Essential College Physics

Andrew F. Rex, Richard Wolfson

Chapter 8

Rotational Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

While standing on the rotating Earth, is your centripetal acceleration greater on the equator or at latitude $45^{\circ}$ north?

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

A bicycle wheel rotates with increasing angular velocity. Compare the tangential acceleration of a point on the rim with that of a point midway along one spoke.

Prabhu Ramji

Numerade Educator

Problem 3

Why does a compact disk turn fastest when information is being read near its inner edge?

Prabhu Ramji

Numerade Educator

Problem 4

Explain why the rotational inertia of a hollow ball is greater than that of a uniform solid ball with the same mass and radius.

Prabhu Ramji

Numerade Educator

Problem 5

Explain why the rotational inertia of a hollow ball is greater than that of a uniform solid ball with the same mass and radius.

Prabhu Ramji

Numerade Educator

Problem 6

Is a baseball bat's rotational inertia greater when rotated about an axis perpendicular to one end or when rotated about its axis of symmetry?

Prabhu Ramji

Numerade Educator

Problem 7

A wheel rolls without slipping, with center-of-mass speed $v_{\mathrm{cm}}$. What's the instantaneous velocity of the bottom of the wheel (in contact with the ground)? What's the instantaneous velocity of the top of the wheel?

Prabhu Ramji

Numerade Educator

Problem 8

Give an example of an object that's in translational equilibrium but not rotational equilibrium.

Prabhu Ramji

Numerade Educator

Problem 9

Earth's core is denser than the near-surface layers. Should its rotational inertia be more or less than $\frac{2}{5} M R^{2}$ ? Explain.

Averell Hause

Carnegie Mellon University

Problem 10

A figure skater spins with arms extended horizontally. Explain what happens, and why, when she brings her arms tight to her body.

Averell Hause

Carnegie Mellon University

Problem 11

A spinning top rotates with its point on the floor and its rotation axis slightly tilted. Describe its subsequent motion. Why doesn't it fall over?

Averell Hause

Carnegie Mellon University

Problem 12

Why is it easier to balance a basketball on your fingertip if it's spinning?

Averell Hause

Carnegie Mellon University

Problem 13

A wheel of mass $M$ and radius $R$ has rotational inertia $I=\frac{9}{10} M R^{2}$ Is it more like a solid disk, or more like a bicycle wheel with most of the mass at the rim?

Narayan Hari

Numerade Educator

Problem 14

A common demonstration involves two unopened soup cans with the same dimensions and mass. One soup is thick (e.g., cream of mushroom), the other a watery liquid (e.g., chicken broth). The two are released simultaneously from rest on an incline. Explain why the chicken broth wins the race to the bottom.

Averell Hause

Carnegie Mellon University

Problem 15

Earth makes one rotation in 24 hours. What's its angular velocity?
(a) $1.16 \times 10^{-5} \mathrm{rad} / \mathrm{s}$
(b) $0.042 \mathrm{rad} / \mathrm{s}$
(c) $1.39 \times 10^{-5} \mathrm{rad} / \mathrm{s}$
(d) $7.27 \times 10^{-5} \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 16

To stop a wheel rotating at $6.0 \mathrm{rev} / \mathrm{s}$ in $9.0 \mathrm{~s}$ requires average angular acceleration (a) $-0.67 \mathrm{rad} / \mathrm{s}^{2}$; (b) $-2.1 \mathrm{rad} / \mathrm{s}^{2}$ (c) $-4.2 \mathrm{rad} / \mathrm{s}^{2} ;$ (d) $-67 \mathrm{rad} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 17

A potter's wheel starting with angular velocity $2.4 \mathrm{rad} / \mathrm{s}$ accelerates in $2.0 \mathrm{~s}$ to $3.6 \mathrm{rad} / \mathrm{s},$ with constant angular acceleration. During this time the wheel turns through an angle of
(a) 6.0 rad;
(b) $12.0 \mathrm{rad} ;$
(c) $0.95 \mathrm{rad}$
(d) $4.8 \mathrm{rad}$.

Narayan Hari

Numerade Educator

Problem 18

A compact disk with radius $6.0 \mathrm{~cm}$ takes 2.5 s to accelerate from rest to $40 \mathrm{rad} / \mathrm{s}$. What's the tangential acceleration of a point on the CD's edge? (a) $0.49 \mathrm{~m} / \mathrm{s}^{2}$; (b) $0.96 \mathrm{~m} / \mathrm{s}^{2}$; (c) $0.22 \mathrm{~m} / \mathrm{s}^{2}$; (d) $6.0 \mathrm{~m} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 19

A wheel rotates with constant angular acceleration. Which of the following is constant? (a) Angular velocity; (b) tangential velocity; (c) tangential acceleration; (d) centripetal acceleration.

Narayan Hari

Numerade Educator

Problem 20

A disk with radius $1.5 \mathrm{~m}$ and rotational inertia $34 \mathrm{~kg} \cdot \mathrm{m}^{2}$ has a 160 -N force applied tangentially to its rim. If the disk starts from rest, the angular velocity at the end of $2.0 \mathrm{~s}$ is (a) $1.22 \mathrm{rad} / \mathrm{s}$;(b) $6.27 \mathrm{rad} / \mathrm{s}$(c) $7.06 \mathrm{rad} / \mathrm{s}$(d) $14.1 \mathrm{rad} / \mathrm{s}$ (e) $25.5 \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 21

A $46-\mathrm{g}$ golf ball has radius $2.13 \mathrm{~cm}$. Assuming uniform density, what's the ball's rotational inertia?
(a) $8.3 \times 10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(b) $2.1 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(c) $1.3 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2}$
(d) $4.3 \times$ $10^{-6} \mathrm{~kg} \cdot \mathrm{m}^{2}$

Narayan Hari

Numerade Educator

Problem 22

What's the rotational kinetic energy of the golf ball in the preceding question when it's rotating at $80 \mathrm{~Hz}$ ?
(a) $0.5 \mathrm{~J}$;
(b) $1.0 \mathrm{~J}$
(c) $2.0 \mathrm{~J} ;$ (d) $4.0 \mathrm{~J} .$

Narayan Hari

Numerade Educator

Problem 23

A bicycle with wheels $69 \mathrm{~cm}$ in diameter is moving at $40 \mathrm{~km} / \mathrm{h}$ If the wheels roll without slipping, their angular velocity is (a) $4 \mathrm{rad} / \mathrm{s}$ (b) $9 \mathrm{rad} / \mathrm{s} ;$ (c) $16 \mathrm{rad} / \mathrm{s}$ (d) $32 \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 24

A solid cylinder has mass $0.55 \mathrm{~kg}$ and radius $3.5 \mathrm{~cm}$. It's rolling with center-of-mass speed $0.75 \mathrm{~m} / \mathrm{s}$. What's its total kinetic energy? (a) $0.07 \mathrm{~J}$(b) $0.16 \mathrm{~J} ;$ (c) $0.23 \mathrm{~J}$(d) $0.30 \mathrm{~J}$.

Narayan Hari

Numerade Educator

Problem 25

A heavy machine wheel has a rotational inertia $25 \mathrm{~kg} \cdot \mathrm{m}^{2}$ and radius $0.75 \mathrm{~m}$. It's initially at rest, and a tangential force of $35 \mathrm{~N}$ is applied at its edge for $5.0 \mathrm{~s}$. What's the resulting angular velocity?
(a) $0.86 \mathrm{rad} / \mathrm{s}$
(b) $1.7 \mathrm{rad} / \mathrm{s}$
(c) $5.3 \mathrm{rad} / \mathrm{s}$
(d) $10.6 \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 26

The Moon (mass $7.35 \times 10^{22} \mathrm{~kg}$ ) takes 27.3 days to complete an essentially circular orbit of radius $3.84 \times 10^{8} \mathrm{~m} .$ Estimate the magnitude of the Moon's orbital angular momentum.
(a) $7.33 \times 10^{25} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$
(b) $2.81 \times 10^{34} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$
(c) $7.33 \times$ $10^{41} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s} ;$ (d) $7.33 \times 10^{51} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$

Narayan Hari

Numerade Educator

Problem 27

In a car's drive train, a flywheel with rotational inertia $26.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$ rotates at $310 \mathrm{rad} / \mathrm{s}$. The clutch engages, pressing the clutch plate $-$ a disk with rotational inertia half that of the flywheel $-$ against the flywheel, so the two rotate as one. Assuming both are otherwise free from torque, what's the rotational speed of the combined system? (a) $155 \mathrm{rad} / \mathrm{s} ;$ (b) $206 \mathrm{rad} / \mathrm{s} ;$ (c) $267 \mathrm{rad} / \mathrm{s}$ (d) $310 \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 28

A rotating wheel may have (a) both centripetal and tangential acceleration; (b) neither centripetal nor tangential acceleration; (c) angular but not centripetal acceleration; (d) neither angular nor centripetal acceleration.

Narayan Hari

Numerade Educator

Problem 29

A wheel has its rotation axis vertical, and turns counterclockwise as viewed from above. The direction of the wheel's angular momentum is (a) straight up; (b) straight down; (c) tangent to the wheel, in the direction of rotation; (d) tangent to the wheel, opposite the direction of rotation.

Averell Hause

Carnegie Mellon University

Problem 30

A bicycle wheel with radius $0.79 \mathrm{~m}$ is in pure rotation (not rolling) at $4.0 \mathrm{rev} / \mathrm{s}$. What distance does a point on the wheel's rim travel in 1 minute?

Narayan Hari

Numerade Educator

Problem 31

Jupiter has radius $7.14 \times 10^{7} \mathrm{~m}$ and makes one rotation every 9 hours, 50 minutes. How far does a point on Jupiter's equator travel each second, due to the planet's rotation?

Narayan Hari

Numerade Educator

Problem 32

A hydroelectric turbine makes 24,500 revolutions in one day. What's its angular velocity?

Narayan Hari

Numerade Educator

Problem 33

Bacterium rotation rate. A typical lab centrifuge spins at 3000 rpm. How's that compare with the $E$. coli bacterium's flagellum described in Section $8.1 ?$

Narayan Hari

Numerade Educator

Problem 34

A compact disk spins with angular velocity $43.8 \mathrm{rad} / \mathrm{s}$. The player is turned off, and 2.45 s later the CD has stopped. Find its average angular acceleration.

Narayan Hari

Numerade Educator

Problem 35

A lawnmower blade accelerates at $98 \mathrm{rad} / \mathrm{s}^{2}$. Starting from rest, what's its angular velocity after 2.5 s have elapsed? Answer in $\mathrm{rad} / \mathrm{s}$ and $\mathrm{rpm}$

Narayan Hari

Numerade Educator

Problem 36

A diver jumps from a 10 -m-high tower, and hopes to complete $3 \frac{1}{2}$ somersaults. What should be his rotation rate?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 37

Tides dissipate energy, slowing Earth's rotation. Some 4 billion years ago, Earth's rotation period is estimated to have been 14 hours. Find Earth's average angular acceleration over this 4 -billion-year period.

Prabhu Ramji

Numerade Educator

Problem 38

A skater spins at 2.50 revolutions per second. She slows with constant angular acceleration, stopping after $1.75 \mathrm{~s}$. (a) What's her angular acceleration? (b) Through how many revolutions did she turn before stopping?

Narayan Hari

Numerade Educator

Problem 39

A dentist's drill accelerates from rest at $615 \mathrm{rad} / \mathrm{s}^{2}$ for $2.10 \mathrm{~s}$ and then runs at constant angular velocity for $7.50 \mathrm{~s}$. Through how many total revolutions has the drill turned?

Narayan Hari

Numerade Educator

Problem 40

Your car's fan belt turns a pulley at $3.40 \mathrm{rev} / \mathrm{s}$. When you step on the gas for $1.30 \mathrm{~s},$ the rate increases steadily to $5.50 \mathrm{rev} / \mathrm{s}$.
(a) What's the pulley's angular acceleration?
(b) Through what angle did the pulley turn while accelerating?

Narayan Hari

Numerade Educator

Problem 41

A centrifuge rotating initially at 9000 rpm slows to 5000 rpm with constant angular acceleration over $3.50 \mathrm{~s}$. (a) What's its angular acceleration? (b) Through how many revolutions does it turn while decelerating? (c) Through what distance does a point on the edge of the centrifuge, at radius of $9.40 \mathrm{~cm},$ turn during this time?

Prabhu Ramji

Numerade Educator

Problem 42

A machine shop grinding wheel accelerates from rest with a constant angular acceleration of $2.3 \mathrm{rad} / \mathrm{s}^{2}$ for $7.5 \mathrm{~s}$ and is then brought to rest with a constant angular acceleration of $-4.2 \mathrm{rad} / \mathrm{s}^{2} .$ Find the total time elapsed and the total number of revolutions turned.

Prabhu Ramji

Numerade Educator

Problem 43

A machine shop grinding wheel accelerates from rest with a length of the day increasing by about $2.3 \mathrm{~ms} /$ century. Find Earth's angular acceleration.

Averell Hause

Carnegie Mellon University

Problem 44

A 1.75 -m-diameter wagon wheel makes one revolution in $3.20 \mathrm{~s}$. What's the tangential speed of a point in its rim?

Narayan Hari

Numerade Educator

Problem 45

A tornado has wind speed $325 \mathrm{~km} / \mathrm{h}$ at a rotation radius of $18 \mathrm{~m}$. What is the angular velocity at this point in the tornado?

Narayan Hari

Numerade Educator

Problem 46

A lab centrifuge with radius $11 \mathrm{~cm}$ turns at $75 \mathrm{rev} / \mathrm{s}$. What should be its angular acceleration for a point at the $11-\mathrm{cm}$ radius to have tangential acceleration that is $1 \%$ of its centripetal acceleration?

Narayan Hari

Numerade Educator

Problem 47

A string is wrapped around a pulley of radius $3.50 \mathrm{~cm}$, and a weight hangs from the other end. The weight falls with a constant acceleration $3.40 \mathrm{~m} / \mathrm{s}^{2}$. (a) What's the angular acceleration of the pulley? (b) If the weight starts from rest $1.30 \mathrm{~m}$ above the floor, what's the pulley's angular velocity when the weight hits the floor?

Narayan Hari

Numerade Educator

Problem 48

A cylindrical space station with diameter $150 \mathrm{~m}$ simulates gravity by rotating about its central axis. (a) If an astronaut on the outer edge is to experience a centripetal acceleration $g / 2,$ what should be the station's angular velocity? (b) What tangential acceleration is required to bring the station to that rate, starting from rest, with a constant acceleration for 60 days?

Narayan Hari

Numerade Educator

Problem 49

To simulate the extreme accelerations during launch, astronauts train in a large centrifuge with diameter $10.5 \mathrm{~m}$. (a) If the centrifuge is spinning so the astronaut on the end of one arm is subjected to a centripetal acceleration of $5.5 g,$ what is the astronaut's tangential velocity at that point? (b) Find the angular acceleration needed to reach the velocity in part (a) after $25 \mathrm{~s}$.

Prabhu Ramji

Numerade Educator

Problem 50

Early DVD burners operated in the same constant-tangential speed mode as described for CDs in Example $8.5 .$ For a $6 \times$ burn (i.e., six times the normal playback data rate), the rotational speed was highest $(8400 \mathrm{rpm})$ at the innermost data track, $2.6 \mathrm{~cm}$ from the rotation axis. (a) What's the corresponding rotation rate near the outer edge, at $5.7 \mathrm{~cm}$ from the axis? (b) If the burn takes $9.0 \mathrm{~min},$ what's the DVD's average angular acceleration? (Newer DVD burners use so-called zoned linear velocities, or even constant angular velocity, to avoid very high rotation rates.)

Averell Hause

Carnegie Mellon University

Problem 51

Information on a DVD (see preceding problem) is stored on a continuous spiral track in the region from $2.6-\mathrm{cm}$ to $5.7-\mathrm{cm}$ radius. Individual turns of the spiral are $0.74 \mu \mathrm{m}$ apart. (a) Find the length of the entire track. (b) If one byte of information has average length $2.3 \mu \mathrm{m}$, how many bytes are on the DVD?

Averell Hause

Carnegie Mellon University

Problem 52

For the situation described in Example $8.6,$ use the tangential and centripetal components to find the magnitude and direction of the acceleration vector.

Narayan Hari

Numerade Educator

Problem 53

An eagle with a 2.1 -m wingspan flaps its wings back and forth 20 times per minute, each stroke extending from $45^{\circ}$ above the horizontal to $45^{\circ}$ below. Downward and upward strokes take the same amount of time. On a given downstroke, what's (a) the average angular velocity of the wing and (b) the average tangential velocity of the wingtip?

Averell Hause

Carnegie Mellon University

Problem 54

Using astronomical data from Appendix E, compute Earth's rotational inertia, assuming the planet is a uniform solid ball.

Narayan Hari

Numerade Educator

Problem 55

Use your answer to the preceding problem to find Earth's rotational kinetic energy.

Averell Hause

Carnegie Mellon University

Problem 56

Find the kinetic energy of Earth's orbital motion around the Sun (see Appendix E). Compare your answer with the rotational kinetic energy found in the preceding problem.

Averell Hause

Carnegie Mellon University

Problem 57

The circular blade of a power saw has kinetic energy $44 \mathrm{~J}$. If its rotation rate drops to half, what's its new kinetic energy?

Narayan Hari

Numerade Educator

Problem 58

A revolving door consists of four rectangular glass slabs, with the long end of each attached to a pole that acts as the rotation axis. Each slab is $2.20 \mathrm{~m}$ tall by $1.25 \mathrm{~m}$ wide and has mass $35.0 \mathrm{~kg} .$
(a) Find the rotational inertia of the entire door. (b) If it's rotating at one revolution every $9.0 \mathrm{~s}$, what's the door's kinetic energy?

Prabhu Ramji

Numerade Educator

Problem 59

A $145-\mathrm{g}$ baseball has radius $3.7 \mathrm{~cm}$. (a) Assuming uniform density, what's its rotational inertia? (b) The ball is pitched at $22 \mathrm{~m} / \mathrm{s}$ with spin rate $20 \mathrm{~Hz}$. Find and compare the ball's translational and rotational kinetic energies.

Prabhu Ramji

Numerade Educator

Problem 60

A wheel consists of 20 thin spokes, each with mass $0.055 \mathrm{~kg}$, attached to a rim of mass $4.2 \mathrm{~kg}$ and radius $0.75 \mathrm{~m}$. Find its rotational inertia.

Narayan Hari

Numerade Educator

Problem 61

A bicycle has 69-cm-diameter wheels. If they roll without slipping when the bicycle is traveling at $50 \mathrm{~km} / \mathrm{h}$, what's the wheels' angular velocity?

Narayan Hari

Numerade Educator

Problem 62

A bicycle has $63.5-\mathrm{cm}$ -diameter wheels turning at $11.4 \mathrm{rev} / \mathrm{s}$. How fast is the bicycle moving?

Narayan Hari

Numerade Educator

Problem 63

The condition for a wheel to roll without slipping is described in the text by $\omega=v_{\mathrm{cm}} / R .$ Describe what's happening with car tires in the cases (a) $\omega>v_{\mathrm{cm}} / r$ and $(\mathrm{b}) \omega<v_{\mathrm{cm}} / r$

Averell Hause

Carnegie Mellon University

Problem 64

Based on the preceding problem, consider a $32.5-\mathrm{cm}$ -radius automobile tire on a car moving at $10.4 \mathrm{~m} / \mathrm{s}$. Describe the motion of the bottom of the tire relative to the road for each of the following angular velocities:
(a) $\omega=25.0 \mathrm{rad} / \mathrm{s}$
(b) $\omega=32.0 \mathrm{rad} / \mathrm{s}$
(c) $\omega=38.7 \mathrm{rad} / \mathrm{s}$

Averell Hause

Carnegie Mellon University

Problem 65

A drag racer has 76.2 -cm-diameter wheels. Haw fast are they turning when the drag racer is doing $140 \mathrm{~km} / \mathrm{h} ?$

Narayan Hari

Numerade Educator

Problem 66

A solid bowling ball with mass $7.2 \mathrm{~kg}$ and diameter $22 \mathrm{~cm}$ rolls without slipping at $6.5 \mathrm{~m} / \mathrm{s} .$ Find its translational kinetic energy, rotational kinetic energy, and total kinetic energy.

Prabhu Ramji

Numerade Educator

Problem 67

A solid cylinder is released from rest on an incline. When it reaches the bottom, what fraction of its total kinetic energy is translational and what fraction is rotational?

Prabhu Ramji

Numerade Educator

Problem 68

Repeat the preceding problem for a solid sphere. Why the difference?

Prabhu Ramji

Numerade Educator

Problem 69

A meterstick pivots freely from one end. If it's released from a horizontal position, find its angular velocity when it passes through the vertical. Treat the stick as a uniform thin rod.

Prabhu Ramji

Numerade Educator

Problem 70

A 1.00 -m-long ramp is inclined at $15^{\circ}$ to the horizontal. A solid ball is released from rest at the top of the ramp. Find
(a) the ball's speed at the bottom of the ramp and (b) its translational acceleration.

Prabhu Ramji

Numerade Educator

Problem 71

Using the work-energy analysis in the text, along with kinematic equations for one-dimensional motion, find the translational acceleration of a solid ball rolling down a ramp inclined at angle $\theta$ expressed in terms of $\theta$ and $g .$ Compare with $a=g \sin \theta,$ the acceleration of an object sliding without friction down the same ramp.

Prabhu Ramji

Numerade Educator

Problem 72

A solid ball is released from rest at the top of a 1.50 -m-long ramp inclined at $10^{\circ} .$ At the bottom, the ball continues along a flat section that's also $1.50 \mathrm{~m}$ long. What's the overall travel time?

Prabhu Ramji

Numerade Educator

Problem 73

Consider the "great rolling race" from Example $8.11 .$ Suppose the ball travels $1.00 \mathrm{~m}$ along the ramp from top to bottom.
(a) How far does the cylinder travel in the same amount of time?
(b) Does the answer to part (a) depend on the inclination angle? Explain.

Prabhu Ramji

Numerade Educator

Problem 74

(a) If you push the outer edge of a 1.05 -m-wide door with a 23. 0-N tangential force, what torque results?
(b) What's the torque if you apply the same magnitude of force, in the same place, but at a $45^{\circ}$ angle?

Narayan Hari

Numerade Educator

Problem 75

An auto mechanic applies a $65-\mathrm{N}$ force near the end of a 35 -cm-long wrench. What's the maximum torque?

Narayan Hari

Numerade Educator

Problem 76

An electric trimmer blade has rotational inertia $1.70 \times$ $10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}$ (a) What torque is needed to accelerate this blade from rest to 640 rpm in $1.50 \mathrm{~s}$ ? (b) How much work was done to accelerate the blade?

Narayan Hari

Numerade Educator

Problem 77

The cellular motor driving the flagellum in the $E$. coli bacterium exerts a torque of typically $400 \mathrm{pN} \cdot \mathrm{nm}$ on the flagellum (see discussion in Section 8.1 ). If this torque results from a force applied tangentially to the outside of the $12-\mathrm{nm}-$ radius flagellum, what is the magnitude of that force?

Narayan Hari

Numerade Educator

Problem 78

A meterstick pivots freely from one end. If it's released from a horizontal position and rotates due to gravity, find (a) its angular acceleration just after it's released and (b) the stick's angular acceleration as a function of the angle $\theta$ it makes with the vertical. Treat the stick as a uniform thin rod.

Prabhu Ramji

Numerade Educator

Problem 79

A meterstick pivots freely from one end. If it's released from ends of a 3.4 -m-long seesaw with mass $25 \mathrm{~kg}$, with the fulcrum at its midpoint. With the seesaw horizontal, find (a) the net torque on the seesaw and (b) its angular acceleration.

Prabhu Ramji

Numerade Educator

Problem 80

A meterstick is initially balanced on a fulcrum at its midpoint. You have four identical masses. Three of them are placed atop the meterstick at the following locations: $25 \mathrm{~cm}, 45 \mathrm{~cm},$ and $95 \mathrm{~cm} .$ Where should the fourth mass be placed in order to balance the meterstick?

Prabhu Ramji

Numerade Educator

Problem 81

A meterstick of negligible mass has a $0.20-\mathrm{kg}$ mass at its $35 \mathrm{~cm}$ mark and a $0.40-\mathrm{kg}$ mass at the $75-\mathrm{cm}$ mark. Where should the fulcrum be so the meterstick is balanced?

Prabhu Ramji

Numerade Educator

Problem 82

Repeat the preceding problem if the meterstick has mass $0.15 \mathrm{~kg}$

Prabhu Ramji

Numerade Educator

Problem 83

Consider again the ladder in Example $8.15 .$ Compute the normal force again, assuming a $75-\mathrm{kg}$ man is standing on the ladder (a) at its midpoint and (b) four-fifths of the way up the ladder.

Prabhu Ramji

Numerade Educator

Problem 84

A 15 -m-long ladder is mounted on a fire truck. The ladder itself has mass $125 \mathrm{~kg}$, and at the top is a $35-\mathrm{kg}$ basket holding a 91 -kg firefighter. If the ladder makes a $60^{\circ}$ angle with the horizontal, what's the net torque about the ladder's base?

Prabhu Ramji

Numerade Educator

Problem 85

In Figure $\mathrm{P} 8.85,$ the meterstick's mass is $0.160 \mathrm{~kg}$ and the string tension is $2.50 \mathrm{~N}$. The system is in equilibrium. Find
(a) the unknown mass $m$ and
(b) the upward force the fulcrum exerts on the stick.

Prabhu Ramji

Numerade Educator

Problem 86

Use data from Appendix $\mathrm{E}$ to compute Earth's angular momentum due to its rotation alone.

Narayan Hari

Numerade Educator

Problem 87

Use data from Appendix $\mathrm{E}$ to compute Earth's angular momentum due to its orbital motion, and compare with your answer to the preceding problem.

Narayan Hari

Numerade Educator

Problem 88

A student sitting on a frictionless rotating stool has rotational inertia $0.95 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about a vertical axis through her center of mass when her arms are tight to her chest. The stool rotates at $6.80 \mathrm{rad} / \mathrm{s}$ and has negligible mass. The student extends her arms until her hands, each holding a $5.0-\mathrm{kg}$ mass, are $0.75 \mathrm{~m}$ from the rotation axis. (a) Ignoring her arm mass, what's her new rotational velocity? (b) Repeat if each arm is modeled as a 0.75-m-long uniform rod of mass of $5.0 \mathrm{~kg}$ and her total body mass is $65 \mathrm{~kg}$.

Jose Carlos

Numerade Educator

Problem 89

A turntable with a rotational inertia $0.225 \mathrm{~kg} \cdot \mathrm{m}^{2}$ is rotating at $3.25 \mathrm{rad} / \mathrm{s} .$ Suddenly, a disk with rotational inertia $0.104 \mathrm{~kg} \cdot \mathrm{m}^{2}$ is dropped onto the turntable with its center on the rotation axis. Assuming no outside forces act, what's the common rotational velocity of the turntable and disk?

Narayan Hari

Numerade Educator

Problem 90

A grinding wheel has rotational inertia $0.355 \mathrm{~kg} \cdot \mathrm{m}^{2}$. (a) Find the constant torque needed to bring it from rest to $43.0 \mathrm{rad} / \mathrm{s} 1 \mathrm{n}$ $3.50 \mathrm{~s}$. (b) Using your answer to part (a), find the wheel's angular momentum change, and show that your answer agrees with the angular momentum computed using $L=I \omega$.

Narayan Hari

Numerade Educator

Problem 91

A merry-go-round with rotational inertia $35 \mathrm{~kg} \cdot \mathrm{m}^{2}$ rotates clockwise at $1.3 \mathrm{rad} / \mathrm{s}$. Find the magnitude and direction of (a) the merry-go-round's angular momentum and (b) the torque needed to stop the merry-go-round in $10 \mathrm{~s}$.

Prabhu Ramji

Numerade Educator

Problem 92

A car drives straight north. What's the direction of its wheels' angular momentum?

Prabhu Ramji

Numerade Educator

Problem 93

A wrench handle points straight up, in the $+y$ -direction. A mechanic applies a force in the $+x$ -direction to the top end of the wrench. What's the direction of the torque on the wrench?

Prabhu Ramji

Numerade Educator

Problem 94

In a simple model of the hydrogen atom, an electron orbits a proton at $2.18 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in a circle of radius $5.29 \times 10^{-11} .$ Find the magnitude and direction of the electron's angular momentum.

Narayan Hari

Numerade Educator

Problem 95

Earth's rotation rate is slowing. What's the direction of the torque needed to cause this? Use data from Problem 43 to estimate the torque's magnitude.

Averell Hause

Carnegie Mellon University

Problem 96

An $83.2-\mathrm{kg}$ propeller blade measures $2.24 \mathrm{~m}$ end to end. Model the blade as a thin rod rotating about its center of mass. It's initially turning at 175 rpm. Find (a) the blade's angular momentum, (b) the tangential speed at the blade tip, and (c) the angular acceleration and torque required to stop the blade in $12.0 \mathrm{~s}$

Prabhu Ramji

Numerade Educator

Problem 97

A string is wrapped around the outer rim of a cylindrical disk with mass $500 \mathrm{~g}$ and radius $12.5 \mathrm{~cm} .$ A student pulls on the string, applying a 23.5-N force tangentially to the cylinder's rim. Find (a) the torque on the cylinder and (b) its angular acceleration, assuming no frictional torque.

Narayan Hari

Numerade Educator

Problem 98

A shot putter holds the $7.26-\mathrm{kg}$ shot still with his arm extended straight, the shot $61.8 \mathrm{~cm}$ from his shoulder joint. Find the torque on the athlete's arm due to the shot if the arm (a) is horizontal, (b) makes a $45^{\circ}$ angle below the horizontal, and (c) is hanging straight down.

Prabhu Ramji

Numerade Educator

Problem 99

Recent advances using microprocessors have caused significant improvements in the safety of power tools such as circular saws. The saw's control unit places a small electric charge on the spinning blade, giving a constant 3 -volt signal to the microprocessor. If the blade contacts human skin, the capacitance changes. The microprocessor senses this change and stops the blade in just $5.0 \mathrm{~ms} .$ (Capacitance will be discussed in Chapter 16.) Assume constant angular acceleration.
(a) If the blade normally spins at 3500 rpm, through what angle does it turn while stopping?
(b) What's the change in the blade's kinetic energy, assuming it's essentially a uniform disk $19.0 \mathrm{~cm}$ in diameter with mass $0.860 \mathrm{~kg} ?$ (c) What torque is required to stop the blade?

Prabhu Ramji

Numerade Educator

Problem 100

Your car tire has radius $31.0 \mathrm{~cm}$. (a) If it rolls without slipping and has angular velocity $79.3 \mathrm{rad} / \mathrm{s}$, what's your car's speed? Now suppose your car has the speed found in part (a), but different angular velocity. Describe what's happening to the tire where it contacts the road when (b) $\omega=91.5 \mathrm{rad} / \mathrm{s}$ and $(\mathrm{c}) \omega=$ $52.0 \mathrm{rad} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 101

The following objects are released simultaneously from rest at the top of a $1.50-\mathrm{m}$ -long ramp inclined at $3.50^{\circ}$ to the horizontal: a solid sphere, a solid cylinder, a hollow cylindrical shell, and a hollow ball. (a) Which wins the race? (b) At the moment the winner reaches the bottom, find the positions of the other three objects.

Prabhu Ramji

Numerade Educator

Problem 102

A solid cylinder is released from rest and rolls without slipping down a $4.0^{\circ}$ incline. Two photogates are connected to a timer that measures the elapsed time for the ball to roll between them. If the first gate is $1.00 \mathrm{~m}$ from the starting point and the second is $0.20 \mathrm{~m}$ past the first, what will the timer read?

Prabhu Ramji

Numerade Educator

Problem 103

A baseball player extends his arm straight upward to catch a $0.145-\mathrm{kg}$ batted baseball moving horizontally at $42.5 \mathrm{~m} / \mathrm{s}$. It's $63.5 \mathrm{~cm}$ from the player's shoulder joint to the point where the ball strikes his hand, and his arm remains stiff while it rotates about the shoulder joint during the catch. The player's hand recoils horizontally a distance of $5.00 \mathrm{~cm}$ while he stops the ball with constant acceleration. What torque does the player's arm exert on the ball?

Prabhu Ramji

Numerade Educator

Problem 104

In Chapter 7 you studied the simple pendulum, in which the mass of the string holding the bob was neglected. Now consider the physical pendulum, with mass distributed throughout its length. An example is the meterstick in Figure GP8.104. (a) With gravity acting on the stick's center of mass, find the torque on the stick as a function of its mass $M,$ length $L,$ the angle $\theta$ and $g$. (b) Use $\tau=I \alpha$ to find the stick's angular acceleration as a function of the same variables. (c) By analogy with the simple pendulum, show that the period for small oscillations of this physical pendulum is approximately $T=2 \pi \sqrt{\frac{2 L}{3 g}}$.

Averell Hause

Carnegie Mellon University

Problem 105

(a) Use the results of the preceding problem to evaluate the period of small oscillations of a 1.00 -m long stick. (b) Find the length of a simple pendulum that would have the same period as you found in part (a). Explain why the length is shorter than $1.00 \mathrm{~m}$.

Averell Hause

Carnegie Mellon University

Problem 106

Consider your arm to be a uniform rod pivoted about one end. (a) Estimate the period of small oscillations if your 75 -cm-long arm hangs freely downward from your shoulder. (b) Should your leg (hanging freely from your hip) have a larger, a smaller, or the same period compared with the period you estimated in part (a)? Explain.

Averell Hause

Carnegie Mellon University

Problem 107

Consider a physical pendulum as described in the preceding problems, but not necessarily a uniform stick. Suppose this physical pendulum has one end fixed, with a rotational inertia $I$ about that end and center of mass a distance $d$ from that end. (a) Show that the period for small oscillations is given by
$$
T=2 \pi \sqrt{\frac{d}{g}} \sqrt{\frac{I}{M d^{2}}}
$$
(b) Show that this general result reduces to the correct one for the uniform stick and the simple pendulum.

Narayan Hari

Numerade Educator

Problem 108

If the polar ice caps melt, adding more liquid water to the oceans, Earth's rotational inertia could increase by as much as an estimated $0.3 \%$. Compute the effect such a change would have on the length of 1 day, assuming Earth's angular momentum remains constant.

Averell Hause

Carnegie Mellon University