Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 8

Potential Energy and Conservation of Energy - all with Video Answers

Educators

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Chapter Questions

Problem 1

A $1000-\mathrm{kg}$ roller coaster is initially at the top of a rise, at point $A$. It then moves $135 \mathrm{ft}$, at an angle of $40.0^{\circ}$ below the horizontal, to a lower point $B$. (a) Choose point $B$ to be the zero level for gravitational potential energy. Find the potential energy of the roller coaster-Earth system at points $A$ and $B$ and the change in its potential energy as the coaster moves. (b) Repeat part (a), setting the zero reference level at point $A$.

Nicholas Mogoi

Nicholas Mogoi

Numerade Educator

Problem 2

A $40.0-\mathrm{N}$ child is in a swing that is attached to ropes $2.00 \mathrm{~m}$ long. Find the gravitational potential energy of the child-Earth system relative to the child's lowest position when (a) the ropes are horizontal, (b) the ropes make a $30.0^{\circ}$ angle with the vertical, and (c) the child is at the bottom of the circular arc.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 3

A $4.00-\mathrm{kg}$ particle moves from the origin to position $C .$ which has coordinates $x=5.00 \mathrm{~m}$ and $y=5.00 \mathrm{~m}$ (Fig. P8.3). One force on it is the force of gravity acting in the negative $y$ direction. Using Equation $7.2$, calculate the work done by gravity as the particle moves from $O$ to $C$ along (a) $O A C,(b) O B C$, and $(c) O C$. Your results should all be identical. Why?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 4

(a) Suppose that a constant force acts on an object. The force does not vary with time, nor with the position or velocity of the object. Start with the general definition for work done by a force
$$
W=\int_{i}^{f} \mathbf{F} \cdot d \mathbf{s}
$$
and show that the force is conservative. (b) As a special case, suppose that the force $\mathbf{F}=(3 \mathbf{i}+4 \mathbf{j}) \mathrm{N}$ acts on a particle that moves from $O$ to $C$ in Figure P8.3. Calculate the work done by $\mathbf{F}$ if the particle moves along each one of the three paths $O A C, O B C$, and $O C .$ (Your three answers should be identical.)

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 5

A force acting on a particle moving in the $x y$ plane is given by $\mathbf{F}=\left(2 y \mathbf{i}+x^{2} \mathbf{j}\right) \mathrm{N}$, where $x$ and $y$ are in meters. The particle moves from the origin to a final position having coordinates $x=5.00 \mathrm{~m}$ and $y=5.00 \mathrm{~m}$, as in Figure $\mathrm{P} 8.3 .$ Calculate the work done by $\mathbf{F}$ along
(a) $O A C,($ b $) O B C,(c) O C$.
(d) Is F conservative or nonconservative? Explain.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 6

At time $t_{i}$, the kinetic energy of a particle in a system is $30.0$ J and the potential energy of the system 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 at time $t_{f} ?(\mathrm{~b})$ If the potential energy of the system at time $t_{f}$ is $5.00 \mathrm{~J}$, are any nonconservative forces acting on the particle? Explain.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 7

A single conservative force acts on a $5.00-\mathrm{kg}$ particle. The equation $F_{x}=(2 x+4) \mathrm{N}$, where $x$ is in meters, $\mathrm{de}-$ scribes this force. As the particle moves along the $x$ axis from $x=1.00 \mathrm{~m}$ to $x=5.00 \mathrm{~m}$, calculate (a) the work done by this force, (b) the change in the potential energy of the system, and (c) the kinetic energy of the particle at $x=5.00 \mathrm{~m}$ if its speed at $x=1.00 \mathrm{~m}$ is $3.00 \mathrm{~m} / \mathrm{s}$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 8

A single constant force $\mathbf{F}=(3 \mathbf{i}+5 \mathbf{j}) \mathrm{N}$ acts on a $4.00-\mathrm{kg}$ particle. (a) Calculate the work done by this force if the particle moves from the origin to the point having the vector position $\mathbf{r}=(2 \mathbf{i}-3 \mathbf{j}) \mathrm{m} .$ Does this result depend on the path? Explain. (b) What is the speed of the particle at $\mathbf{r}$ if its speed at the origin is $4.00 \mathrm{~m} / \mathrm{s}^{2}(\mathrm{c})$ What is the change in the potential energy of the system?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 9

A single conservative force acting on a particle varies as $\mathbf{F}=\left(-A x+B x^{2}\right) \mathbf{i} \mathrm{N}$, where $A$ and $B$ are constants and $x$ is in meters. (a) Calculate the potential energy function $U(x)$ associated with this force, taking $U=0$ at $x=0 .$ (b) Find the change in potential energy and change in kinetic energy as the particle moves from $x=2.00 \mathrm{~m}$ to $x=3.00 \mathrm{~m}$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 10

A particle of mass $0.500 \mathrm{~kg}$ is shot from $P$ as shown in Figure $\mathrm{P} 8.10 .$ The particle has an initial velocity $\mathbf{v}_{i}$ with a horizontal component of $30.0 \mathrm{~m} / \mathrm{s} .$ The particle rises to a maximum height of $20.0 \mathrm{~m}$ above $P$. Using the law of conservation of energy, determine (a) the vertical component of $\mathbf{v}_{i}$, (b) the work done by the gravitational force on the particle during its motion from $P$ to $B$, and
(c) the horizontal and the vertical components of the velocity vector when the particle reaches $B$.

Prashant Bana

Numerade Educator

Problem 11

A $3.00-\mathrm{kg}$ mass starts from rest and slides a distance $d$ down a frictionless $90.0^{\circ}$ incline. While sliding, it comes into contact with an unstressed spring of negligible mass, as shown in Figure $\mathrm{P} 8.11 .$ The mass slides an additional $0.200 \mathrm{~m}$ as it is brought momentarily to rest by compression of the spring $(k=400 \mathrm{~N} / \mathrm{m}) .$ Find the initial separation $d$ between the mass and the spring.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 12

A mass $m$ starts from rest and slides a distance $d$ down a frictionless incline of angle $\theta .$ While sliding, it contacts an unstressed spring of negligible mass, as shown in Figure $\mathrm{P8} .11 .$ The mass slides an additional distance $x$ as it
is brought momentarily to rest by compression of the spring (of force constant $k$ ). Find the initial separation $d$ between the mass and the spring.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 13

A particle of mass $m=5.00 \mathrm{~kg}$ is released from point $@$ and slides on the frictionless track shown in Figure P8.13. Determine (a) the particle's speed at points (B) and (C) and (b) the net work done by the force of gravity in moving the particle from $\triangle$ to $(\mathrm{C})$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 14

A simple, $2.00-$ m-long pendulum is released from rest when the support string is at an angle of $25.0^{\circ}$ from the vertical. What is the speed of the suspended mass at the bottom of the swing?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 15

A bead slides without friction around a loop-the-loop (Fig. $\mathrm{P} 8.15$ ). If the bead is released from a height $h=$ $3.50 R$, what is its speed at point $A$ ? How great is the normal force on it if its mass is $5.00 \mathrm{~g}$ ?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 16

A $120-\mathrm{g}$ mass is attached to the bottom end of an unstressed spring. The spring is hanging vertically and has a spring constant of $40.0 \mathrm{~N} / \mathrm{m}$. The mass is dropped.
(a) What is its maximum speed?
(b) How far does it
drop before coming to rest momentarily?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 17

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

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 18

Dave Johnson, the bronze medalist at the 1992 Olympic decathlon in Barcelona, leaves the ground for his high jump with a vertical velocity component of $6.00 \mathrm{~m} / \mathrm{s}$. How far up does his center of gravity move as he makes the jump?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 19

A $0.400-\mathrm{kg}$ ball is thrown straight up into the air and reaches a maximum altitude of $20.0 \mathrm{~m}$. Taking its initial position as the point of zero potential energy and using energy methods, find (a) its initial speed, (b) its total mechanical energy, and (c) the ratio of its kinetic energy to the potential energy of the ball-Earth system when the ball is at an altitude of $10.0 \mathrm{~m}$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 20

In the dangerous "sport" of bungee-jumping, a daring student jumps from a balloon with a specially designed elastic cord attached to his ankles, as shown in Figure P8.20. The unstretched length of the cord is $25.0 \mathrm{~m}$, the student weighs $700 \mathrm{~N}$, and the balloon is $36.0 \mathrm{~m}$ above the surface of a river below. Assuming that Hooke's law describes the cord, calculate the required force constant if the student is to stop safely $4.00 \mathrm{~m}$ above the river.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 21

Two masses are connected by a light string passing over a light frictionless pulley, as shown in Figure P8.21. The $5.00-\mathrm{kg}$ mass is released from rest. Using the law of conservation of energy, (a) determine the speed of the $3.00-$ kg mass just as the $5.00-\mathrm{kg}$ mass hits the ground and (b) find the maximum height to which the $3.00-\mathrm{kg}$ mass rises.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 22

Two masses are connected by a light string passing over a light frictionless pulley, as shown in Figure $\mathrm{P8} .21$. The mass $m_{1}$ (which is greater than $m_{2}$ ) is released from rest. Using the law of conservation of energy, (a) determine the speed of $m_{2}$ just as $m_{1}$ hits the ground in terms of $m_{1}, m_{2}$, and $h$, and $(b)$ find the maximum height to which mo rises.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 23

A $20.0-\mathrm{kg}$ cannon ball is fired from a cannon with a 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 law of conservation of mechanical energy to find (a) the maximum height reached by each ball and
(b) the total mechanical energy at the maximum height for each ball. Let $y=0$ at the cannon.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 24

A $2.00-\mathrm{kg}$ ball is attached to the bottom end of a length of 10 -lb $(44.5-\mathrm{N})$ fishing line. The top end of the fishing line is held stationary. The ball is released from rest while the line is taut and horizontal $\left(\theta=90.0^{\circ}\right) . \mathrm{At}$ what angle $\theta$ (measured from the vertical) will the fishing line break?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 25

The circus apparatus known as the trapeze consists of a bar suspended by two parallel ropes, each of length $\ell$. The trapeze allows circus performers to swing in a vertical circular arc (Fig. P8.25). Suppose a performer with mass $m$ and holding the bar steps off an elevated platform, starting from rest with the ropes at an angle of $\theta_{i}$ with respect to the vertical. Suppose the size of the performer's body is small compared with the length $\ell$, that she does not pump the trapeze to swing higher, and that air resistance is negligible. (a) Show that when the ropes make an angle of $\theta$ with respect to the vertical, the performer must exert a force
$$
F=m g\left(3 \cos \theta-2 \cos \theta_{i}\right)
$$
in order to hang on. (b) Determine the angle $\theta_{i}$ at which the force required to hang on at the bottom of the swing is twice the performer's weight.

Stephen Zaffke

Numerade Educator

Problem 26

After its release at the top of the first rise, a rollercoaster car moves freely with negligible friction. The roller coaster shown in Figure $\mathrm{P} 8.26$ has a circular loop of radius $20.0 \mathrm{~m}$. The car barely makes it around the loop: At the top of the loop, the riders are upside down and feel weightless. (a) Find the speed of the roller coaster car at the top of the loop (position 3$)$. Find the speed of the roller coaster car (b) at position 1 and
(c) at position 2. (d) Find the difference in height between positions 1 and 4 if the speed at position 4 is $10.0 \mathrm{~m} / \mathrm{s}$

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 27

A light rigid rod is $77.0 \mathrm{~cm}$ long. Its top end is pivoted on a low-friction 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?

Surjit Tewari

Numerade Educator

Problem 28

A $70.0-\mathrm{kg}$ diver steps off a $10.0-\mathrm{m}$ tower and drops straight down into the water. If he comes to rest $5.00 \mathrm{~m}$ beneath the surface of the water, determine the average resistance force that the water exerts on the diver.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 29

A force $F_{x}$, shown as a function of distance in Figure P8.29, acts on a $5.00-\mathrm{kg}$ mass. If the particle starts from rest at $x=0 \mathrm{~m}$, determine the speed of the particle at. $x=2.00,4.00$, and $6.00 \mathrm{~m}$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 30

A softball pitcher swings a ball of mass $0.250 \mathrm{~kg}$ around a vertical circular path of radius $60.0 \mathrm{~cm}$ before releasing it from her hand. The pitcher maintains a component of force on the ball of constant magnitude $30.0 \mathrm{~N}$ in the direction of motion around the complete path. The speed of the ball at the top of the circle is $15.0 \mathrm{~m} / \mathrm{s}$. If the ball is released at the bottom of the circle, what is its speed upon release?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 31

The coefficient of friction between the $3.00-\mathrm{kg}$ block and the surface in Figure $\mathrm{P} 8.31$ is $0.400 .$ The system starts from rest. What is the speed of the $5.00 \mathrm{~kg}$ ball when it has fallen $1.50 \mathrm{~m}$ ?

Keshav Singh

Numerade Educator

Problem 32

A $2000-\mathrm{kg}$ car starts from rest and coasts down from the top of a $5.00$ -m-long driveway that is sloped at an angle of $20.0^{\circ}$ with the horizontal. If an average friction force of $4000 \mathrm{~N}$ impedes the motion of the car, find the speed of the car at the bottom of the driveway.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 33

A $5.00-\mathrm{kg}$ block is set into motion up an inclined plane with an initial speed of $8.00 \mathrm{~m} / \mathrm{s}$ (Fig. $\mathrm{P8} .33$ ). The block comes to rest after traveling $3.00 \mathrm{~m}$ along the plane, which is inclined at an angle of $30.0^{\circ}$ to the horizontal. For this motion determine (a) the change in the block's kinetic energy, (b) the change in the potential energy, and (c) the frictional force exerted on it (assumed to be constant). (d) What is the coefficient of kinetic friction?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 34

A boy in a wheelchair (total mass, $47.0 \mathrm{~kg}$ ) wins a race with a skateboarder. He has a speed of $1.40 \mathrm{~m} / \mathrm{s}$ at the crest of a slope $2.60 \mathrm{~m}$ high and $12.4 \mathrm{~m}$ long. At the bottom of the slope, his speed is $6.20 \mathrm{~m} / \mathrm{s}$. If air resistance and rolling resistance can be modeled as a constant frictional force of $41.0 \mathrm{~N}$, find the work he did in pushing forward on his wheels during the downhill ride.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 35

A parachutist of mass $50.0 \mathrm{~kg}$ jumps out of a balloon at a height of $1000 \mathrm{~m}$ and lands on the ground with a speed of $5.00 \mathrm{~m} / \mathrm{s} .$ How much energy was lost to air friction during this jump?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 36

An $80.0-\mathrm{kg}$ sky diver jumps out of a balloon at an altitude of $1000 \mathrm{~m}$ and opens the parachute at an altitude of $200.0 \mathrm{~m}$. (a) Assuming that the total retarding force on the diver is constant at $50.0 \mathrm{~N}$ with the parachute closed and constant at $3600 \mathrm{~N}$ with the parachute open, what is the speed of the diver when he lands on the ground? (b) Do you think the sky diver will get hurt? Explain. (c) At what height should the parachute be opened so that the final speed of the sky diver 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.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 37

A toy cannon uses a spring to project a $5.30-\mathrm{g}$ soft rubber ball. The spring is originally compressed by $5.00 \mathrm{~cm}$ and has a stiffness constant of $8.00 \mathrm{~N} / \mathrm{m}$. When it is fired, the ball moves $15.0 \mathrm{~cm}$ through the barrel of the cannon, and there is a constant frictional force of $0.0820 \mathrm{~N}$ between the barrel and 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?

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 38

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

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 39

A $3.00-\mathrm{kg}$ block starts at a height $h=60.0 \mathrm{~cm}$ on a plane that has an inclination angle of $30.0^{\circ}$, as shown in Figure $\mathrm{P} 8.39 .$ Upon reaching the bottom, the block slides along a horizontal surface. If the coefficient of friction on both surfaces is $\mu_{k}=0.200$, how far does the block slide on the horizontal surface before coming to rest? (Hint: Divide the path into two straight-line parts.)

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 40

A $75.0-\mathrm{kg}$ sky diver is falling with a terminal speed of $60.0 \mathrm{~m} / \mathrm{s}$. Determine the rate at which he is losing mechanical energy.

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 41

The potential energy of a two-particle system separated by a distance $r$ is given by $U(r)=A / r$, where $A$ is a constant. Find the radial force $\mathbf{F}_{t}$ that each particle exerts on the other.

Bettina Hanlon

Numerade Educator

Problem 42

A potential energy function for a two-dimensional force is of the form $U=3 x^{3} y-7 x$. Find the force that acts at the point $(x, y)$.

Jacob Adamczyk

Numerade Educator

Problem 43

A particle moves along a line where the potential energy depends on its position $r$, as graphed in Figure P8.43. In the limit as $r$ increases without bound, $U(r)$ approaches $+1$ J. (a) Identify each equilibrium position for this particle. Indicate whether each is a point of stable, unstable, or neutral equilibrium. (b) The particle will be bound if its total energy is in what range? Now suppose the particle has energy $-3 J .$ Determine
(c) the range of positions where it can be found,
(d) its maximum kinetic energy, (e) the location at which it has maximum kinetic energy, and (f) its binding energy - that is, the additional energy that it would have to be given in order for it to move out to $r \rightarrow \infty$.

Mayukh Banik

Numerade Educator

Problem 44

A right circular cone can be balanced on a horizontal surface in three different ways. Sketch these three equilibrium configurations and identify them as positions of stable, unstable, or neutral equilibrium.

Surjit Tewari

Numerade Educator

Problem 45

For the potential energy curve shown in Figure $\mathrm{P} 8.45$,
(a) determine whether the force $F_{x}$ is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for $F_{x}$ versus $x$ from $x=0$ to $x=9.5 \mathrm{~m}$.

Donald Albin

Numerade Educator

Problem 46

A hollow pipe has one or two weights attached to its inner surface, as shown in Figure $\mathrm{P} 8.46 .$ Characterize each configuration as being stable, unstable, or neutral equilibrium and explain each of your choices ("CM" indicates center of mass).

Stephen Zaffke

Numerade Educator

Problem 47

A particle of mass $m$ is attached between two identical springs on a horizontal frictionless tabletop. The springs have spring constant $k$, and each is initially unstressed. (a) If the mass is pulled a distance $x$ along a direction perpendicular to the initial configuration of the springs, as in Figure $\mathrm{P} 8.47$, show that the potential energy of the system is
$$
U(x)=k x^{2}+2 k L\left(L-\sqrt{x^{2}+L^{2}}\right)
$$
(Hint: See Problem 66 in Chapter 7.) (b) Make a plot of $U(x)$ versus $x$ and identify all equilibrium points. Assume that $L=1.20 \mathrm{~m}$ and $k=40.0 \mathrm{~N} / \mathrm{m} .(\mathrm{c})$ If the
mass is pulled $0.500 \mathrm{~m}$ to the right and then released, what is its speed when it reaches the equilibrium point $x=0 ?$

Stephen Zaffke

Numerade Educator

Problem 48

Find the energy equivalents of (a) an electron of mass $9.11 \times 10^{-31} \mathrm{~kg}$, (b) a uranium atom with a mass of $4.00 \times 10^{-25} \mathrm{~kg},(\mathrm{c})$ a paper clip of mass $2.00 \mathrm{~g}$, and
(d) the Earth (of mass $5.99 \times 10^{24} \mathrm{~kg}$ ).

Bettina Hanlon

Numerade Educator

Problem 49

The expression for the kinetic energy of a particle moving with speed $v$ is given by Equation $7.19$, which can be written as $K=\gamma m c^{2}-m c^{2}$, where $\gamma=\left[1-(v / c)^{2}\right]^{-1 / 2}$.
The term $\gamma m c^{2}$ is the total energy of the particle, and the term $m c^{2}$ is its rest energy. A proton moves with a speed of $0.990 c$, where $c$ is the speed of light. Find (a) its rest energy, (b) its total energy, and (c) its kinetic energy.

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 50

A block slides down a curved frictionless track and then up an inclined plane as in Figure $\mathrm{P} 8.50 .$ The coefficient of kinetic friction between the block and the incline is $\mu_{k} .$ Use energy methods to show that the maximum height reached by the block is
$$
y_{\max }=\frac{h}{1+\mu_{k} \cot \theta}
$$

Mayukh Banik

Numerade Educator

Problem 51

Close to the center of a campus is a tall silo topped with a hemispherical cap. The cap is frictionless when wet. Someone has somehow balanced a pumpkin at the highest point. The line from the center of curvature of the cap to the pumpkin makes an angle $\theta_{i}-0^{\circ}$ with the vertical. On 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?

Samuel Smith

Numerade Educator

Problem 52

A $200-\mathrm{g}$ particle is released from rest at point (A) along the horizontal diameter on the inside of a frictionless, hemispherical bowl of radius $R=30.0 \mathrm{~cm}$ (Fig. P8.52). Calculate (a) the gravitational potential energy when the particle is at point $($ a relative to point (B), (b) the kinetic energy of the particle at point (B), (c) its speed at point (B), and (d) its kinetic energy and the potential energy at point (C).

Bettina Hanlon

Numerade Educator

Problem 53

The particle described in Problem 52 (Fig. $\mathrm{P8} .52$ ) is released from rest at $@$, and the surface of the bowl is rough. The speed of the particle at (B) is $1.50 \mathrm{~m} / \mathrm{s}$.
(a) What is its kinetic energy at (B)? (b) How much energy is lost owing to friction as the particle moves from
(A) to (B)? (c) Is it possible to determine $\mu$ from these results in any simple manner? Explain.

Samuel Smith

Numerade Educator

Problem 54

The mass of a car is $1500 \mathrm{~kg}$. The shape of the body is such that its aerodynamic drag coefficient is $D=0.390$ and the frontal area is $2.50 \mathrm{~m}^{2}$. Assuming that the drag force is proportional to $v^{2}$ and neglecting other sources of friction, calculate the power the car requires to maintain a speed of $100 \mathrm{~km} / \mathrm{h}$ as it climbs a long hill sloping at $3.20^{\circ}$.

Samuel Smith

Numerade Educator

Problem 55

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

Numerade Educator

Problem 56

A child's pogo stick (Fig. $\mathrm{P8} .56$ ) stores energy in a spring $\left(k=2.50 \times 10^{4} \mathrm{~N} / \mathrm{m}\right) .$ At position $\mathrm{A}\left(x_{\mathrm{A}}=\right.$
$-0.100 \mathrm{~m}$ ), the spring compression is a maximum and the child is momentarily at rest. At position (B) $\left(x_{\mathrm{B}}=0\right)$, the spring is relaxed and the child is moving upward. At position (C), the child is again momentarily at rest at the top of the jump. Assuming that the combined mass of the child and the pogo stick is $25.0 \mathrm{~kg}$, (a) calculate the total energy of the system if both potential energies are zero at $x=0$, (b) determine $x_{c},(c)$ calculate the speed of the child at $x=0,(\mathrm{~d})$ determine the value of $x$ for which the kinetic energy of the system is a maximum, and (e) calculate the child's maximum upward speed.

Samuel Smith

Numerade Educator

Problem 57

A $10.0-\mathrm{kg}$ block is released from point (A) in Figure P8.57. The track is frictionless except for the portion between (B) and (C), which has a length of $6.00 \mathrm{~m}$. The block travels down the track, hits a spring of force constant $k=2250 \mathrm{~N} / \mathrm{m}$, and compresses the spring $0 . .300 \mathrm{~m}$ from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between (B) and (C).

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 58

A 2.00-kg block situated on a rough incline is connected to a spring of negligible mass having a spring constant of $100 \mathrm{~N} / \mathrm{m}$ (Fig. $\mathrm{P} 8.58$ ). The pulley is frictionless. The block is released from rest when the spring is unstretched. The block moves $20.0 \mathrm{~cm}$ down the incline before coming to rest. Find the coefficient of kinetic friction between block and incline.

Mayukh Banik

Numerade Educator

Problem 59

Suppose the incline is frictionless for the system described in Problem 58 (see Fig. $\mathrm{P} 8.58$ ). The block is released from rest with the spring initially unstretched. (a) How far does it move down the incline before coming to rest? (b) What is its acceleration at its lowest point? Is the acceleration constant? (c) Describe the energy transformations that occur during the descent.

Samuel Smith

Numerade Educator

Problem 60

The potential energy function for a system is given by $U(x)=-x^{3}+2 x^{2}+3 x$ (a) Determine the force $F_{x}$ as a function of $x$. (b) For what values of $x$ is the force equal to zero? (c) Plot $U(x)$ versus $x$ and $F_{x}$ versus $x$, and indicate points of stable and unstable equilibrium.

Mayukh Banik

Numerade Educator

Problem 61

A $20.0-\mathrm{kg}$ block is connected to a $30.0-\mathrm{kg}$ block by a string that passes over a frictionless pulley. The $30.0-\mathrm{kg}$ block is connected to a spring that has negligible mass and a force constant of $250 \mathrm{~N} / \mathrm{m}$, as shown in Figure P8.61. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The $20.0-\mathrm{kg}$ block is pulled $20.0 \mathrm{~cm}$ down the incline (so that the $30.0-\mathrm{kg}$ block is $40.0 \mathrm{~cm}$ above the floor) and is released from rest. Find the speed of each block when the $30.0-\mathrm{kg}$ block is $20.0 \mathrm{~cm}$ above the floor (that is, when the spring is unstretched).

Samuel Smith

Numerade Educator

Problem 62

A $1.00-\mathrm{kg}$ mass slides to the right on a surface having a coefficient of friction $\mu=0.250$ (Fig. P8.62). The mass has a speed of $v_{i}=3.00 \mathrm{~m} / \mathrm{s}$ when it makes contact with a light spring that has a spring constant $k=50.0 \mathrm{~N} / \mathrm{m}$. The mass comes to rest after the spring has been compressed a distance $d$. The mass is then forced toward the left by the spring and continues to move in that direction beyond the spring's unstretched position. Finally, the mass comes to rest at a distance $D$ to the left of the unstretched spring. Find (a) the distance of compression $d,(\mathrm{~b})$ the speed $v$ of the mass at the unstretched position when the mass is moving to the left, and
(c) the distance $D$ between the unstretched spring and the point at which the mass comes to rest.

MS

Michael Shaikhet

Numerade Educator

Problem 63

A block of mass $0.500 \mathrm{~kg}$ is pushed against a horizontal spring of negligible mass until the spring is compressed a distance $\Delta x$ (Fig. P8.63). The spring constant is $450 \mathrm{~N} / \mathrm{m} .$ When it is released, the block travels along a frictionless, horizontal surface to point $B$, at the bottom of a vertical circular track of radius $R=1.00 \mathrm{~m}$, and continues to move up the track. The speed of the block at the bottom of the track is $v_{B}=12.0 \mathrm{~m} / \mathrm{s}$, and the block experiences an average frictional force of $7.00 \mathrm{~N}$ while sliding up the track. (a) What is $\Delta x ?$ (b) What speed do you predict for the block at the top of the track? (c) Does the block actually reach the top of the track, or does it fall off before reaching the top?

Samuel Smith

Numerade Educator

Problem 64

A uniform chain of length $8.00 \mathrm{~m}$ initially lies stretched out on a horizontal table. (a) If the coefficient of static friction between the chain and the 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 all of it leaves the table, given that the coefficient of kinetic friction between the chain and the table is $0.400 .$

Mayukh Banik

Numerade Educator

Problem 65

An object of mass $m$ is suspended from a post on top of a cart by a string of length $L$ as in Figure $\mathrm{P} 8.66 \mathrm{a} .$ The cart and object are initially moving to the right at constant speed $v_{i} .$ The cart comes to rest after colliding and sticking to a bumper as in Figure $\mathrm{P} 8.65 \mathrm{~b}$, and the suspended object swings through an angle $\theta .$ (a) Show that the speed is $v_{i}=\sqrt{2 g L(1-\cos \theta)}$. (b) If $L=1.20 \mathrm{~m}$
and $\theta=85.0^{\circ}$, find the initial speed of the cart. (Hint:
The force exerted by the string on the object does no work on the object.)

Eduard Sanchez

Numerade Educator

Problem 66

A child slides without friction from a height $h$ along a curved water slide (Fig. $\mathrm{P8} .66$ ). She is launched from a height $h / 5$ into the pool. Determine her maximum airborne height $y$ in terms of $h$ and $\theta$.

Maxime Rossetti

Maxime Rossetti

Numerade Educator

Problem 67

A ball having mass $m$ is connected by a strong string of length $L$ to a pivot point and held in place in a vertical position. A wind exerting constant force of magnitude $F$ is blowing from left to right as in Figure P8.67a. (a) If the ball is released from rest, show that the maximum height $H$ it reaches, as measured from its initial height, is
$$
H=\frac{2 L}{1+(m g / F)^{2}}
$$
Check that the above formula is valid both when $0 \leq H \leq L$ and when $L \leq H \leq 2 L .$ (Hint: First determine the potential energy associated with the constant wind force.) (b) Compute the value of $H$ using the values $m=2.00 \mathrm{~kg}, L=2.00 \mathrm{~m}$, and $F=14.7 \mathrm{~N}$. (c) Using these same values, determine the equilibrium height of the ball. (d) Could the equilibrium height ever be greater than $L$ ? Explain.

Mayukh Banik

Numerade Educator

Problem 68

A ball is tied to one end of a string. The other end of the string is fixed. The ball is set in motion around a vertical circle without friction. At the top of the circle, the ball has a speed of $v_{i}=\sqrt{R g}$, as shown in Figure $\mathrm{P8} .68 .$ At what angle $\theta$ should the string be cut so that the ball will travel through the center of the circle?

Mayukh Banik

Numerade Educator

Problem 69

A ball at the end of a string whirls around in a vertical circle. If the ball's total energy remains constant, show that the tension in the string at the bottom is greater than the tension at the top by a value six times the weight of the ball.

Mayukh Banik

Numerade Educator

Problem 70

A pendulum comprising a string of length $L$ and a sphere swings in the vertical plane. The string hits a peg located a distance $d$ below the point of suspension (Fig. P8.70). (a) Show that if the sphere is released from a height below that of the peg, it will return to this height after striking the peg. (b) Show that if the pendulum is released from the horizontal position $\left(\theta=90^{\circ}\right)$ and is to swing in a complete circle centered on the peg, then the minimum value of $d$ must be $3 L / 5$.

Mayukh Banik

Numerade Educator

Problem 71

Jane, whose mass is $50.0 \mathrm{~kg}$, needs to swing across a river (having width $D$ ) filled with man-eating crocodiles to save Tarzan from danger. However, 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. $\mathrm{P8} .71$ ). Taking $D=50.0 \mathrm{~m}, F=$ $110 \mathrm{~N}, L=40.0 \mathrm{~m}$, and $\theta=50.0^{\circ}$, (a) with what minimum speed must Jane begin her swing to just make it to the other side? (Hint: First determine the potential energy associated with the wind force.) (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 that Tarzan has a mass of $80.0 \mathrm{~kg}$

Surjit Tewari

Numerade Educator

Problem 72

A child starts from rest and slides down the frictionless slide shown in Figure P8.72. In terms of $R$ and $H$, at what height $h$ will he lose contact with the section of radius $R$ ?

Vishal Gupta

Numerade Educator

Problem 73

A $5.00-\mathrm{kg}$ block free to move on a horizontal, frictionless surface is attached to one end of a light horizontal spring. The other end of the spring is fixed. The spring is compressed $0.100 \mathrm{~m}$ from equilibrium and is then released. The speed of the block is $1.20 \mathrm{~m} / \mathrm{s}$ when it passes the equilibrium position of the spring. The same experiment is now repeated with the frictionless surface replaced by a surface for which $\mu_{k}=0.300$. Determine the speed of the block at the equilibrium position of the spring. 100

MS

Michael Shaikhet

Numerade Educator

Problem 74

A $50.0-\mathrm{kg}$ block and a $100-\mathrm{kg}$ block are connected by a string as in Figure $\mathrm{P} 8.74$. The pulley is frictionless and of negligible mass. The coefficient of kinetic friction between the $50.0-\mathrm{kg}$ block and the incline is $\mu_{k}=0.250 .$ Determine the change in the kinetic energy of the $50.0-\mathrm{kg}$ block as it moves from (A) to (B), a distance of $20.0 \mathrm{~m} .$

Vishal Gupta

Numerade Educator