University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 6

Potential Energy and Energy Conservation - all with Video Answers

Educators

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Chapter Questions

Problem 1

A block of mass $5.0 \mathrm{~kg}$ slides without friction at a speed of $8.0 \mathrm{~m} / \mathrm{s}$ on a horizontal table surface until it strikes and sticks to a mass of $4.0 \mathrm{~kg}$ attached to a horizontal spring (with spring constant of $k=2000.0 \mathrm{~N} / \mathrm{m}$ ), which in turn is attached to a wall. How far is the spring compressed before the masses come to rest?
a) $0.40 \mathrm{~m}$
b) $0.54 \mathrm{~m}$
c) $0.30 \mathrm{~m}$
d) $0.020 \mathrm{~m}$
e) $0.67 \mathrm{~m}$

Akshaya Rs

Akshaya Rs

Numerade Educator

Problem 2

A pendulum swings in a vertical plane. At the bottom of the swing, the kinetic energy is $8 \mathrm{~J}$ and the gravitational potential energy is 4 J. At the highest position of its swing, the kinetic and gravitational potential energies are
a) kinetic energy $=0 \mathrm{~J}$ and gravitational potential energy $=4 \mathrm{~J}$
b) kinetic energy $=12 \mathrm{~J}$ and gravitational potential energy $=0 \mathrm{~J}$
c) kinetic energy $=0 \mathrm{~J}$ and gravitational potential energy $=12 \mathrm{~J}$
d) kinetic energy $=4$ J and gravitational potential energy $=8 \mathrm{~J}$
e) kinetic energy $=8 \mathrm{~J}$ and gravitational potential energy $=4$ J.

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 3

A ball of mass $0.5 \mathrm{~kg}$ isreleased from rest at point $A$, which is $5 \mathrm{~m}$ above the bottom of a tank of oil, as shown in the figure. At $B$, which is $2 \mathrm{~m}$ above the bottom of the tank, the ball has a speed of $6 \mathrm{~m} / \mathrm{s}$. The work done on the ball by the force of fluid friction is
a) +15 J.
b) $+9 \mathrm{~J}$
c) $-15 \mathrm{~J}$.
d) $-9 J$.
e) $-5.7 \mathrm{~J}$

Akshaya Rs

Numerade Educator

Problem 4

A child throws three identical marbles from the same height above the ground so that they land on the flat roof of a building. The marbles are launched with the same initial speed. The first marble, marble $\mathrm{A}$, is thrown at an angle of $75^{\circ}$ above horizontal, while marbles $\mathrm{B}$ and $\mathrm{C}$ are thrown with launch angles of $60^{\circ}$ and $45^{\circ}$, respectively. Neglecting air resistance, rank the marbles according to the speeds with which they hit the roof.
a) $A<B<C$
b) $C<B<A$
c) $\mathrm{A}$ and $\mathrm{C}$ have the same speed; B has a lower speed.
d) $\mathrm{B}$ has the highest speed; $\mathrm{A}$ and $C$ have the same speed.
e) $\mathrm{A}, \mathrm{B},$ and $\mathrm{C}$ all hit the roof with the same speed.

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 5

Which of the following is not a valid potential energy function for the spring force $F=-k x ?$
a) $\left(\frac{1}{2}\right) k x^{2}$
b) $\left(\frac{1}{2}\right) k x^{2}+10 \mathrm{~J}$
c) $\left(\frac{1}{2}\right) k x^{2}-10 \mathrm{~J}$
d) $-\left(\frac{1}{2}\right) k x^{2}$
e) None of the above is valid.

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 6

You use your hand to stretch a spring to a displacement $x$ from its equilibrium position and then slowly bring it back to that position. Which is true?
a) The spring's $\Delta U$ is positive.
b) The spring's $\Delta U$ is negative.
c) The hand's $\Delta U$ is positive.
d) The hand's $\Delta U$ is negative.
e) None of the above statements is true.

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 7

In question $6,$ what is the work done by the hand?
a) $-\left(\frac{1}{2}\right) k x^{2}$
b) $+\left(\frac{1}{2}\right) k x^{2}$
c) $\left(\frac{1}{2}\right) m v^{2},$ where $v$ is the speed of the hand
d) zero
e) none of the above

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 8

Which of the following is not a unit of energy?
a) newton-meter
b) joule
c) kilowatt-hour
d) $\operatorname{kg} \mathrm{m}^{2} / \mathrm{s}^{2}$
e) all of the above

Akshaya Rs

Numerade Educator

Problem 9

A spring has a spring constant of $80 \mathrm{~N} / \mathrm{m}$. How much potential energy does it store when stretched by $1.0 \mathrm{~cm} ?$
a) $4.0 \cdot 10^{-3}$ J
b) $0.40 \mathrm{~J}$
c) 80
d) $800 \mathrm{~J}$
e) $0.8 \mathrm{~J}$

Akshaya Rs

Numerade Educator

Problem 10

Can the kinetic energy of an object be negative? Can the potential energy of an object be negative?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 11

a) If you jump off a table onto the floor, is your mechanical energy conserved? If not, where does it go? b) A car moving down the road smashes into a tree. Is the mechanical energy of the car conserved? If not, where does it go?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 12

How much work do you do when you hold a bag of groceries while standing still? How much work do you do when carrying the same bag a distance $d$ across the parking lot of the grocery store?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 13

An arrow is placed on a bow, the bowstring is pulled back, and the arrow is shot straight up into the air; the arrow then comes back down and sticks into the ground. Describe all of the changes in work and energy that occur.

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 14

Two identical billiard balls start at the same height and the same time and roll along different tracks, as shown in the figure.
a) Which ball has the highest speed at the end?
b) Which one will get to the end first?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 15

A girl of mass $49.0 \mathrm{~kg}$ is on a swing, which has a mass of $1.0 \mathrm{~kg} .$ Suppose you pull her back until her center of mass is $2.0 \mathrm{~m}$ above the ground. Then you let her $\mathrm{go},$ and she swings out and returns to the same point. Are all forces acting on the girl and swing conservative?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 16

Can a potential energy function be defined for the force of friction?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 17

Can the potential energy of a spring be negative?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 18

One end of a rubber band is tied down and you pull on the other end to trace a complicated closed trajectory. If you were to measure the elastic force $F$ at every point and took its scalar product with the local displacements, $\vec{F} \cdot \Delta \vec{r},$ and then summed all of these, what would you get?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 19

Can a unique potential energy function be identified with a particular conservative force?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 20

In skvdiving, the vertical velocity component of the skydiver is typically zero at the moment he or she leaves the plane; the vertical component of the velocity then increases until the skydiver reaches terminal speed (see Chapter 4 ). Let's make a simplified model of this motion. We assume that the horizontal velocity component is zero. The vertical velocity component increases linearly, with acceleration $a_{y}=-g,$ until the skydiver reaches terminal velocity, after which it stays constant. Thus, our simplified model assumes free fall without air resistance followed by falling at constant speed. Sketch the kinetic energy, potential energy, and total energy as a function of time for this model.

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 21

A projectile of mass $m$ is launched from the ground at $t=0$ with a speed $v_{0}$ and at an angle $\theta_{0}$ above the horizontal. Assuming that air resistance is negligible, write the kinetic, potential, and total energies of the projectile as explicit functions of time.

Prashant Bana

Numerade Educator

Problem 22

The energy height, $H$, of an aircraft of mass $m$ at altitude $h$ and with speed $v$ is defined as its total energy (with the zero of the potential energy taken at ground level) divided by its weight. Thus, the energy height is a quantity with units of length.
a) Derive an expression for the energy height, $H$, in terms of the quantities $m, h$, and $v$.
b) A Boeing 747 jet with mass $3.5 \cdot 10^{5} \mathrm{~kg}$ is cruising in level flight at $250.0 \mathrm{~m} / \mathrm{s}$ at an altitude of $10.0 \mathrm{~km} .$ Calculate the value of its energy height. Note: The energy height is the maximum altitude an aircraft can reach by "zooming" (pulling into a vertical climb without changing the engine thrust). This maneuver is not recommended for a 747 , however.

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 23

A body of mass $m$ moves in one dimension under the influence of a force, $F(x)$, which depends only on the body's position.
a) Prove that Newton's Second Law and the law of conservation of energy for this body are exactly equivalent.
b) Explain, then, why the law of conservation of energy is considered to be of greater significance than Newton's Second Law.

Paul Gabriel

Numerade Educator

Problem 24

The molecular bonding in a diatomic molecule such as the nitrogen $\left(\mathrm{N}_{2}\right)$ molecule can be modeled by the Lennard Jones potential, which has the form
$$
U(x)=4 U_{0}\left(\left(\frac{x_{0}}{x}\right)^{12}-\left(\frac{x_{0}}{x}\right)^{6}\right)
$$
where $x$ is the separation distance between the two nuclei and $x_{0}$, and $U_{0}$ are constants. Determine, in terms of these constants, the following:
a) the corresponding force function;
b) the equilibrium separation $x_{0}$, which is the value of $x$ for which the two atoms experience zero force from each other; and
c) the nature of the interaction (repulsive or attractive) for separations larger and smaller than $x_{0}$.

Paul Gabriel

Numerade Educator

Problem 25

A particle of mass $m$ moving in the $x y$ -plane is confined by a two-dimensional potential function, $U(x, y)=\frac{1}{2} k\left(x^{2}+y^{2}\right)$
a) Derive an expression for the net force, $\vec{F}=F_{x} \hat{x}+F_{y} \hat{y}$.
b) Find the equilibrium point on the $x y$ -plane.
c) Describe qualitatively the effect of net force.
d) What is the magnitude of the net force on the particle at the coordinate (3.00,4.00) in $\mathrm{cm}$ if $k=10.0 \mathrm{~N} / \mathrm{cm} ?$
e) What are the turning points if the particle has $10.0 \mathrm{~J}$ of total mechanical energy?

Paul A.

California State Polytechnic University, Pomona

Problem 26

For a rock dropped from rest from a height $h$, to calculate the speed just before it hits the ground, we use the conservation of mechanical energy and write $m g h=\frac{1}{2} m v^{2}$ The mass cancels out, and we solve for $v$. A very common error made by some beginning physics students is to assume, based on the appearance of this equation, that they should set the kinetic energy equal to the potential energy at the same point in space. For example, to calculate the speed $v_{1}$ of the rock at some height $y_{1}<h,$ they often write $m g y_{1}=\frac{1}{2} m v_{1}^{2}$ and solve for $v_{1} .$ Explain why this approach is wrong.

Paul Gabriel

Numerade Educator

Problem 27

What is the gravitational potential energy of a $2.0-\mathrm{kg}$ book $1.5 \mathrm{~m}$ above the floor?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 28

a) If the gravitational potential energy of a 40.0 -kg rock is 500 . J relative to a value of zero on the ground, how high is the rock above the ground?
b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 29

A rock of mass $0.773 \mathrm{~kg}$ is hanging from a string of length $2.45 \mathrm{~m}$ on the Moon, where the gravitational acceleration is a sixth of that on Earth. What is the change in gravitational potential energy of this rock when it is moved so that the angle of the string changes from $3.31^{\circ}$ to $14.01^{\circ} ?$ (Both angles are measured relative to the vertical.)

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 30

A 20.0 -kg child is on a swing attached to ropes that are $L=1.50 \mathrm{~m}$ long. Take the zero of the gravitational potential energy to be at the position of the child when the ropes are horizontal.
a) Determine the child's gravitational potential energy when the child is at the lowest point of the circular trajectory.
b) Determine the child's gravitational potential energy when the ropes make an angle of $45.0^{\circ}$ relative to the vertical.
c) Based on these results, which position has the higher potential energy?

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 31

A $1.50 \cdot 10^{3}-\mathrm{kg}$ car travels $2.50 \mathrm{~km}$ up an incline at constant velocity. The incline has an angle of $3.00^{\circ}$ with respect to the horizontal. What is the change in the car's potential energy? What is the net work done on the car?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 32

A constant force of $40.0 \mathrm{~N}$ is needed to keep a car traveling at constant speed as it moves $5.0 \mathrm{~km}$ along a road. How much work is done? Is the work done on or by the car?

Griffin Goodwin

Griffin Goodwin

Numerade Educator

Problem 33

A piñata of mass $3.27 \mathrm{~kg}$ is attached to a string tied to a hook in the ceiling. The length of the string is $0.81 \mathrm{~m},$ and the piñata is released from rest from an initial position in which the string makes an angle of $56.5^{\circ}$ with the vertical. What is the work done by gravity by the time the string is in a vertical position for the first time?

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 34

A particle is moving along the $x$ -axis subject to the potential energy function $U(x)=1 / x+x^{2}+x-1$
a) Express the force felt by the particle as a function of $x$.
b) Plot this force and the potential energy function.
c) Determine the net force on the particle at the coordinate $x=2.00 \mathrm{~m}$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 35

Calculate the force $F(y)$ associated with each of the following potential energies:
a) $U=a y^{3}-b y^{2}$
b) $U=U_{0} \sin (c y)$

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 36

The potential energy of a certain particle is given by $U=10 x^{2}+35 z^{3}$. Find the force vector exerted on the particle.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 37

A ball is thrown up in the air, reaching a height of $5.00 \mathrm{~m}$. Using energy conservation considerations, determine its initial speed.

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 38

A cannonball of mass $5.99 \mathrm{~kg}$ is shot from a cannon at an angle of $50.21^{\circ}$ relative to the horizontal and with an initial speed of $52.61 \mathrm{~m} / \mathrm{s}$. As the cannonball reaches the highest point of its trajectory, what is the gain in its potential energy relative to the point from which it was shot?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 39

A basketball of mass $0.624 \mathrm{~kg}$ is shot from a vertical height of $1.2 \mathrm{~m}$ and at a speed of $20.0 \mathrm{~m} / \mathrm{s}$. After reaching its maximum height, the ball moves into the hoop on its downward path, at $3.05 \mathrm{~m}$ above the ground. Using the principle of energy conservation, determine how fast the ball is moving just before it enters the hoop.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 40

A classmate throws a $1.0-\mathrm{kg}$ book from a height of $1.0 \mathrm{~m}$ above the ground straight up into the air. The book reaches a maximum height of $3.0 \mathrm{~m}$ above the ground and begins to fall back. Assume that $1.0 \mathrm{~m}$ above the ground is the reference level for zero gravitational potential energy. Determine
a) the gravitational potential energy of the book when it hits the ground.
b) the velocity of the book just before hitting the ground.

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 41

Suppose you throw a 0.052 -kg ball with a speed of $10.0 \mathrm{~m} / \mathrm{s}$ and at an angle of $30.0^{\circ}$ above the horizontal from a building $12.0 \mathrm{~m}$ high.
a) What will be its kinetic energy when it hits the ground?
b) What will be its speed when it hits the ground?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 42

A uniform chain of total mass $m$ is laid out straight on a frictionless table and held stationary so that one-third of its length, $L=1.00 \mathrm{~m},$ is hanging vertically over the edge of the table. The chain is then released. Determine the speed of the chain at the instant when only one-third of its length remains on the table.

Paul A.

California State Polytechnic University, Pomona

Problem 43

a) If you are at the top of a toboggan run that is $40.0 \mathrm{~m}$ high, how fast will you be going at the bottom, provided you can ignore friction between the sled and the track?
b) Does the steepness of the run affect how fast you will be going at the bottom?
c) If you do not ignore the small friction force, does the steepness of the track affect the value of the speed at the bottom?

Paul Gabriel

Numerade Educator

Problem 44

A block of mass $0.773 \mathrm{~kg}$ on a spring with spring constant $239.5 \mathrm{~N} / \mathrm{m}$ oscillates vertically with amplitude $0.551 \mathrm{~m}$. What is the speed of this block at a distance of $0.331 \mathrm{~m}$ from the equilibrium position?

Christopher Dzorkpata

Christopher Dzorkpata

Numerade Educator

Problem 45

A spring with $k=10.0 \mathrm{~N} / \mathrm{cm}$ is initially stretched $1.00 \mathrm{~cm}$ from its equilibrium length.
a) How much more energy is needed to further stretch the spring to $5.00 \mathrm{~cm}$ beyond its equilibrium length?
b) From this new position, how much energy is needed to compress the spring to $5.00 \mathrm{~cm}$ shorter than its equilibrium position?

Paul Gabriel

Numerade Educator

Problem 46

A 5.00 -kg ball of clay is thrown downward from a height of $3.00 \mathrm{~m}$ with a speed of $5.00 \mathrm{~m} / \mathrm{s}$ onto a spring with $k=$ $1600 . \mathrm{N} / \mathrm{m} .$ The clay compresses the spring a certain maximum amount before momentarily stopping
a) Find the maximum compression of the spring.
b) Find the total work done on the clay during the spring's compression.

Alex Garger

Numerade Educator

Problem 47

A horizontal slingshot consists of two light, identical springs (with spring constants of $30.0 \mathrm{~N} / \mathrm{m}$ ) and a light cup that holds a 1.00 -kg stone. Each spring has an equilibrium length of $50.0 \mathrm{~cm}$. When the springs are in equilibrium, they line up vertically. Suppose that the cup containing the mass is pulled to $x=70.0 \mathrm{~cm}$ to the left of the vertical and then released. Determine
a) the system's total mechanical energy.
b) the speed of the stone at $x=0$

LB

Laura Bravo

Numerade Educator

Problem 48

Suppose the stone in Problem 6.47 is instead launched vertically and the mass is a lot smaller $(m=0.100 \mathrm{~kg})$. Take the zero of the gravitational potential energy to be at the equilibrium point.
a) Determine the total mechanical energy of the system.
b) How fast is the stone moving as it passes the equilibrium point?

LB

Laura Bravo

Numerade Educator

Problem 49

A 80.0 -kg fireman slides down a 3.00 -m pole by applying a frictional force of $400 .$ N against the pole with his hands. If he slides from rest, how fast is he moving once he reaches the ground?

Paul Gabriel

Numerade Educator

Problem 50

A large air-filled 0.100 -kg plastic ball is thrown up into the air with an initial speed of $10.0 \mathrm{~m} / \mathrm{s}$. At a height of $3.00 \mathrm{~m}$ the ball's speed is $3.00 \mathrm{~m} / \mathrm{s}$. What fraction of its original energy has been lost to air friction?

Paul Gabriel

Numerade Educator

Problem 51

How much mechanical energy is lost to friction if a 55.0-kg skier slides down a ski slope at constant speed of $14.4 \mathrm{~m} / \mathrm{s}$ ? The slope is $123.5 \mathrm{~m}$ long and makes an angle of $14.7^{\circ}$ with respect to the horizontal.

Paul Gabriel

Numerade Educator

Problem 52

A truck of mass 10,212 kg moving at a speed of $61.2 \mathrm{mph}$ has lost its brakes. Fortunately, the driver finds a runaway lane, a gravel-covered incline that uses friction to stop a truck in such a situation; see the figure. In this case, the incline makes an angle of $\theta=40.15^{\circ}$ with the horizontal, and the gravel has a coefficient of friction of 0.634 with the tires of the truck. How far along the incline $(\Delta x)$ does the truck travel before it stops?

LB

Laura Bravo

Numerade Educator

Problem 53

A snowboarder of mass $70.1 \mathrm{~kg}$ (including gear and clothing), starting with a speed of $5.1 \mathrm{~m} / \mathrm{s}$, slides down a slope at an angle $\theta=37.1^{\circ}$ with the horizontal. The coefficient of kinetic friction is $0.116 .$ What is the net work done on the snowboarder in the first 5.72 s of descent?

Paul Gabriel

Numerade Educator

Problem 54

The greenskeepers of golf courses use a stimpmeter to determine how "fast" their greens are. A stimpmeter is a straight aluminum bar with a V-shaped groove on which a golf ball can roll. It is designed to release the golf ball once the angle of the bar with the ground reaches a value of $\theta=20.0^{\circ} .$ The golf ball $($ mass $=1.62 \mathrm{oz}=0.0459 \mathrm{~kg})$ rolls 30.0 in down the bar and then continues to roll along the green for several feet. This distance is called the "reading." The test is done on a level part of the green, and stimpmeter readings between 7 and $12 \mathrm{ft}$ are considered acceptable. For a stimpmeter reading of $11.1 \mathrm{ft},$ what is the coefficient of friction between the ball and the green? (The ball is rolling and not sliding, as we usually assume when considering friction, but this does not change the result in this case.)

LB

Laura Bravo

Numerade Educator

Problem 55

A 1.00 -kg block is pushed up and down a rough plank of length $L=2.00 \mathrm{~m},$ inclined at $30.0^{\circ}$ above the horizontal. From the bottom, it is pushed a distance $L / 2$ up the plank, then pushed back down a distance $L / 4,$ and finally pushed back up the plank until it reaches the top end. If the coefficient of kinetic friction between the block and plank is $0.300,$ determine the work done by the block against friction.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 56

A 1.00 -kg block initially at rest at the top of a 4.00 -m incline with a slope of $45.0^{\circ}$ begins to slide down the incline. The upper half of the incline is frictionless, while the lower half is rough, with a coefficient of kinetic friction $\mu_{\mathrm{k}}=0.300$.
a) How fast is the block moving midway along the incline, before entering the rough section?
b) How fast is the block moving at the bottom of the incline?

LB

Laura Bravo

Numerade Educator

Problem 57

A spring with a spring constant of $500 . \mathrm{N} / \mathrm{m}$ is used to propel a 0.500 -kg mass up an inclined plane. The spring is compressed $30.0 \mathrm{~cm}$ from its equilibrium position and launches the mass from rest across a horizontal surface and onto the plane. The plane has a length of $4.00 \mathrm{~m}$ and is inclined at $30.0^{\circ} .$ Both the plane and the horizontal surface have a coefficient of kinetic friction with the mass of $0.350 .$ When the spring is compressed, the mass is $1.50 \mathrm{~m}$ from the bottom of the plane.
a) What is the speed of the mass as it reaches the bottom of the plane?
b) What is the speed of the mass as it reaches the top of the plane?
c) What is the total work done by friction from the beginning to the end of the mass's motion?

Brian Francisco

Brian Francisco

Numerade Educator

Problem 58

The sled shown in the figure leaves the starting point with a velocity of $20.0 \mathrm{~m} / \mathrm{s}$. Use the work-energy theorem to calculate the sled's speed at the end of the track or the maximum height it reaches if it stops before reaching the end.

LB

Laura Bravo

Numerade Educator

Problem 59

On the segment of roller coaster track shown in the figure, a cart of mass $237.5 \mathrm{~kg}$ starts at $x=0$ with a speed of $16.5 \mathrm{~m} / \mathrm{s}$. Assuming that dissipation of energy due to friction is small enough to be ignored, where is the turning point of this trajectory?

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 60

A 70.0 -kg skier moving horizontally at $4.50 \mathrm{~m} / \mathrm{s}$ encounters a $20.0^{\circ}$ incline.
a) How far up the incline will the skier move before she momentarily stops, ignoring friction?
b) How far up the incline will the skier move if the coefficient of kinetic friction between the skies and snow is $0.100 ?$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 61

A 0.200 -kg particle is moving along the $x$ -axis, subject to the potential energy function shown in the figure, where $U_{\mathrm{A}}=50.0 \mathrm{~J}, U_{\mathrm{B}}=0 \mathrm{~J}, U_{\mathrm{C}}=25.0 \mathrm{~J}, U_{\mathrm{D}}=10.0 \mathrm{~J},$ and $U_{\mathrm{E}}=60.0 \mathrm{~J}$
along the path. If the particle was initially at $x=4.00 \mathrm{~m}$ and had a total mechanical energy of $40.0 \mathrm{~J}$, determine:
a) the particle's speed at $x=3.00 \mathrm{~m}$
b) the particle's speed at $x=4.50 \mathrm{~m}$, and
c) the particle's turning points.

Manish Jain

Numerade Educator

Problem 62

A ball of mass $1.84 \mathrm{~kg}$ is dropped from a height $y_{1}=$$1.49 \mathrm{~m}$ and then bounces back up to a height of $y_{2}=0.87 \mathrm{~m}$ How much mechanical energy is lost in the bounce? The effect of air resistance has been experimentally found to be negligible in this case, and you can ignore it.

Paul Gabriel

Numerade Educator

Problem 63

A car of mass $987 \mathrm{~kg}$ is traveling on a horizontal segment of a freeway with a speed of $64.5 \mathrm{mph}$. Suddenly, the driver has to hit the brakes hard to try to avoid an accident up ahead. The car does not have an ABS (antilock braking system), and the wheels lock, causing the car to slide some distance before it is brought to a stop by the friction force between the car's tires and the road surface. The coefficient of kinetic friction is $0.301 .$ How much mechanical energy is lost to heat in this process?

Paul Gabriel

Numerade Educator

Problem 64

Two masses are connected by a light string that goes over a light, frictionless pulley, as shown in the figure. The 10.0 -kg mass is released and falls through a vertical distance of $1.00 \mathrm{~m}$ before hitting the ground. Use conservation of mechanical energy to determine:
a) how fast the 5.00 -kg mass is moving just before the 10.0 -kg mass hits the ground; and
b) the maximum height attained by the 5.00 -kg mass.

Manish Jain

Numerade Educator

Problem 65

In 1896 in Waco, Texas, William George Crush, owner of the K-T (or "Katy") Railroad, parked two locomotives at opposite ends of a 6.4 -km-long track, fired them up, tied their throttles open, and then allowed them to crash head-on at full speed in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assuming that each locomotive weighed $1.2 \cdot 10^{6} \mathrm{~N}$ and its acceleration along the track was a constant $0.26 \mathrm{~m} / \mathrm{s}^{2},$ what was the total kinetic energy of the two locomotives just before the collision?

Paul Gabriel

Numerade Educator

Problem 66

A baseball pitcher can throw a 5.00 -oz baseball with a speed measured by a radar gun to be 90.0 mph. Assuming that the force exerted by the pitcher on the ball acts over a distance of two arm lengths, each 28.0 in, what is the average force exerted by the pitcher on the ball?

Paul Gabriel

Numerade Educator

Problem 67

A 1.50 -kg soccer ball has a speed of $20.0 \mathrm{~m} / \mathrm{s}$ when it is $15.0 \mathrm{~m}$ above the ground. What is the total energy of the ball?

Paul Gabriel

Numerade Educator

Problem 68

If it takes an average force of $5.5 \mathrm{~N}$ to push a $4.5-\mathrm{g}$ dart $6.0 \mathrm{~cm}$ into a dart gun, assuming the barrel is frictionless, how fast will the dart exit the gun?

Paul Gabriel

Numerade Educator

Problem 69

A high jumper approaches the bar at $9.0 \mathrm{~m} / \mathrm{s}$. What is the highest altitude the jumper can reach, if he does not use any additional push off the ground and is moving at $7.0 \mathrm{~m} / \mathrm{s}$ as he goes over the bar?

Paul Gabriel

Numerade Educator

Problem 70

A roller coaster is moving at $2.00 \mathrm{~m} / \mathrm{s}$ at the top of the first hill $(h=40.0 \mathrm{~m}) .$ Ignoring friction and air resistance, how fast will the roller coaster be moving at the top of a subsequent hill, which is $15.0 \mathrm{~m}$ high?

Paul Gabriel

Numerade Educator

Problem 71

You are on a swing with a chain $4.0 \mathrm{~m}$ long. If your maximum displacement from the vertical is $35^{\circ},$ how fast will you be moving at the bottom of the arc?

Paul Gabriel

Numerade Educator

Problem 72

A truck is descending a winding mountain road. When the truck is $680 \mathrm{~m}$ above sea level and traveling at $15 \mathrm{~m} / \mathrm{s}$, its brakes fail. What is the maximum possible speed of the truck at the foot of the mountain, $550 \mathrm{~m}$ above sea level?

Paul Gabriel

Numerade Educator

Problem 73

Tarzan swings on a taut vine from his tree house to a limb on a neighboring tree, which is located a horizontal distance of $10.0 \mathrm{~m}$ from and $4.00 \mathrm{~m}$ below his starting point. Amazingly the vine neither stretches nor breaks; Tarzan's trajectory is thus a portion of a circle. If Tarzan starts with zero speed, what is his speed when he reaches the limb?

Paul Gabriel

Numerade Educator

Problem 74

The graph shows the component $(F \cos \theta)$ of the net force that acts on a $2.0-\mathrm{kg}$ block as it moves along a flat horizontal surface. Find
a) the net work done on the block.
b) the final speed of the block if it starts from rest at $s=0$.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 75

A 3.00 -kg model rocket is launched vertically upward with sufficient initial speed to reach a height of $1.00 \cdot 10^{2} \mathrm{~m}$ even though air resistance (a nonconservative force) performs $-8.00 \cdot 10^{2} \mathrm{~J}$ of work on the rocket. How high would the rocket have gone, if there were no air resistance?

Paul Gabriel

Numerade Educator

Problem 76

A 0.500 -kg mass is attached to a horizontal spring with $k=100 . \mathrm{N} / \mathrm{m}$. The mass slides across a frictionless surface. The spring is stretched $25.0 \mathrm{~cm}$ from equilibrium, and then the mass is released from rest.
a) Find the mechanical energy of the system.
b) Find the speed of the mass when it has moved $5.00 \mathrm{~cm}$.
c) Find the maximum speed of the mass.

LB

Laura Bravo

Numerade Educator

Problem 77

You have decided to move a refrigerator (mass $=81.3 \mathrm{~kg}$, including all the contents) to the other side of the room. You slide it across the floor on a straight path of length $6.35 \mathrm{~m}$, and the coefficient of kinetic friction between floor and fridge is $0.437 .$ Happy about your accomplishment, you leave the apartment. Your roommate comes home, wonders why the fridge is on the other side of the room, picks it up (you have a strong roommate!), carries it back to where it was originally, and puts it down. How much net mechanical work have the two of you done together?

LB

Laura Bravo

Numerade Educator

Problem 78

A 1.00 -kg block compresses a spring for which $k=$ 100. $\mathrm{N} / \mathrm{m}$ by $20.0 \mathrm{~cm}$ and is then released to move across a horizontal, frictionless table, where it hits and compresses another spring, for which $k=50.0 \mathrm{~N} / \mathrm{m}$. Determine
a) the total mechanical energy of the system,
b) the speed of the mass while moving freely between springs, and
c) the maximum compression of the second spring.

LB

Laura Bravo

Numerade Educator

Problem 79

A 1.00 -kg block is resting against a light, compressed spring at the bottom of a rough plane inclined at an angle of $30.0^{\circ}$; the coefficient of kinetic friction between block and plane is $\mu_{\mathrm{k}}=0.100 .$ Suppose the spring is compressed $10.0 \mathrm{~cm}$ from its equilibrium length. The spring is then released, and the block separates from the spring and slides up the incline a distance of only $2.00 \mathrm{~cm}$ beyond the spring's normal length before it stops. Determine
a) the change in total mechanical energy of the system and
b) the spring constant $k$.

LB

Laura Bravo

Numerade Educator

Problem 80

A 0.100 -kg ball is dropped from a height of $1.00 \mathrm{~m}$ and lands on a light (approximately massless) cup mounted on top of a light, vertical spring initially at its equilibrium position. The maximum compression of the spring is to be $10.0 \mathrm{~cm}$.
a) What is the required spring constant of the spring?
b) Suppose you ignore the change in the gravitational energy of the ball during the 10 -cm compression. What is the percentage difference between the calculated spring constant for this case and the answer obtained in part (a)?

Manish Jain

Numerade Educator

Problem 81

A mass of $1.00 \mathrm{~kg}$ attached to a spring with a spring constant of $100 .$ N/m oscillates horizontally on a smooth frictionless table with an amplitude of $0.500 \mathrm{~m} .$ When the mass is $0.250 \mathrm{~m}$ away from equilibrium, determine:
a) its total mechanical energy;
b) the system's potential energy and the mass's kinetic energy;
c) the mass's kinetic energy when it is at the equilibrium point.
d) Suppose there was friction between the mass and the table so that the amplitude was cut in half after some time. By what factor has the mass's maximum kinetic energy changed?
e) By what factor has the maximum potential energy changed?

Manish Jain

Numerade Educator

Problem 82

Bolo, the human cannonball, is ejected from a $3.50 \mathrm{~m}$ long barrel. If Bolo $(m=80.0 \mathrm{~kg})$ has a speed of $12.0 \mathrm{~m} / \mathrm{s}$ at the top of his trajectory, $15.0 \mathrm{~m}$ above the ground, what was the average force exerted on him while in the barrel?

Paul Gabriel

Numerade Educator

Problem 83

A 1.00 -kg mass is suspended vertically from a spring with $k=100 . \mathrm{N} / \mathrm{m}$ and oscillates with an amplitude of $0.200 \mathrm{~m} .$ At the top of its oscillation, the mass is hit in such a way that it instantaneously moves down with a speed of $1.00 \mathrm{~m} / \mathrm{s}$. Determine
a) its total mechanical energy,
b) how fast it is moving as it crosses the equilibrium point, and
c) its new amplitude.

Manish Jain

Numerade Educator

Problem 84

A runner reaches the top of a hill with a speed of $6.50 \mathrm{~m} / \mathrm{s}$ He descends $50.0 \mathrm{~m}$ and then ascends $28.0 \mathrm{~m}$ to the top of the next hill. His speed is now $4.50 \mathrm{~m} / \mathrm{s}$. The runner has a mass of $83.0 \mathrm{~kg} .$ The total distance that the runner covers is $400 . \mathrm{m}$ and there is a constant resistance to motion of $9.00 \mathrm{~N}$. Use energy considerations to find the work done by the runner over the total distance.

LB

Laura Bravo

Numerade Educator

Problem 85

A package is dropped on a horizontal conveyor belt. The mass of the package is $m,$ the speed of the conveyor belt is $v$, and the coefficient of kinetic friction between the package and the belt is $\mu_{\mathrm{k}}$
a) How long does it take for the package to stop sliding on the belt?
b) What is the package's displacement during this time?
c) What is the energy dissipated by friction?
d) What is the total work supplied by the system?

Manish Jain

Numerade Educator

Problem 86

A father exerts a $2.40 \cdot 10^{2} \mathrm{~N}$ force to pull a sled with his daughter on it (combined mass of $85.0 \mathrm{~kg}$ ) across a horizontal surface. The rope with which he pulls the sled makes an angle of $20.0^{\circ}$ with the horizontal. The coefficient of kinetic friction is $0.200,$ and the sled moves a distance of $8.00 \mathrm{~m}$. Find
a) the work done by the father,
b) the work done by the friction force, and
c) the total work done by all the forces.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 87

A variable force acting on a 0.100 - $\mathrm{kg}$ particle moving in the $x y$ -plane is given by $F(x, y)=\left(x^{2} \hat{x}+y^{2} \hat{y}\right) \mathrm{N},$ where $x$ and $y$ are in meters. Suppose that due to this force, the particle moves from the origin, $O$, to point $S$, with coordinates $(10.0 \mathrm{~m},$ $10.0 \mathrm{~m}$ ). The coordinates of points $P$ and $Q$ are $(0 \mathrm{~m}, 10.0 \mathrm{~m})$ and $(10.0 \mathrm{~m}, 0 \mathrm{~m})$ respectively. Determine the work performed by the force as the particle moves along each of the following paths:
a) OPS
b) OQS
c) OS
d) $O P S Q O$
e) $O Q S P O$

Manish Jain

Numerade Educator

Problem 88

In problem 6.87 , suppose there was friction between the 0.100 -kg particle and the $x y$ -plane, with $\mu_{\mathrm{k}}=0.100$. Determine the net work done by all forces on this particle when it takes each of the following paths:
a) OPS
b) OQS
c) OS
d) OPSQO
e) OQSPO

Manish Jain

Numerade Educator