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 $10.0 \mathrm{~N}$. At what position of the block does its maximum speed occur in this situation?

Jacob Schulze
Jacob Schulze
Numerade Educator
02:29

Problem 28

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 bottom of the path. The pitcher maintains a component of force on the $0.180-\mathrm{kg}$ ball of constant magnitude $12.0 \mathrm{~N}$ 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 $25.0 \mathrm{~m} / \mathrm{s}$

Surjit Tewari
Surjit Tewari
Numerade Educator
02:23

Problem 29

Jonathan is riding a bicycle and encounters a hill of height $7.30 \mathrm{~m}$. At the base of the hill, he is traveling at $6.00 \mathrm{~m} / \mathrm{s}$ When he reaches the top of the hill, he is traveling at $1.00 \mathrm{~m} / \mathrm{s}$. Jonathan and his bicycle together have a mass of $85.0 \mathrm{~kg} .$ 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?

Surjit Tewari
Surjit Tewari
Numerade Educator
01:31

Problem 30

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?

Surjit Tewari
Surjit Tewari
Numerade Educator
05:39

Problem 31

As the driver steps on the gas pedal, a car of mass $1160 \mathrm{~kg}$ 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 $t=2.50 \mathrm{~s} ?$ (b) What is the minimum average power output of the engine over this time interval? (c) Why is the value in part (b) described as the minimum value?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:41

Problem 32

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 cylinderas shown in Figure $\mathrm{P} 8.32$. 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.

Jacob Schulze
Jacob Schulze
Numerade Educator
05:36

Problem 33

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
$\frac{1}{2}(46.0 \mathrm{~kg})(2.40 \mathrm{~m} / \mathrm{s})^{2}+(46.0 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(2.80 \mathrm{~m}+x)=$
$$
\frac{1}{2}\left(1.94 \times 10^{4} \mathrm{~N} / \mathrm{m}\right) x^{2}
$$
(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
05:13

Problem 34

Review. Why is the following situation impossible? A new highspeed 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 cir-
cle so that the passengers are upside down for a short time interval. The radius of the circular section is $12.0 \mathrm{~m},$ and the coaster enters the bottom of the circular section at a speed of $22.0 \mathrm{~m} / \mathrm{s}$. Assume the coaster moves without friction on the track and model the coaster as a particle.

Jacob Schulze
Jacob Schulze
Numerade Educator
03:35

Problem 35

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 $\mathrm{P} 8.35 .$ (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 $x_{i} / 2=3.00 \mathrm{~cm} ?$ (d) Why isn't the answer to part (c) half the answer to part (b)?

Manish Kumar
Manish Kumar
Numerade Educator
06:09

Problem 36

More than 2900 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 inan 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.

Jacob Schulze
Jacob Schulze
Numerade Educator
05:19

Problem 37

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
03:03

Problem 38

Review. 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. P8.38). 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.

Surjit Tewari
Surjit Tewari
Numerade Educator
05:11

Problem 39

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
Ren Jie Tuieng
Numerade Educator
06:34

Problem 40

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. $\mathrm{P} 8.40$ ). (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 $\left(\theta=90^{\circ}\right)$ 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
Ren Jie Tuieng
Numerade Educator
04:54

Problem 41

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 ballEarth 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.

Jacob Schulze
Jacob Schulze
Numerade Educator
01:51

Problem 42

You are working in the distribution center of a large online shopping site. Efforts are being made to increase the number of packages per unit time that are being loaded onto a conveyor belt to be carried to waiting trucks. But the motor driving the conveyor belt is having difficulty keeping up with the increased demands. Your supervisor has asked you to determine the requirements for a new motor that can provide enough power to keep the conveyor belt moving smoothly under the increased loading rate. You are given the following information: The design goal is to have $50.0-\mathrm{kg}$ packages loaded onto the belt at several locations at an average rate of 5.00 packages per second. The belt moves at a horizontal speed of $1.35 \mathrm{~m} / \mathrm{s}$. Humans at the various locations along the belt place the package on the belt so that it is initially at rest relative to the floor of the building just before being dropped from negligible height onto the belt. Your task is to determine the minimum power the driving motor must have to accelerate these packages and keep the belt moving at constant speed.

Jacob Schulze
Jacob Schulze
Numerade Educator
06:30

Problem 43

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 $x=0 .$ (b) What is the maximum value of the coefficient of friction that would allow the block to
return to $x=0 ?$

Jacob Schulze
Jacob Schulze
Numerade Educator
12:10

Problem 44

Starting from rest, a 64.0 -kg person bungee jumps from a tethered hot-air balloon $65.0 \mathrm{~m}$ above the ground. The bungee cord has negligible mass and unstretched length $25.8 \mathrm{~m} .$ 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 $81.0 \mathrm{~N} / \mathrm{m},$ 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.

Jacob Schulze
Jacob Schulze
Numerade Educator
06:13

Problem 45

Review. A uniform board of length $L$ is sliding along a smooth, frictionless, horizontal plane as shown in Figure P8.45a. 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 $\mathrm{P} 8.45 \mathrm{~b}$. Find the initial speed $v$ of the board.

Jacob Schulze
Jacob Schulze
Numerade Educator
15:50

Problem 46

A uniform chain of length $8.00 \mathrm{~m}$ 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 $3.00 \mathrm{~m}$ 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 .

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
03:56

Problem 47

What If? Consider the roller coaster described in Problem $34 .$ Because of some friction between the coaster and
the track, the coaster enters the circular section at a speed of $15.0 \mathrm{~m} / \mathrm{s}$ rather than the $22.0 \mathrm{~m} / \mathrm{s}$ in Problem $34 .$ Is this situation more or less dangerous for the passengers than that in Problem 34 ? Assume the circular section is still frictionless.

Jacob Schulze
Jacob Schulze
Numerade Educator