A bicycle odometer (which8cul8 n/(u3muq21hz* ja ouh g measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?h z/ *u2u8ulaghc jn 1(mq8uo3

Suppose a disk rotates at constant angular velocity. Does a pointo xkm-iur-r*nbd q r;n7;(ru ; on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or n--rrr *;u n(idm;k7x r ou;qbtangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sing;ydxohjss-38c au/a0jzr 0;s- ; j hlcle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? (bcjlmqx1*aj ne, qw t t7c3-2Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up uq es+f bi;+ik2j8 2 q(wipv1nwith your hands behind your head than when your arms are stretched out in fro i82+ isnfq2 kv1p;w+q ueji(bnt of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at t/4u xfvjcyg*4he rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small frontyfx4/4v*c ujg sprocket or a large front sprocket? Why?

Mammals that depend 0p+7tpkrbg/p on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thi/rg +p7b p0pkts distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l qij0ar)( 62v ww(ygxfong, narrow beam?

If the net force on a system is zero, is the net torque also zero?yp76+8j ssrke dvo 80onpwr67 If the net torque on a system is zero, is the net fo7sp6o8nyk+v8 s0o 6r rwped7j rce zero?

Two inclines have the same height but make different a-sa u5mj mhr6efe:i +*ngles with the horizontal. The same steel ball is rolled down each incline. On *mmsr5-+u e hfai6ej:which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from.j* b8b;wkk/d-n h rtb rest) down an incline. One sphere has twice the radius and twice the mass of the ;tk8* w /bdkjrnb.bh-other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sa*16bzxn rd 1iqf x+9kih5zyp+me mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has thiz5i+91pdx1fx *q+kyn6 rzh be greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or r m acxox(ajv-kx 1)-;lotating objects eventually sxjx l(x; o-aam1-kcv) low down and stop. Explain.

If there were a great migratid3o. f-fx3ba-n t 75g i1jsgowon of people toward the Earth’s equator, how would this affect the length of7b wgs3t f3g.x5aoo1fjd in-- the day?

Can the diver of Fig. 8–29 do a somersault without having any initia**z, m9; -vc/ghey1 dc dnzdecl rotation when she ledeg zdcm9v* -;*czhn y1,/edcaves the board?

The moment of inertia vnq*wwo 4f/nio2gqq7)0c p-h1in;b jof a rotating solid disk about an axis through its center of mass iq2nqi v oo;n7q w/-ncjf)gh * 1pwb0i4s $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdfrlc mu83b8*hn, ez2sn4b *)av2 l hcving a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your ang8vb*hnv38,fm2 b uz arlhens)c4cl*2 ular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical av+5jm ysr o :6ea/;)5to asgnpnd have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which n6op+:vroj atge/ms;5s ay5) is which.

The angular velocity of a wheel rotating oopbs8- 5fio( ;qb8c sgn a horizontal axle points west. In wha8-;8bsqscf (gooi5bp t direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large/n))u-f-wy miwop0co freely rotating turntable. What happens if you walk toward the/y-n -mw0 )wf ooic)up center?

A shortstop may leap into the air to catch a ball and throw it (ul0vk.vr jq* m*he (e9y*ac dquickly. As he throwhcq.*9(l*ay0* (vrk edmu ejvs the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angur6swn1w2ervo ozy,:6, c y1 xzesd6o ok :18ljlar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller c:rk o61d1zyr,n6:oo8 oezjexsywv 6 , ls 2w1can operate to keep the helicopter stable.

  Express the following angles in radians: (a0 ejbmpfsu id.v4q 5)+) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calcula ca3,2mmnj 2za(e6chs4m) tvbte, using the informatijc4ehm6( m2t2,)ncvbza3 s amon inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380o+c9jdw l 0h0 s,tx9sz,000 km from Earth. The beam diverges at anwh d+0tj99z cs0,lsxo angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotat+r 7j1-,eup:vhh uzhc e at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angularu:hph,rczh u v-e 7+j1 acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level w7;g (:z hv9ueugpy*f floor 3.5 m to another child. If the ball makes 15.0 reey7;hg(zpvwu : 9g*ufvolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter y00( ocxs 3m1m7 bgjaltravels 8.0 km. How many revolutions do the wheels m1g3sb moxml 0 a7yj0c(ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calc8ynu7 n9ks -ynulate its angular velocity in n-yusny79 8kn $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one s,1 m6 bhs+fgdcomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fro6+bsm,h1dgf sm the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the E/sjljkhrvg +wr/agg 2/j) p1 o,z)b.parth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi+/uaz2j 1 optpnt
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0x :o7* mfjngu0 qo uj;r//7qvw cm from the axis of rotation is to experience an acce/uonjuwo7 m:;rxq0*/7 jv qgf leration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centeru2jm u. 2hcjmdbfvel.oq:j d)(b-0(z from 130 rpm to 280 rpm in 4.0 s. Dzm h- 2bq2juodc) ( jfvde(jl0mu.:.betermine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius .h ahv5/ xlu(u$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard th/x kscb w(e-a z*w9em+k+hw)re Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-- (9abmkr /es*he+w+xww)kzcmin time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly fro9ij uclvhjl- /1f0 rtua)0 .com rest to 15,000 rpm in 220 s. Through how manc.-90h cilol v/a 1)jjru 0utfy revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate oldf()t3b1fx
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlingph1 ah5vra,z7rda6f jy:5 o -rundt3 vx8n+(t “human ce-oaf+hrdnj85d35t7:avnx1 puy rztv r (ah,6 ntrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 24o0 owf x-jq;p00 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travelfo 0ow-0q;jp xed in this time?    m

  A cooling fan is turned ou8,ew4 g dyg,6o7fsmc f9px 0kff when it is running at 850rev/min It turns 1500 revolutions before it yf, kpwc m,fdog9u4760xs g8e comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fr9x0aerw /m *qvom 95km/h to 45km/h The tires have a diameter /w0vex9m* raq of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car makb; khunqv6k(9e 65 revolutions as the car reduces its speed uniformly from 95km/h 9(;uh6kb kvqn to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all p4ji *idj,pu53t4w8bszu6+phkk /ix her weight on each pedal when climbing a hill. The ubhpk 8pkix pzusw6j 4 *d+i,3 j5/ti4pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a4n,93v4o6ya rjfn)kv) p t6-nocbd ya door 74 cm wide. What is the magnitude of the torque if the force i yaa6,6onfbj)p 4dr o kntv4c9)yv-3 ns exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel8h,): quk j/tkuliul8brt h.3 shown in Fig. 8–39. Assume that a fric:lh)ku qt brku8j uih/t,l8 .3tion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to y -qa1-e6yxr+rvvib:the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the qa--vvr61x: ry iy+b emagnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torq2v sx.(dj+ 2 oyb1hrazk2tk, yue of 38 b.ak1,rkt2 zvd+jxo(yy 22h s $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg spheree3a 6k,bgrp w)wqtx 4i lmm ;.qwsui,5im5 l9/ of radius 0.648 m when the axis of rotation is th6 5t;smm uw wx/ieq9w5,g 3iaik,q )lpm.4blrrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whp)r t i9x b*:0dl)sfdbeel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ig0ltpf )) ixs*b bd:9rdnored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, lighnha02wd :a(dc 6qq ry0t rod is rotated in a horizontal circle of radius 1.2 m. Caly0a0wd qa6q (dh:nrc 2culate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whk6v9qjs)enb j59w8hh- nqrr6 eel rotating at constant angular speed (Fig. 8–42). The friction forn6rjkhvq9b9 rwnje5-68s)qh ce between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of rs hkekw/-j8cffvo4 czm)r5b2 2(p :t( wc- ippoint objects shown in Fig. 8–43 cvrwm p28 :ke- t)4fhbzop-rsi(/5wj2 (fcc kabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whosr o)vv) 1pqjq5yzojhb5x6 0wm5s qm8, e total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, e4;jifx4z(ncm 8z .ae ingineers fire four tangential rockets as shown in Fig. 8–44. If the satellit8zx4m i(i4 .fnec ;jaze has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mab m9/3k b9 ga*u,8xddzt.wih9b : ujvdss of 0.580 kgzj/t3b89*i.: g 9xvabbu9 ,dhkmduw d. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bats:v6bz cue 4 i7*xtbh9, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rou- 1umpx3 wfo:pnd and is able to accelerate it from rest to a frequency of 15 rpm ow -3x:1ppfmu in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,3007c coojk:2vb1hre (7b rpm is shut off and is eventually brought uniformly to rest by a frictional torqu 2:c1bo7rh(jvcoe7 bke of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kgow t1 wdw,lfj12, sk5v ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of qh 7l yx(nj, vy;3xse;the forearm, which rotates about the elbow joinq7vn; y ,esy;(l xh3xjt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, asqby,7zvsw-0a/g p+; 7jgtix jlu .ri/ shown in Fig. 8–46x7 gp+/-qy;t/biw j7 ju zvirs.ag,l0.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,x/k(/v 8vdu.;yw jcszeoq*q: $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer fr+pb.h66kc mb tbiq 5w;om rest within four full turns (revolutioncbb 6m+ ikw.t qb5;6phs) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hasvw09e9x fzg xyhu94.2nzld6 i y-stp, a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2:sc75puzs yyv1 z5is) 80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg a dmuw*w(/xrju 4qu;b2nd radius 9.0 cm rolls without slipping down a lane at 3.3uwjd x qu* w;bmu4/r2( $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respec bw+j(( ry4vt cl0zcv;t to the Sun as the sum of two te wlcvtyrj4(;+0zc ( bvrms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of6 u oqoest9ueoq/n7:p3 ol5w6 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolut :o63wt o7qupl9q/ 6sne oue5oion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and masnt.piy fg:h ,3s 1.80 kg starts from rest and rolls without slipping down a 30.0 3ntp,y.f h:ig$^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starteq g5co7kzg1m ; q3) p3o:neaas to fall and its lower end does not slip. What will be the speed of the upper end of the pole juste o)ac ;gp:noe5q3azkg 3qm17 before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m; nkgq+:bsvr/)umg 5aomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at bmqu5r +knva:g/ sg );an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unif 0kil0e 7 t )sew4dfb7isby(l*orm cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? lbt i4fy7 0ede 7ks)l*b(s0i w   $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a37xkx *vyxg(d rate of 1.3rev/syx ( *xv3d7kgx If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reducf ze66fvgbc2k(5oo+p3 i av-te her moment of inertia by kf23 t(b i+6o vfaz5-cvo gpe6a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1(6 a3j rg8cwju/ f slda+up0g1.0 revj3f 8au r610c/pjg w+gd lsu(a every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through inxe0u7a(j9 q qeo2es: g-lh urjw05l0 ts center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the s2eol-l0rjju0ae9 q7(x gquh : n5we0frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular v z, 7fk4yun5g 7 5ts7zaq6pho33lyvcmomentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of hm6q iiqx14yf8/avm/the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped ongnlv7y0pu2 c(-n d -+v+dkgwnto an identical disk rotating at angular lkpddvcgwng2-vy+nn7( +u0- speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.d8(lg3jqlg4 -ik 2aq6h aw;pqw8j ;xb4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating m9 cng)7 osy gm8:-cbqxerry-go-round platform of radius 3.0 m and moment of oc8gy7sb x-9g: m)cnqinertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular vea 3e mbvhi /zg/7z 31abs8atc)locity of 0g3a3icazz) mb87/hbt/e a s1 v.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventua / sitskh(e9ac lp, 0.;srkvn1lly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Ast.n/hk; (sk9 1,pialecss0vr suming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excmqpha0zm,)/ g* p gqadn1:)sless of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass .fs7d*y f-q3bvo *whs$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a ( -sj-tr26ch* r vdwaxplatform, initially at rest, that can rotate freely without friction. The momath2s-x-rw* 6vd (rc jent of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at thbv2 t673ob);3: 1jz yajroeuuwrg3 f le edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a m 3bjy1ojlwe2)uz736brfg : aov3u r;toment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on i( 86 -ounn,akmesue( the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behioe6u nmk8i(, u enas-(nd the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces c*0qjhhf/1az*+edjc8d ni 6pthe Earth. Determine the ratio of the Moon’s spin angular momentum (abouf d*pjai+cq61/hn je dz8hc* 0t its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at wegod30hj t j-5;mk6nis a)d9a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of s3uc:dshvrxz cx )6f*t (b*ju89 ,kloa uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6k cz dx3* :8vb6tjs)ru*9,l ocxsufh(.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two + mola+dh 9o55 ug;ms0v lk7cisolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear splamo5uo 7g clk9+;5+d mhv 0sieed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular s dr+aa+hp 0v+ubd7 ge(peed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gfo0m,jc 0fw+ 03sbp;b6hdetf reat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at ad0f,w6o0hm p;3 b0cjb s+fet fll times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored i7f z5)nkdha1in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kgin hzkf a71)5d, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates ev fqti:1fs)5k/2mmg an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizonz,je 6 inud:73rrnvude +)p6dtal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of massbp6s obhf* 2y-q0 i7sa M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod7b6 aq o0spyh-*i fsb2. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. Ipjao 2g0,k t;ot is standing vertically on the floor, and we want to exert a horizontal force F at its axle so thaja;2 g pt,ook0t it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flatuclnyz7 l45/b fhn3 d; road is making a turn with a radius The forces acting on the cyclist and cycle are the normal force d4u/z;75bl 3nhfycnl $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into ar q dcoyk1x*r1ntp4).g,vjt+9 x s q;e sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleratinxy4k g.xtj rpn,o1t9)v 1+e; q*scqdr g it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mv)ydcri h524 pwty 3dm0bj3wx9b*o7xaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.wd3coy 7bi053rvjbwmh t 2)d4y *x9p x   

  You are designing a clutch assembly which consists of;,+j3r2ulce g6dqx f t two cylindrical plates, of masscx2,j lr 3+gt;d u6eqf $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the7m/; te) o /pd5b;l5m /tsav8acrtvtl2e hq/u looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop wiaatm5et to )/;2q/hudelp; /scrlb8 /m vvt75thout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ask1ze.x/vsprf7ezc 7*sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforml c0fyx:0 itvlew5b/x (y from 90km/h to 6fte0vwl:x/i5 0c yxb( 0km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s