College Physics

Roger A. Freedman; Todd Ruskell; Philip R. Kesten

Chapter 8

Rotational Motion - all with Video Answers

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Chapter Questions

Problem 1

Define the SI unit radian. The unit appears in some physical quantities (for example, the angular velocity of a turntable is $3.5 \mathrm{rad} / \mathrm{s}$ ) and it is omitted in others (for example, the translational velocity at the rim of a turntable is $0.35 \mathrm{~m} / \mathrm{s}$ ). Because the formula relating rotational and translational quantities involves multiplying by a radian $(\mathrm{v}=\mathrm{r} \omega),^{,}(v=r \omega)$, discuss when it is appropriate to include radians and when the unit should be dropped.

Prabhu Ramji

Prabhu Ramji

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Problem 2

Why is it critical to define the axis of rotation when you set out to find the moment of inertia of an object?

Prabhu Ramji

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Problem 3

The four solids shown in Figure 8-33 have equal heights, widths, and masses. The axes of rotation are located at the center of each object and are perpendicular to the plane of the paper. Rank the moments of inertia from greatest to least.

Prabhu Ramji

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Problem 4

What is the ratio of rotational kinetic energy for two balls, each tied to a light string and spinning in a circle with a radius equal to the length of the string? The first ball has a mass $m$ and a string of length $L$, and rotates at a rate of $\omega$. The second ball has a mass $2 m$ and a string of length $2 L$, and rotates at a rate of $2 \omega$.

Prabhu Ramji

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Problem 5

In Chapter 4 you learned that the mass of an object determines how that object responds to an applied force. Write a rotational analog to that idea based on the concepts of this chapter.

Prabhu Ramji

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Problem 6

Describe what a "torque wrench" is (look up the definition, if not known) and discuss any difficulties that a Canadian auto or bicycle mechanic might have working with an American mechanic's tools (and vice versa).

Alex Garger

Numerade Educator

Problem 7

In describing rotational motion it is often useful to develop an analogy with translational motion. First, write a set of equations describing translational motion. Then write the rotational analogs (for example, $\theta=\theta 0 \ldots$ $\theta=\theta_{0} \ldots$ of the translational equations (for example, $\mathrm{x}=\mathrm{x} 0+\mathrm{v} 0 \mathrm{xt}+12 \mathrm{axt} 2$
$x=x_{0}+v_{0 x} t+\frac{1}{2} a_{x} t^{2}$ ) using the following legend:
$\mathrm{x} \Leftrightarrow \theta \mathrm{v} \mathrm{x} \Leftrightarrow \omega \mathrm{zax} \Leftrightarrow \alpha \mathrm{z} \mathrm{Fx} \Leftrightarrow \mathrm{tzm} \Leftrightarrow \operatorname{Ipx} \Leftrightarrow$ LzKtranslational $\Leftrightarrow$ Krotational
$x \Leftrightarrow \theta \quad v_{x} \Leftrightarrow \omega_{z} \quad a_{x} \Leftrightarrow \alpha_{z} \quad F_{x} \Leftrightarrow \tau_{z} \quad m \Leftrightarrow I$
$$
p_{x} \Leftrightarrow L_{z} \quad K_{\text {translational }} \Leftrightarrow K_{\text {rotational }}
$$

Prabhu Ramji

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Problem 8

A student cannot open a door at her school. She pushes with ever-greater force, and still the door will not budge! Knowing that the door does push open, is not locked, and a minimum torque is required to open the door, give a few reasons why this might be occurring.

Prabhu Ramji

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Problem 9

Rank the torques exerted on the bolts in A-D (Figure 8-34) from least to greatest. Note that the forces in $\mathrm{B}$ and $\mathrm{D}$ make an angle of $45^{\circ}$ with the wrench. Assume the wrenches and the magnitude of the force $F$ are identical.

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Problem 10

A hollow cylinder rolls without slipping up an incline, stops, and then rolls back down. Which of the following graphs in Figure 8-35 shows the (a) angular acceleration and (b) angular velocity for the motion? Assume that up the ramp is the positive direction.

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Problem 11

Consider a situation in which a merry-go-round, starting from rest, speeds up in the counterclockwise direction. It eventually reaches and maintains a maximum angular velocity. After a short time the merry-goround then starts to slow down and eventually stops. Assume the accelerations experienced by the merry-go-round have constant magnitudes.
(a) Which graph in Figure 8-36 describes the angular velocity as the merrygo-round speeds up? (b) Which graph describes the angular position as the merry-go-round speeds up? (c) Which graph describes the angular velocity as the merry-go-round travels at its maximum velocity? (d) Which graph describes the angular position as the merry-go-round travels at its maximum velocity? (e) Which graph describes the angular velocity as the merry-goround slows down? (f) Which graph describes the angular position as the merry-go-round slows down? (g) Draw a graph of the torque experienced by the merry-go-round as a function of time during the scenario described in the problem.

Prabhu Ramji

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Problem 12

Using the rotational concepts of this chapter, explain why a uniform solid sphere beats a uniform solid cylinder which beats a ring when the three objects "race" down an inclined plane while rolling without slipping.

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Problem 13

Describe any inconsistencies in the following statement: "The units of torque are $\mathrm{N} \cdot \mathrm{m}$, but that's not the same as the units of energy.'

Alex Garger

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Problem 14

What are the units of the following quantities: (a) rotational kinetic energy, (b) moment of inertia, and (c) angular momentum?

Prabhu Ramji

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Problem 15

Explain which physical quantities change when an ice skater moves her arms in and out as she rotates in a pirouette. What causes her angular velocity to change, if it changes at all?

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Problem 16

Which quantity is larger: the angular momentum of Earth rotating on its axis each day or the angular momentum of Earth revolving about the Sun each year? Try to determine the answer without using a calculator.

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Problem 17

Explain how an object moving in a straight line can have a nonzero angular momentum.

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Problem 18

Analyze the following statement and determine if there are any physical inconsistencies: While rotating a ball on the end of a string of length $L$, the rotational kinetic energy remains constant as long as the length and angular speed are fixed. When the ball is pulled inward and the length of the string is shortened, the rotational kinetic energy will remain constant due to conservation of energy, but the angular momentum will not because there is an external force acting on the ball to pull it inward. The moment of inertia and angular speed will, of course, remain the same throughout the process because the ball is rotating in the same plane throughout the motion.

Alex Garger

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Problem 19

A freely rotating turntable moves at a steady angular velocity. A glob of cookie dough falls straight down and attaches to the very edge of the turntable. Describe which quantities (angular velocity, angular acceleration, torque, rotational kinetic energy, moment of inertia, or angular momentum) are conserved during the process and describe qualitatively what happens to the motion of the turntable.

Prabhu Ramji

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Problem 20

What are the units of angular velocity $\left(\omega^{\rightarrow}\right)^{(\vec{\omega})}$ ? Why are factors of $2 \pi$ present in many equations describing rotational motion?

Prabhu Ramji

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Problem 21

While watching two people on a seesaw, you notice that the person at the top always leans backward, while the person at the bottom always leans forward. (a) Why do the riders do this? (b) Assuming they are sitting equidistant from the pivot point of the seesaw, what, if anything, can you say about the relative masses of the two riders? SSM

Prabhu Ramji

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Problem 22

Referring to the time-lapse photograph of a falling cat in Figure 8-32, do you think that a cat will fall on her feet if she does not have a tail? Explain your answer using the concepts of this chapter.

Hubert Agamasu

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Problem 23

A solid sphere of radius $R$, a solid cylinder of radius $R$, and a rod of length $R$ all have the same mass, and all three are rotating with the same angular velocity. The sphere is rotating around an axis through its center. The cylinder is rotating around its long axis, and the rod is rotating around an axis through its center but perpendicular to the rod. Which one has the greatest rotational kinetic energy?
A. the sphere
B. the cylinder
C. the rod
D. the rod and cylinder have the same rotational kinetic energy
E. they all have the same kinetic energy

Prabhu Ramji

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Problem 24

How would a flywheel's (spinning disk's) kinetic energy change if its moment of inertia were five times larger but its angular speed were five times smaller?
A. $0.1$ times as large as before
B. $0.2$ times as large as before
C. same as before
D. 5 times as large as before
E. 10 times as large as before

Nicholas Mogoi

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Problem 25

You have two steel spheres; sphere 2 has twice the radius of sphere $1 .$ What is the ratio of the moment of inertia $I_{2}: I_{1}$ measured about an axis through the center of the spheres?
A. 2
B. 4
C. 8
D. 16
E. 32

Prabhu Ramji

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Problem 26

A solid ball, a solid disk, and a hoop, all with the same mass and the same radius, are set rolling without slipping up an incline, all with the same initial energy. Which goes farthest up the incline?
A. the ball
B. the disk
C. the hoop
D. the hoop and the disk roll to the same height, farther than the ball
E. they all roll to the same height

Prabhu Ramji

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Problem 27

A solid ball, a solid disk, and a hoop, all with the same mass and the same radius, are set rolling without slipping up an incline, all with the same initial linear speed. Which goes farthest up the incline?
A. the ball
B. the disk
C. the hoop
D. the hoop and the disk roll to the same height, farther than the ball
E. they all roll to the same height

Prabhu Ramji

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Problem 28

Todd and Susan are riding on a merry-go-round. Todd rides on a horse toward the outside of the circular platform, and Susan rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed, Todd's angular speed is
A. exactly half as much as Susan's.
B. larger than Susan's.
C. smaller than Susan's.
D. the same as Susan's.
E. exactly twice as much as Susan's.

Prabhu Ramji

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Problem 29

Todd and Susan are riding on a merry-go-round. Todd rides on a horse toward the outer edge of a circular platform. and Susan rides on a horse toward the center of the circular platform. When the merry-go-round is rotating at a constant angular speed $\omega$, Todd's speed $v$ is
A. exactly half as much as Susan's.
B. larger than Susan's.
C. smaller than Susan's
D. the same as Susan's.
E. exactly twice as much as Susan's.

Prabhu Ramji

Numerade Educator

Problem 30

While a gymnast is in the air during a leap, which of the following quantities must remain constant for her?
A. position
B. velocity
C. momentum
D. angular velocity
E. angular momentum about her center of mass

Prabhu Ramji

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Problem 31

The moment of inertia of a thin ring about its symmetry axis is $\mathrm{ICM}=\mathrm{MR} 2 I_{\mathrm{CM}}=M R^{2}$. What is the moment of inertia if you twirl a large ring around your finger, so that in essence it rotates about a point on the ring, about an axis parallel to the symmetry axis?
A. $5 M R^{2}$
B. $2 M R^{2}$
C. $M R^{2}$
D. $1.5 M R^{2}$
E. $0.5 M R^{2}$

Prabhu Ramji

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Problem 32

You give a quick push to a ball at the end of a massless, rigid rod, causing the ball to rotate clockwise in a horizontal circle (Figure 8-37). The rod's pivot is frictionless. After the push has ended, the ball's angular velocity A. steadily increases.
B. increases for a while, then remains constant.
C. decreases for a while, then remains constant.
D. remains constant.
E. steadily decreases.

Prabhu Ramji

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Problem 33

Estimate the angular speed of a car moving around a cloverleaf on-ramp of a typical freeway. Cloverleaf ramps extend through approximately threequarters of a circle to connect two perpendicular freeways.

Alex Garger

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Problem 34

A fan is designed to last for a certain time before it will have to be replaced (planned obsolescence). The fan has one speed (at a maximum of $750 \mathrm{rpm}$ ), which it reaches in $2 \mathrm{~s}$ starting from rest. It takes the fan $10 \mathrm{~s}$ to stop rotating once it is turned off. If the manufacturer specifies that the fan will operate up to 1 billion rotations, estimate how many days you will be able to use the fan.

Prabhu Ramji

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Problem 35

Estimate the torque you apply when you open a door in your house, and specify the axis to which your estimate refers.

Alex Garger

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Problem 36

Make a rough estimate of the moment of inertia of a pencil that is spun about its center by a nervous student during an exam.

Prabhu Ramji

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Problem 37

Estimate the moment of inertia of a figure skater as she rotates about the vertical axis that passes straight down through the center of her body into the ice. Make this estimation for the extreme parts of a pirouette (arms fully extended and arms drawn in tightly).

Manish Jain

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Problem 38

Estimate the angular displacement (in radians and degrees) of Earth in one day of its orbit around the Sun.

Prabhu Ramji

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Problem 39

Estimate the angular speed of the apparent passage of the Sun across the sky of Earth (from dawn until dusk).

Prabhu Ramji

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Problem 40

Estimate the angular acceleration of a lone sock that is inside a washing machine that starts from rest and reaches the maximum speed of its spin cycle in typical fashion.

Alex Garger

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Problem 41

Estimate the angular momentum about the center of rotation for a "skipit ball" that is spun around on the ankle of a small child (the child hops over the ball as it swings around and around her feet). SSM

Alex Garger

Numerade Educator

Problem 42

Using a spreadsheet and the data below, calculate the average angular speed of the rotating object over the first 10 s. Calculate the average angular acceleration from 15 to $25 \mathrm{~s}$. If the object has a moment of inertia of $0.25 \mathrm{~kg}$ $\mathrm{m} 2^{0.25 \mathrm{~kg} \cdot \mathrm{m}^{2}}$ about the axis of rotation, calculate the average torque during the following time intervals: $0<t<10 \mathrm{~s}, 0<t<10 \mathrm{~s}, 10 \mathrm{~s}<\mathrm{t}<15 \mathrm{~s}$, $10 \mathrm{~s}<t<15 \mathrm{~s}$, and $15 \mathrm{~s}<\mathrm{t}<25 \mathrm{~s} .15 \mathrm{~s}<t<25 \mathrm{~s}$.
$$
\begin{array}{cc}
t(\mathrm{~s}) & \theta(\mathrm{rad}) \\
\hline 0 & 0 \\
1 & 0.349 \\
2 & 0.700
\end{array}
$$
$$
\begin{array}{l}
\begin{array}{cc}
3 & 1.05 \\
4 & 1.40 \\
5 & 1.75 \\
6 & 2.10 \\
7 & 2.44 \\
8 & 2.80 \\
9 & 3.14 \\
10 & 3.50 \\
11 & 3.50 \\
12 & 3.49 \\
13 & 3.50 \\
14 & 3.51 \\
15 & 3.51 \\
16 & 3.98 \\
17 & 5.01 \\
18 & 6.48 \\
19 & 8.53 \\
20 & 11.0 \\
21 & 14.1 \\
22 & 17.6 \\
23 & 21.6 \\
24 & 26.2 \\
25 & 31.0 \\
\hline
\end{array}\\
\begin{array}{l}
\hline
\end{array}
\end{array}
$$

Manish Jain

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Problem 43

What is the angular speed, in $\mathrm{rad} / \mathrm{s}$, of an object that completes $2.00$ rev every $12.0$ s? Example $8-2$

Prabhu Ramji

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Problem 44

A car rounds a curve with a translational speed of $12.0 \mathrm{~m} / \mathrm{s}$. If the radius of the curve is $7.00 \mathrm{~m}$, calculate the angular speed in $\mathrm{rad} / \mathrm{s}$. Example $8-2$

Prabhu Ramji

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Problem 45

Convert the following: Example 8-2
45.0 rev/min= ______rad/s3313rpm=_______rad/s2? rev/s=_______rad/s
$45.0 \mathrm{rev} / \mathrm{min}=$______$\mathrm{rad} / \mathrm{s}$
$33 \frac{1}{3} \mathrm{rpm}=$______$\mathrm{rad} / \mathrm{s}$
$2 \pi \mathrm{rev} / \mathrm{s}=$_______$\mathrm{rad} / \mathrm{s}$

Prabhu Ramji

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Problem 46

Calculate the angular speed of the Moon as it orbits Earth. The Moon completes one orbit about Earth in $27.4$ days and the Earth-Moon distance is $3.84 \times 108 \mathrm{~m} 3.84 \times 10^{8} \mathrm{~m} .$ SSM $\underline{\text { Example } 8-1}$

Prabhu Ramji

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Problem 47

If a $0.250$ -kg point object rotates at $3.00$ rev/s about an axis that is $0.500$ m away, what is the kinetic energy of the object? Example $8-3$

Prabhu Ramji

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Problem 48

What is the rotational kinetic energy of an object that has a moment of inertia of $0.280 \mathrm{~kg} \cdot \mathrm{m} 2^{0.280 \mathrm{~kg} \cdot \mathrm{m}^{2}}$ about the axis of rotation when its angular speed is $4.00 \mathrm{rad} / \mathrm{s}$ ? Example $8-2$

Prabhu Ramji

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Problem 49

What is the moment of inertia of an object that rotates at $13.0$ rev $/ \min$ about an axis and has a rotational kinetic energy of $18.0$ J? Example $8-2$

Prabhu Ramji

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Problem 50

What is the angular speed of a rotating wheel that has a moment of inertia of $0.330 \mathrm{~kg} \cdot \mathrm{m} 2^{0.330 \mathrm{~kg} \cdot \mathrm{m}^{2}}$ and a rotational kinetic energy of $2.75 \mathrm{~J}$ ? Give your answer in both rad/s and rev/min. SSM Example 8-2

Prabhu Ramji

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Problem 51

What is the combined moment of inertia for the three point objects about the axis $O$ in $\underline{\text { Figure } 8-38 \text { ? }}$ Example $8-5$

Prabhu Ramji

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Problem 52

What is the combined moment of inertia of three point objects $\left(\mathrm{m} 1=1.00 \mathrm{~kg}, m_{1}=1.00 \mathrm{~kg}, \mathrm{~m} 2=1.50 \mathrm{~kg}, m_{2}=1.50 \mathrm{~kg}, \mathrm{~m} 3=2.00 \mathrm{~kg}\right.$
$m_{3}=2.00 \mathrm{~kg}$ ) tied together with massless strings and rotating about the axis $O$ as shown in Figure 8-39? Example 8-4

Prabhu Ramji

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Problem 53

A baton twirler in a marching band complains that her baton is defective (Figure 8-40). The manufacturer specifies that the baton should have an overall length of $\mathrm{L}=60.0 \mathrm{~cm} L=60.0 \mathrm{~cm}$ and a total mass between 940 and $950 \mathrm{~g}$ (there is one 350 -g object on each end). Also according to the manufacturer, the moment of inertia about the central axis passing through the baton should fall between $0.0750$ and $0.0800 \mathrm{~kg} \cdot \mathrm{m} 2^{0.0800 \mathrm{~kg} \cdot \mathrm{m}^{2}}$. The twirler (who has completed a class in physics) claims this is impossible. Who's right? Explain your answer. Example $8-8$

Prabhu Ramji

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Problem 54

What is the moment of inertia of a steering wheel about the axis that passes through its center? Assume the rim of the wheel has a radius $R$ and a mass $M$. Assume that there are five radial spokes that connect in the center as shown in Figure 8-41. The spokes are thin rods of uniform mass density with length $R$ and mass $12 \mathrm{M}, \frac{1}{2} M$, evenly spaced around the wheel. SSM Example $8-8$

Prabhu Ramji

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Problem 55

Using the parallel-axis theorem, calculate the moment of inertia for a solid, uniform sphere about an axis that is tangent to its surface (Figure 8-42). Example 8-8

Prabhu Ramji

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Problem 56

Calculate the moment of inertia for a uniform, solid cylinder (mass M, radius $R$ ) if the axis of rotation is tangent to the side of the cylinder as shown in $\underline{\text { Figure } 8-43 . \text { Example } 8-8}$

Prabhu Ramji

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Problem 57

Calculate the moment of inertia for a thin uniform rod that is $1.25 \mathrm{~m}$ long and has mass of $2.25 \mathrm{~kg}$. The axis of rotation passes through the rod at a point one-third of the way from the left end (Figure 8-44). Example 8-7

Prabhu Ramji

Numerade Educator

Problem 58

Calculate the moment of inertia of a thin plate that is $5.00 \mathrm{~cm} \times 7.00 \mathrm{~cm}$ $5.00 \mathrm{~cm} \times 7.00 \mathrm{~cm} \mathrm{~cm}$ in area and has a uniform mass density of $1.50 \mathrm{~g} / \mathrm{cm} 2$
$1.50 \mathrm{~g} / \mathrm{cm}^{2}$. The axis of rotation is located at the left side, as shown in Figure 8-45. Example 8-8

Prabhu Ramji

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Problem 59

Calculate the radius of a solid uniform sphere of mass $M$ that has the same moment of inertia about an axis through its center of mass as a second solid uniform sphere of radius $R$ and mass $M$ which has the axis of rotation passing tangent to the surface and parallel to the center of mass axis (Figure $\underline{\text { 8-46 }}$ ). Example 8-8

Prabhu Ramji

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Problem 60

Two uniform, solid spheres (one has a mass $M$ and a radius $R$ and the other has a mass $M$ and a radius $2 R$ ) are connected by a thin, uniform rod of length $3 R$ and mass $M$ (Figure 8-47). Find the moment of inertia about the axis through the center of the rod. SSM Example $8-7$

Prabhu Ramji

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Problem 61

What is the moment of inertia of the sphere-rod system shown in Figure 8-48 where the sphere is of uniform mass density and has a radius $R$ and a mass $M$ and the rod is thin and massless, and has a length $L$ ? The sphere-rod system is spun about an axis A.Example 8-7

Prabhu Ramji

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Problem 62

A bowling ball that has a radius of $11.0 \mathrm{~cm}$ and a mass of $5.00$ $\mathrm{kg}$ rolls without slipping on a level lane at $2.00 \mathrm{rad} / \mathrm{s}$. Calculate the ratio of the translational kinetic energy to the rotational kinetic energy of the bowling ball. SSM Example 8-10

Prabhu Ramji

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Problem 63

Earth is approximately a solid sphere, has a mass of $5.98 \times 1024 \mathrm{~kg}, 5.98 \times 10^{24} \mathrm{~kg}$, a radius of $6.38 \times 106 \mathrm{~m}, 6.38 \times 10^{6} \mathrm{~m}$, and completes one rotation about its central axis each day. Calculate the rotational kinetic energy of Earth as it spins on its axis. Example $8-9$

Prabhu Ramji

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Problem 64

Calculate the translational kinetic energy of Earth as it orbits the Sun once each year (the Earth-Sun distance is $1.50 \times 1011 \mathrm{~m}$ $1.50 \times 10^{11} \mathrm{~m}$ ). Calculate the ratio of the translational kinetic energy to the rotational kinetic energy calculated in the previous problem. Example $8-9$

Prabhu Ramji

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Problem 65

A potter's flywheel is made of a $5.00$ -cm-thick, round slab of concrete that has a mass of $60.0 \mathrm{~kg}$ and a diameter of $35.0 \mathrm{~cm}$. This disk rotates about an axis that passes through its center, perpendicular to its round area. Calculate the angular speed of the slab about its center if the rotational kinetic energy is $15.0 \mathrm{~J}$. Express your answer in both $\mathrm{rad} / \mathrm{s}$ and rev/min. Example $8-9$

Prabhu Ramji

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Problem 66

A flying disk (160 g, $25.0 \mathrm{~cm}$ in diameter) spins at a rate of $3.00 \times 102 \mathrm{rpm}^{3.00 \times 10^{2}}$ rpm rpm with its center balanced on a fingertip. What is the rotational kinetic energy of the Frisbee if the disc has $70.0 \%$ of its mass on the outer edge (basically a thin ring 25.0-cm in diameter) and the remaining $30.0 \%$ is a nearly flat disk 25.0-cm in diameter? SSM Example 89

Prabhu Ramji

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Problem 67

A uniform, solid cylinder of radius $5.00 \mathrm{~cm}$ and mass $3.00 \mathrm{~kg}$ starts from rest at the top of an inclined plane that is $2.00 \mathrm{~m}$ long and tilted at an angle of $25.0^{\circ}$ with the horizontal and rolls without slipping down the ramp. What is the cylinder's speed at the bottom of the ramp? Example $8-10$

Prabhu Ramji

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Problem 68

A uniform, solid sphere of radius $5.00 \mathrm{~cm}$ and mass $3.00 \mathrm{~kg}$ starts with a translational speed of $2.00 \mathrm{~m} / \mathrm{s}$ at the top of an inclined plane that is $2.00 \mathrm{~m}$ long and tilted at an angle of $25.0^{\circ}$ with the horizontal and rolls without slipping down the ramp. What is the sphere's speed at the bottom of the ramp? Example $8-10$

Prabhu Ramji

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Problem 69

A spherical marble that has a mass of $50.0 \mathrm{~g}$ and a radius of $0.500 \mathrm{~cm}$ rolls without slipping down a loop-the-loop track that has a radius of $20.0 \mathrm{~cm}$. The marble starts from rest and just barely clears the loop to emerge on the other side of the track. What is the minimum height that the marble must start from to make it around the loop? Example $8-10$

Prabhu Ramji

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Problem 70

A billiard ball of mass $160 \mathrm{~g}$ and radius $2.50 \mathrm{~cm}$ starts with a translational speed of $2.00 \mathrm{~m} / \mathrm{s}$ at point $A$ on the track as shown in Figure 8 -
49. If point $B$ is at the top of a hill that has a radius of curvature of $60 \mathrm{~cm}$, what is the normal force acting on the ball at point $B$ ? Assume the billiard ball rolls without slipping on the track. Example $8-10$

Prabhu Ramji

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Problem 71

Suppose a roulette wheel is spinning at 1 rev/s. (a) How long will it take for the wheel to come to rest if it experiences an angular acceleration of $-0.02$ $\mathrm{rad} / \mathrm{s}^{2}$ ? (b) How many rotations will it complete in that time? Example 8-11

Prabhu Ramji

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Problem 72

A spinning top completes $6.00 \times 1036.00 \times 10^{3}$ rotations before it starts to topple over. The average speed of the rotations is $8.00 \times 102 \mathrm{rpm}$ $8.00 \times 10^{2}$ rpm. Calculate how long the top spins before it begins to topple.

Prabhat Tyagi

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Problem 73

A child pushes a merry-go-round that has a diameter of $4.00 \mathrm{~m}$ and goes from rest to an angular speed of $18.0$ rpm in a time of $43.0 \mathrm{~s}$. (a) Calculate the average angular acceleration (in $\mathrm{rad} / \mathrm{s}^{2}$ ) of the merry-go-round. (b) Calculate the angular displacement (in rad) of the merry-go-round during this time interval. (c) What is the maximum tangential speed of the child if she rides on the edge of the platform? Example $8-11$

Prabhu Ramji

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Problem 74

Allison twirls an umbrella around its central axis so that it completes $24.0$ rotations in $30.0 \mathrm{~s}$. (a) If the umbrella starts from rest, calculate the angular acceleration (in $\mathrm{rad} / \mathrm{s}^{2}$ ) of a point on the outer edge. (b) What is the maximum tangential speed of a point on the edge if the umbrella has a radius of $55.0 \mathrm{~cm}$ ? Example $8-11$

Prabhu Ramji

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Problem 75

Prior to the music CD, stereo systems had a phonographic turntable on which vinyl disk recordings were played. A particular phonographic turntable starts from rest and achieves a final constant angular speed of $3313^{33 \frac{1}{3}}$ rpm in a time of $4.5 \mathrm{~s}$. (a) How many rotations did the turntable undergo during that time? (b) The classic Beatles album Abbey Road is 47 min and $7 \mathrm{~s}$ in duration. If the turntable requires $8 \mathrm{~s}$ to come to rest once the album is over, calculate the total number of rotations for the complete start-up, playing, and slow-down of the album. Example $8-11$

Prabhu Ramji

Numerade Educator

Problem 76

A CD player varies its speed as it changes circular tracks on the CD. A CD player is rotating at 300 rpm. To read another track the angular speed is increased to 450 rpm in a time of $0.75$ s. Calculate the average angular acceleration in $\mathrm{rad} / \mathrm{s}^{2}$ during the change. SSM Example $8-11$

Prabhu Ramji

Numerade Educator

Problem 77

A communication satellite circles Earth in a geosynchronous orbit such that the satellite remains directly above the same point on the surface of Earth. (a) What angular displacement (in radians) does the satellite undergo in $1 \mathrm{~h}$ of its orbit? (b) Calculate the angular speed of the satellite in rev/min and rad/s. Example 8-11

Prabhu Ramji

Numerade Educator

Problem 78

What is the torque about your shoulder axis if you hold a $10.0-\mathrm{kg}$ barbell in one hand straight out and at shoulder height? Assume your hand is $75 \mathrm{~cm}$ from your shoulder and neglect the torque due to the weight of your arm. SSM Example 8-12

Prabhu Ramji

Numerade Educator

Problem 79

A driver applies a horizontal force of $20.0 \mathrm{~N}$ (to the right) to the top of a steering wheel, as shown in Figure 8-50. The steering wheel has a radius of $18.0 \mathrm{~cm}$ and a moment of inertia of $0.0970 \mathrm{~kg} \cdot \mathrm{m}^{2}$. Calculate the angular acceleration of the steering wheel about the central axis due to this force. $\underline{\text { Example } 8-12}$

Prabhu Ramji

Numerade Educator

Problem 80

Medical When the palmaris longus muscle in the forearm is flexed, the wrist moves back and forth (Figure 8-51). If the muscle generates a force of 45.0 $\mathrm{N}$ and it is acting with an effective lever arm of $22.0 \mathrm{~cm}$, what is the torque that the muscle produces on the wrist? Curiously, many people lack this muscle. Some studies correlate the absence of the muscle with carpal tunnel syndrome. Example 8-12

Prabhu Ramji

Numerade Educator

Problem 81

A torque wrench is used to tighten a nut on a bolt. The wrench is $25 \mathrm{~cm}$ long, and a force of $120 \mathrm{~N}$ is applied at the end of the wrench as shown in Figure 8-52. Calculate the torque about the axis that passes through the bolt. $\underline{\text { Example } 8-12}$

Prabhu Ramji

Numerade Educator

Problem 82

An $85.0$ -cm-wide door is pushed open with a force of $F=75.0 \mathrm{~N}$. Calculate the torque about an axis that passes through the hinges in each of the cases in $\underline{\text { Figure } 8-53 . \text { SSM }} \underline{\text { Example } 8-12}$

Prabhu Ramji

Numerade Educator

Problem 83

A robotic arm lifts a barrel of radioactive waste (Figure 8-54). If the maximum torque delivered by the arm about the axis $O$ is $3.00 \times 103 \mathrm{~N} \cdot \mathrm{m}$ $3.00 \times 10^{3} \mathrm{~N} \cdot \mathrm{m}$ and the distance $r$ in the diagram is $3.00 \mathrm{~m}$, what is the maximum mass of the barrel? Example 8-12

Prabhu Ramji

Numerade Educator

Problem 84

A typical adult can deliver about $10 \mathrm{~N} \cdot \mathrm{m}$ of torque when attempting to open a twist-off cap on a bottle. What is the maximum force that the average person can exert with his fingers if most bottle caps are about $2 \mathrm{~cm}$ in diameter? Example $8-12$

Prabhu Ramji

Numerade Educator

Problem 85

A potter's wheel is initially at rest. A constant external torque of $75.0$ $\mathrm{N} \cdot \mathrm{m}$ is applied to the wheel for $15.0 \mathrm{~s}$, giving the wheel an angular speed of $5.00 \times 102$ rev $/ \min ^{5} .00 \times 10^{2}$ rev $/ \mathrm{min}$. (a) What is the moment of inertia of the wheel? (b) The external torque is then removed, and a brake is applied. If it takes the wheel $2.00 \times 102 \mathrm{~s} 2.00 \times 10^{2} \mathrm{~s}$ to come to rest after the brake is applied, what is the magnitude of the torque exerted by the brake? Example $\underline{8-12}$

Prabhu Ramji

Numerade Educator

Problem 86

A solid cylindrical pulley with a mass of $1.00 \mathrm{~kg}$ and a radius of $0.25 \mathrm{~m}$ is free to rotate about its axis. An object of mass $0.250 \mathrm{~kg}$ is attached to the pulley with a light string (Figure 8-55). Assuming the string does not stretch or slip, calculate the tension in the string and the angular acceleration of the pulley. Example $8-13$

Prabhu Ramji

Numerade Educator

Problem 87

Figure 8-56 shows a solid, uniform cylinder of mass $7.00 \mathrm{~kg}$ and radius $0.450 \mathrm{~m}$ with a light string wrapped around it. A $3.00-\mathrm{N}$ tension force is applied to the string, causing the cylinder to roll without slipping across a level surface as shown. (a) What is the angular acceleration of the cylinder?
(b) Calculate the magnitude and direction of the frictional force that acts on the cylinder. Example $8-14$

Prabhu Ramji

Numerade Educator

Problem 88

A crane winch is lifting a 2300 -kg mass. The winch can be modeled as a cable of negligible mass wound around a solid uniform cylinder with a $12-\mathrm{cm}$ radius and a mass of $320 \mathrm{~kg}$ that rotates around its central axis. How much torque is required to accelerate the 2300 -kg mass straight upward at $0.35$ $\mathrm{m} / \mathrm{s}^{2}$ ? Assume there is no friction in the system and that the winch radius remains constant. Example $8-12$

Prabhu Ramji

Numerade Educator

Problem 89

A block with mass $m 1=2.00 \mathrm{~kg} m_{1}=2.00 \mathrm{~kg}$ rests on a frictionless table. It is connected with a light string over a pulley to a hanging block of mass $\mathrm{m} 2=4.00 \mathrm{~kg} m_{2}=4.00 \mathrm{~kg}$. The pulley is a uniform disk with a radius of $4.00 \mathrm{~cm}$ and a mass of $0.500 \mathrm{~kg}$ (Figure 8-57). (a) Calculate the acceleration of each block and the tension in each segment of the string. (b) How long does it take the blocks to move a distance of $2.25 \mathrm{~m} ?$ (c) What is the angular speed of the pulley at this time? Example $8-13$

Prabhu Ramji

Numerade Educator

Problem 90

A yo-yo with a mass of $0.0750 \mathrm{~kg}$ and a rolling radius of $\mathrm{r}=2.50 \mathrm{~cm}$ $r=2.50 \mathrm{~cm}$ rolls down a string with a linear acceleration of $6.50 \mathrm{~m} / \mathrm{s}^{2}$ (\underline{Figure } 8-58). (a) Calculate the tension in the string and the angular acceleration of the yo-yo. (b) What is the moment of inertia of this yo-yo? Example $8-14$

Prabhu Ramji

Numerade Educator

Problem 91

What is the angular momentum about the central axis of a thin disk that is $18.0 \mathrm{~cm}$ in diameter, has a mass of $2.50 \mathrm{~kg}$, and rotates at a constant $1.25$ $\mathrm{rad} / \mathrm{s} ? \underline{\text { Example } 8-15}$

Prabhu Ramji

Numerade Educator

Problem 92

What is the angular momentum of a $0.300-\mathrm{kg}$ tetherball when it whirls around the central pole at $60.0 \mathrm{rpm}$ and at a radius of $125 \mathrm{~cm} ?$ Example $8-16$

Prabhu Ramji

Numerade Educator

Problem 93

Calculate the angular momentum of Earth as it orbits the Sun. Recall that the mass of Earth is $5.98 \times 1024 \mathrm{~kg}, 5.98 \times 10^{24} \mathrm{~kg}$, the distance between Earth and the Sun is $1.50 \times 1011 \mathrm{~m}, 1.50 \times 10^{11} \mathrm{~m}$, and the time for one orbit is $365.3$ days. Example $8-16$

Prabhu Ramji

Numerade Educator

Problem 94

Calculate the angular momentum of Earth as it spins on its central axis once each day. Assume Earth is approximately a uniform, solid sphere that has a mass of $5.98 \times 1024 \mathrm{~kg}^{5.98 \times 10^{24} \mathrm{~kg}}$ and a radius of $6.38 \times 106 \mathrm{~m} 6.38 \times 10^{6} \mathrm{~m} . \underline{\text { Example } 8-15}$

Prabhu Ramji

Numerade Educator

Problem 95

What is the speed of an electron in the lowest energy orbital of hydrogen, of radius equal to $5.29 \times 10-11 \mathrm{~m} 5.29 \times 10^{-11} \mathrm{~m}$ ? The mass of an electron is $9.11 \times 10-31 \mathrm{~kg}, 9.11 \times 10^{-31} \mathrm{~kg}$, and its angular momentum in this orbital is $1.055 \times 10-34 \mathrm{~J} \cdot \mathrm{s} 1.055 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} . \underline{\text { Example } 8-16}$

Prabhu Ramji

Numerade Educator

Problem 96

What is the angular momentum of a $70.0$ -kg person riding on a Ferris wheel that has a diameter of $35.0 \mathrm{~m}$ and rotates once every $25.0 \mathrm{~s}$ ?

Prabhu Ramji

Numerade Educator

Problem 97

A professor sits on a rotating stool that spins at $10.0$ rpm while she holds a $1.00-\mathrm{kg}$ weight in each of her hands. Her outstretched arms are $0.750$ $\mathrm{m}$ from the axis of rotation, which passes through her head into the center of the stool. When she draws the weights in toward her body, her angular speed increases to $20.0$ rpm. Neglecting the mass of her arms, how far are the weights from the rotational axis at the increased speed? Example $8-15$

Prabhu Ramji

Numerade Educator

Problem 98

A $2.15-\mathrm{kg}, 16.0$ -cm radius, high-end turntable is rotating freely at $33.3$ rpm when a naughty child drops $11.0 \mathrm{~g}$ of chewing gum onto it $10.0 \mathrm{~cm}$ from the rotation axis. Assuming that the gum sticks where it lands, and that the turntable can be modeled as a solid, uniform disk, what is the new angular speed of the turntable? Example $8-16$

Prabhu Ramji

Numerade Educator

Problem 99

A giant toroidal space station is being evacuated. Model the $1.0 \times 1010$ $\mathrm{kg}^{1.0} \times 10^{10}-\mathrm{kg}$ station as a cylindrical hoop of radius $1.8 \mathrm{~km}$ rotating at a rate of $2 \pi \mathrm{rad} / \mathrm{min}$. If each wave of evacuees consists of one thousand $4.0 \times 105-\mathrm{kg}^{4.0} \times 10^{5}-\mathrm{kg}$ escape pods whose launchers are pointed radially outward from the station rim, how much does the angular speed of the station change with the first wave of evacuees? Express your answer in rad/sec.

Prabhu Ramji

Numerade Educator

Problem 100

A chef is tossing $0.500 \mathrm{~kg}$ of pizza crust dough. With each toss the dough has an initial angular speed of $5.20 \mathrm{rad} / \mathrm{s}$. During one particular toss, the dough starts out uniformly distributed throughout a $20.0$ -cm diameter disk and expands to $22.0 \mathrm{~cm}$ in diameter. Assuming the mass remains uniformly distributed, what is the angular speed of the dough when the chef catches it?

Prabhu Ramji

Numerade Educator

Problem 101

A $50.0$ -g meter stick is balanced at its midpoint $(50.0 \mathrm{~cm})$. Then a $0.100-\mathrm{kg}$ and a $0.200-\mathrm{kg}$ mass are hung with light string from the $10.0$ -cm and $70.0$ -cm points, respectively. (Figure 8-59). Calculate the clockwise and counterclockwise torques acting on the board due to the four forces shown about an axis pointing out of the page at the following points: (a) the $0-\mathrm{cm}$ point, (b) the 50 -cm point, and (c) the 100 -cm point. (d) Is the meter stick still balanced after the two masses have been added? Example 8-17

Manish Jain

Numerade Educator

Problem 102

A 325 -kg merry-go-round with a radius of $1.40 \mathrm{~m}$ is spinning counterclockwise as viewed from above at $4.70 \mathrm{rad} / \mathrm{s}$. A $36.0$ -kg child is hanging on tightly $1.25 \mathrm{~m}$ from the rotation axis. Her father applies friction to the outer rim to slow the merry-go-round to a stop in $5 \mathrm{~s}$. How much torque must he apply? Model the merry-go-round as a solid disk. Give both the magnitude and the direction of the torque. Example $8-17$

Prabhu Ramji

Numerade Educator

Problem 103

A circular revolving door has a radius of $\mathrm{R}=1.00 \mathrm{~m}, R=1.00 \mathrm{~m}$, total moment of inertia of $\mathrm{I}=119 \mathrm{~kg} \cdot \mathrm{m} 2^{I}=119 \mathrm{~kg} \cdot \mathrm{m}^{2}$ and is free to rotate on
frictionless bearings as shown in Figure 8-60. Each of the door "arms" makes a tight seal with the frame, so that each arm provides $\mathrm{fk}=45.0 \mathrm{~N} f_{\mathrm{k}}=45.0 \mathrm{~N}$ of frictional force when the door rotates. If an $85.0$ -kg adult pushes perpendicularly to one of the doors at a distance $\mathrm{r}=0.600 \mathrm{~m} r=0.600 \mathrm{~m}$ from the rotation axis, what force must the person exert to make the door spin at a constant rate, assuming all four arms are in contact with the frame? Example $8-12$

Prabhu Ramji

Numerade Educator

Problem 104

A tightrope walker is walking between two buildings using a $15.0-\mathrm{m}$ long, $18.0$ -kg pole for balance (Figure 8-61). The daredevil grips the pole with each hand $0.600 \mathrm{~m}$ from the center of the pole. A $0.540$ -kg bird lands on the very end of the left-hand side of the pole. Assuming the daredevil applies forces with each hand in a direction perpendicular to the pole, how much force must each hand exert to counteract the torque of the bird? The $+x$ axis points in the walking direction, through the center of the pole. What are the directions of the torque vectors due to the bird, the left hand, and the right hand? Example $8-17$

Prabhu Ramji

Numerade Educator

Problem 105

A rigid, uniform, $4.00$ -m long, $40.0$ -kg beam rests on a pivot placed $3.00 \mathrm{~m}$ from one end (Figure 8-62). An 80.0-kg man walks up the beam from the end resting on the ground. How far along the beam must the man walk before the low end lifts off the ground? What is the direction of the net torque vector on the beam until the man reaches that critical point?

Prabhu Ramji

Numerade Educator

Problem 106

A yo-yo, with an inner cylinder radius of $\mathrm{r}=2.5 \mathrm{~cm} r=2.5 \mathrm{~cm}$ and outer disk radius of $\mathrm{R}=7.5 \mathrm{~cm}, R=7.5 \mathrm{~cm}$, is at rest on a $30.0^{\circ}$ incline. It is held in place by friction and the tension in the string that is wrapped around its inner cylinder, as shown in Figure 8-63. If the mass of the yo-yo is $420 \mathrm{~g}$, what is the direction and magnitude of the friction force? Example $8-14$

Prabhu Ramji

Numerade Educator

Problem 107

A baton is constructed by attaching two small objects that each have a mass $M$ to the ends of a uniform rod that has a length $L$ and a mass $M$. Find an expression for the moment of inertia of the baton when it is rotated around a point $(3 / 8) L$ from one end. $\underline{\text { Example } 8-8}$

Prabhu Ramji

Numerade Educator

Problem 108

The outside diameter of the playing area of an optical Blu-ray disc is $11.75 \mathrm{~cm}$, and the inside diameter is $4.50 \mathrm{~cm}$. When viewing movies, the disc rotates so that a laser maintains a constant linear speed relative to the disc of $7.50 \mathrm{~m} / \mathrm{s}$ as it tracks over the playing area. (a) What are the maximum and minimum angular speeds (in rad/s and rpm) of the disc? (b) At which location of the laser on the playing area do these speeds occur? (c) What is the average angular acceleration of a Blu-ray disc as it plays an $8.0$ -h set of movies? Example $8-11$

Prabhu Ramji

Numerade Educator

Problem 109

A table saw has a 25.0-cm-diameter blade that rotates at a rate of 7000 rpm. It is equipped with a safety mechanism that can stop the blade within $5.00 \mathrm{~ms}$ if something like a finger is accidentally placed in contact with the blade. (a) What average angular acceleration occurs if the saw starts at 7000 rpm and comes to rest in this time? (b) How many rotations does the blade complete during the stopping period? Example $8-11$

Prabhu Ramji

Numerade Educator

Problem 110

In 1932 Albert Dremel of Racine, Wisconsin, created his rotary tool that has come to be known as a dremel. (a) Suppose a dremel starts from rest and achieves an operating speed of 35,000 rev/min. If it requires $1.20 \mathrm{~s}$ for the tool to reach operating speed and it is held at that speed for $45.0 \mathrm{~s}$, how many rotations has the bit made? (b) Suppose it requires another $8.50 \mathrm{~s}$ for the tool to return to rest. What are the average angular accelerations for the start-up and the slow-down periods? (c) How many rotations does the tool complete from start to finish? $\underline{\text { Example } 8-11}$

Alex Garger

Numerade Educator

Problem 110

In 1932 Albert Dremel of Racine, Wisconsin, created his rotary tool that has come to be known as a dremel. (a) Suppose a dremel starts from rest and achieves an operating speed of 35,000 rev/min. If it requires $1.20 \mathrm{~s}$ for the tool to reach operating speed and it is held at that speed for $45.0 \mathrm{~s}$, how many rotations has the bit made? (b) Suppose it requires another $8.50 \mathrm{~s}$ for the tool to return to rest. What are the average angular accelerations for the start-up and the slow-down periods? (c) How many rotations does the tool complete from start to finish? Example $8-11$

Alex Garger

Numerade Educator

Problem 111

On average, both arms and hands together account for $13 \%$ of a person's mass, while the head is $7.0 \%$ and the trunk and legs account for $80 \%$. We can model a spinning skater with his arms outstretched as a vertical cylinder (head + trunk + legs) with two solid uniform rods (arms + hands) extended horizontally. Suppose a 62.0-kg skater is $1.80 \mathrm{~m}$ tall, has arms that are each $65.0 \mathrm{~cm}$ long (including the hands), and a trunk that can be modeled as being $35.0 \mathrm{~cm}$ in diameter. If the skater is initially spinning at $70.0 \mathrm{rpm}$ with his arms outstretched, what will his angular velocity be (in rpm) when he pulls in his arms until they are at his sides parallel to his trunk?

Manish Jain

Numerade Educator

Problem 112

Because of your success in physics class you are selected for an internship at a prestigious bicycle company in its research and development division. Your first task involves designing a wheel made of a hoop that has a mass of $1.00 \mathrm{~kg}$ and a radius of $50.0 \mathrm{~cm}$, and spokes with a mass of $10.0 \mathrm{~g}$ each. The wheel should have a total moment of inertia $0.280 \mathrm{~kg} \cdot \mathrm{m}^{2}$. (a) How many spokes are necessary to construct the wheel? (b) What is the mass of the wheel? SSM Example $8-8$

Alex Garger

Numerade Educator

Problem 113

Two beads that each have a mass $M$ are attached to a thin rod that has a length $2 L$ and a mass $M / 8$ (Figure 8-64). Each bead is initially a distance $L / 4$ from the center of the rod. The whole system is set into uniform rotation about the center of the rod, with initial angular frequency $\omega \mathrm{i}=20 \pi \mathrm{rad} / \mathrm{s}$ $\omega_{\mathrm{i}}=20 \pi \mathrm{rad} / \mathrm{s}$. If the beads are then allowed to slide to the ends of the rod, what will the angular frequency become? Example $8-15$

Alex Garger

Numerade Educator

Problem 114

A uniform disk that has a mass $\mathrm{M}=0.300 \mathrm{~kg} M=0.300 \mathrm{~kg}$ and a radius $\mathrm{R}=0.270 \mathrm{~m} R=0.270 \mathrm{~m}$ rolls up a ramp of angle $\theta=55.0^{\circ} \theta=55.0^{\circ}$ with initial
speed $\mathrm{v}=4.8 \mathrm{~m} / \mathrm{s}^{v}=4.8 \mathrm{~m} / \mathrm{s}$. If the disk rolls without slipping, how far up the ramp does it go? Example $8-14$

Alex Garger

Numerade Educator

Problem 115

In a new model of a machine, a spinning solid spherical part of radius $R$ must be replaced by a ring of the same mass which is to have the same kinetic energy. Both parts need to spin at the same rate, the sphere about an axis through its center and the ring about an axis perpendicular to its plane at its center. (a) What should the radius of the ring be in terms of $R$ ? (b) Will both parts have the same angular momentum? If not, which one will have more?

Alex Garger

Numerade Educator

Problem 116

Many 2.5-in-diameter (6.35-cm) computer hard disks spin at a constant 7200 rpm operating speed. The disks have a mass of about $7.50 \mathrm{~g}$ and are essentially uniform throughout with a very small hole at the center. If they reach their operating speed $2.50 \mathrm{~s}$ after being turned on, what average torque does the disk drive supply to the disk during the acceleration? Example $8-12$

Alex Garger

Numerade Educator

Problem 117

At the 1984 Olympics, the great diver Greg Louganis won one of his 10 gold medals for the reverse $312^{3 \frac{1}{2}}$ somersault tuck dive. In the dive, Louganis began his $312^{3} \frac{1}{2}$ turns with his body tucked in at a maximum height of approximately $2.0 \mathrm{~m}$ above the platform, which itself was $10.0 \mathrm{~m}$ above the water. He spun uniformly $312^{3} \frac{1}{2}$ times and straightened out his body just as he reached the water. A reasonable approximation is to model the diver as a thin uniform rod $2.0 \mathrm{~m}$ long when he is stretched out and as a uniform solid cylinder of diameter $0.75 \mathrm{~m}$ when he is tucked in. (a) What was Louganis's average angular speed as he fell toward the water with his body tucked in? Hint: How long did it take him to reach the water from his highest point? (b) What was his angular speed just after he stretched out? (c) How much did Louganis's rotational kinetic energy change while extending his body if his mass was 75 kg? Example $8-15$

Alex Garger

Numerade Educator

Problem 118

The bones of the forearm (radius and ulna) are hinged to the humerus at the elbow (Figure 8-65). The biceps muscle connects to the bones of the forearm about $2 \mathrm{~cm}$ beyond the joint. Assume the forearm has a mass of $2 \mathrm{~kg}$ and a length of $0.4 \mathrm{~m}$. When the humerus and the biceps are nearly vertical and the forearm is horizontal, if a person wishes to hold an object of mass $M$ so that her forearm remains motionless, what is the relationship between the force exerted by the biceps muscle and the mass of the object? Example 8-17

Alex Garger

Numerade Educator

Problem 119

The femur of a human leg (mass $10 \mathrm{~kg}$, length $0.9 \mathrm{~m}$ ) is in traction (Figure 8-66). The center of gravity of the leg is one-third of the distance from the pelvis to the bottom of the foot. Two objects, with masses $m_{1}$ and $m_{2}$, are hung at the ends of the leg using pulleys to provide upward support. A third object of $8 \mathrm{~kg}$ is hung to provide tension along the leg. The body provides tension as well. (a) What is the mathematical relationship between $m_{1}$ and $m_{2}$ ? Is this relationship unique in the sense that there is only one combination of $m_{1}$ and $m_{2}$ that maintains the leg in static equilibrium? (b) How does the relationship change if the tension force due to $m_{1}$ is applied at the leg's center of mass? Example $8-17$

Hubert Agamasu

Numerade Educator

Problem 120

It is estimated that 60,000 tons of meteors and other space debris accumulate on Earth each year. Assume the debris is accumulated uniformly across the surface of Earth. (a) How much does Earth's rotation rate change per year as a result of this accumulation? (That is, find the change in angular velocity.) (b) How long would it take the accumulation of debris to change the rotation period by 1 s? SSM Example $8-15$

Manish Jain

Numerade Educator

Problem 121

Suppose we decided to use the rotation of Earth as a source of energy. (a) What is the maximum amount of energy we could obtain from this source? (b) By the year 2025 the projected rate at which the world uses energy is expected to be $6.6 \times 1020 \mathrm{~J} / \mathrm{y}^{6.6} \times 10^{20} \mathrm{~J} / \mathrm{y}$. If energy use continues at that rate, for how many years would the spin of Earth supply our energy needs? Does this seem long enough to justify the effort and expense involved? (c) How long would it take before our day was extended to $48 \mathrm{~h}$ instead of 24 h? Assume that Earth is uniform throughout. Example $8-2$

Manish Jain

Numerade Educator

Problem 122

In a little over 5 billion years, our Sun will collapse to a white dwarf approximately $16,000 \mathrm{~km}$ in diameter. (Ignore the fact that the Sun will lose mass as it ages.) (a) What will our Sun's angular momentum and rotation rate be as a white dwarf? (Express your answers as multiples of its present-day values.) (b) Compared to its present value, will the Sun's rotational kinetic energy increase, decrease, or stay the same when it becomes a white dwarf? If it does change, by what factor will it change? The radius of the Sun is presently $6.96 \times 108 \mathrm{~m} 6.96 \times 10^{8} \mathrm{~m}$. Example $8-15$

Manish Jain

Numerade Educator

Problem 123

(a) If all the people in the world ( 7 billion) lined up along the equator, would Earth's rotation rate increase or decrease? Justify your answer. (b) How would the rotation rate change if all people were no longer on Earth? Assume the average mass of a human is $70.0 \mathrm{~kg}$. Example $8-15$

Manish Jain

Numerade Educator

Problem 124

A $1.00 \times 103-\mathrm{kg} 1.00 \times 10^{3}$ -kg merry-go-round (a flat, solid cylinder) supports 10 children, each with a mass of $50.0 \mathrm{~kg}$, located at the axis of rotation (thus you may assume the children have no angular momentum at that location). Describe a plan to move the children such that the angular velocity of the merry-go-round decreases to one-half its initial value.

Alex Garger

Numerade Educator

Problem 125

One way for pilots to train for the physical demands of flying at high speeds is with a device called the "human centrifuge." It involves having the pilots travel in circles at high speeds so that they can experience forces greater than their own weight. The diameter of the NASA device is $17.8 \mathrm{~m}$.
(a) Suppose a pilot starts at rest and accelerates at a constant rate so that he undergoes 30 rev in 2 min. What is his angular acceleration (in $\left.\mathrm{rad} / \mathrm{s}^{2}\right)$ ? (b) What is his angular velocity (in $\mathrm{rad} / \mathrm{s}$ ) at the end of that time? (c) After the 2min period, the centrifuge moves at a constant speed. The $g$ -force experienced is the centripetal force keeping the pilot moving along a circular path. What is the $g$ -force experienced by the pilot? (1 $\mathrm{g}=\operatorname{mass} \times 9.80 \mathrm{~m} / \mathrm{s} 2$ $1 g=$ mass $\times 9.80 \mathrm{~m} / \mathrm{s}^{2}$ ) (d) The pilot can tolerate $12 \mathrm{~g}$ 's in the horizontal direction. How long would it take the centrifuge to reach that state if it starts at the angular speed found in part (c) and accelerates at the rate found in part
(a)?

Alex Garger

Numerade Educator

Problem 126

A flywheel of mass $35.0 \mathrm{~kg}$ and diameter $60.0 \mathrm{~cm}$ spins at 400 rpm when it experiences a sudden power loss. The flywheel slows due to friction in its bearings during the $20.0 \mathrm{~s}$ the power is off. If the flywheel makes 200 complete revolutions during the power failure, (a) at what rate is the flywheel spinning when the power comes back on? (b) How long would it have taken for the flywheel to come to a complete stop?

Hubert Agamasu

Numerade Educator

Problem 127

A 620 -g basketball rolls without slipping across a level floor at a speed of $1.0 \mathrm{~m} / \mathrm{s}$. At what speed would a 40.0-g glass marble travel, as it rolls without slipping, if it has the same kinetic energy as the basketball? Model the basketball as a thin spherical shell and the marble as a uniform solid sphere. Example 8-10

Alex Garger

Numerade Educator