A bicycle odometer (which measures distance traveled) is attachem9rdkumf ebh0;s726m j . zk-ud near the wheel hu92 .zekb r76dsum;0mjf-h mk ub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the ri/p25d1 0y3 tesmyajv fm have radial and/or tan1 e /vps0y3a2jd5ytfmgential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be,fyk:0r4f m/gw 1mmum described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.qj-am an2/ 7yk

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 with your hands behind youn vz /ll027jcwr head than when your arms are stretched out in front of you? A cl0/7vn2 wjzl diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the v98vgj1 /9l ,x7 vnpc,snhct zrear wheel and three at the pedal cranks. In which gear inv8 jv/ 7cnpxt, c1zl,9gh9vss it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs wiuyf*- s:(4d* qpfkxow y/)md yth flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dxd*ky:f*ofdwy) 4 s-uyq/mp(ynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (zx.ofw ar.1-s5fr43dztyke 2 Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also 5 pcg wv;rl9:uk1/rp(g4ne+ : ktrvf bzero? If the net torque on a system is zero, is the net force z;l1nrk+9vp/pbf:kucew: 4 r( gt5 grvero?

Two inclines have the same height but make different a2az:o 4:jrh:wo,(c,hy *ki0 s mwjego ngles with the horizontal. The same steel ball is rolled down each incline. On whi:(4k h zw:h,*rm wasyj2jico g:0o,o ech incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start roeuon5t,p/ kt)o6 kn(aqp3a 7np.mh.d lling (from rest) down an incline. One sphere has twice t , ad5q3kpn..p6hoot)a(/km e7npnu the radius and twice the mass of the 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 samen/s)3buzj)q ,yp-i yq radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Whi/uqp qj3 )yiyzs, )bn-ch has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum cj7a oheb35,ghax) 6uand angular momentum are conserved. Yet most moving or rotating objects evea5 u3hch aje6x,g )o7bntually slow down and stop. Explain.

If there were a great migration of people toward th8 pb:(r5 hbi-hwtqc0le Earth’s equator, how would this affect the length of the dawq 8t0i5:hp(-hbcb lry?

Can the diver of Fig. 8–29 do a somersault without having any initialo ),7n*e2odbys o9 paq rotation when she leaves the board?pod asbo )9*no7y qe,2

The moment of inertia of a rotating solid disk aboeko)o;l*, mzw6 hr /:xegaw f4ut an axis through its center of mass is4/;*hwefw6zk) o,am r o:lxeg $\frac{1}{2}WR^2$ (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 stoolhb 4v34m y/hxv6 bi*ah1( otqy holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? E6 h1 baxqv bhy4y*i (o3vth/m4xplain.

Two spheres look identical and have the td,wzw1(bc 5fb2y g togc.s1*same mass. However, one is hollow and the other is solid. Destt( ow.g1zf bcdwy5*1 gb2,cscribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points westi+l 3/xhhoh t3 ni*c6t. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points l3 +hhnio *ht3i6x/ct 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 freely rotating turntable. Whaqg/e7n:zo ii r091z ogt happens if you walk toward the cente10 9:e7rniqgi/ oozz gr?

A shortstop may leap into the air to catch a ball and throw it quickly. wl-b j,a2zwf v-0))pt85dsi val7uyn As he throws the ball, the upper part of his body rotatesdf7-b8 )p iltaj0 ,wu)w2nzv l5av-ys. 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 angula+ hda;xg,r 4omr momentum, discuss why a helicopter must have more than one r g+;,omda4 xhrotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) hpg6)huuawo1 x /* ue830 $^{\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 a 8ao:) nmo o. fd*siwnz*vzq;4n amazing coincidence. Calculate, using the information inside the Frovzosd. 8 *qo:ozwi n*)m;4afnnt 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, 380,000 km from Earth. The beam ditcz27 p oo68uuverges at an a 6po7cut 8ozu2ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderu qqm;kiv;ul gtnjh1q) 4w)-0jl x.+ b rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slx0) inhm-1+ g;t.w;kul ulb 4qjqj )vqow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a le-oz) e;4 bhz,xbmdte 4vel floor 3.5 m to another child. If the ball makes 15.0 revolez,bmt-d4x4bhz )o;e utions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revxn92 jh 9lgokp.x pr1/olutions do the wheeor/ 2 xjl9pnphxg9 .k1ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its7laiz axem 5 (1px;j -xkn91js angular velocity ajl 5xjak(sx m1in7z ;x9p-1 ein $rad/s$ $\omega$ =    $rad/sec$
(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 complerrucqm89j( pr 2 w r;9nloon)2te revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cenro9l 9)wpor28c;r(n q mrj n2uter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular veley4yin rw5i w5vm/b bu,n))0n v -kc6bocity of the Earth (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(tf5tspr e75d a point
(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 7.0 cm from the kab/1zu6mi5v- y 0. m * ffp;low3mtieaxis of rotation is to experience akm1z.feub; 5*mtv3 / 0ofia6-imlwp yn acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 139 :mvnz;6i2ytpj rb6d 0 rpm to 280 rpm in 4.0 s. Determinety6vm92j i: pdz6;nbr
(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 v4d+zg/ i/vcc$R_1$ 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 abk5 1ku+ f8oaaxoard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s ener8a1+ukoaxf 5 kgy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min 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 uni ywj1m4 bj9* dhe94lfoformly from rest to 15,000 rpm in 220 s. Through how many revolutions*d hjfew49o1j 4bm9yl did it turn in this time?    $rev$

  An automobile engine slowt9; 5 qp3nnjw*(8mcmqx-fp vws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(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 streby l 3bthc7i (8ghln:9sses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 completlg9c:(ybi38th h7lnb e 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 diamf,.ohl 08z e/fh4vso qeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel ef/,0lhs4z vho o.q8f have traveled in this time?    m

  A cooling fan is turnxsr/ 0cnz k49ad4b(bced off when it is running at 850rev/min It turns 1500 revolutions before it come(bzx44s /kra cdc9nb0s 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 n 0kwl5/l/;n dnk,e shrevolutions as the car reduces its speed uniformly from 95km/h to 4wlkns;dhe,n / n5/kl05km/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

  The tires of a car make 65 revolutions as the car reduces its speed u-*ilgnsqv mi0z3d ,m9niformly from 95km/h to 45km/h The tireiz dvinql,3m0g9-m *ss 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 her weight on ea y )m)fjgh7 tgvqy )fx4okwj;l6s;t1 8ch pedal when climbing a hill. The pedals rotate in a circlefjtl)yj1;;m g))g7o t4svxq8w k h6yf 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 ts phc v 8l8v8w7i+za0mc s5v,ihe end of a door 74 cm wide. What is the magnit mv7vsls+ hwp8v , i50ica8cz8ude of the torque if the force is 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 wheel shown in Fig. 8–394ba dvqlv07*v . Assume that a friction tl v4vdqv7*ab0 orque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arw2eyvv4x ctde 6y,..;q4w ad ce attached to 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 magnitude and directioe..qdacyt ;ewv2,wd644y vxc n of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a 7wy y-8yqj6httorque of 38 yyj8wq 7th y-6$m \cdot N$ 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 spherewd3 0+;)loh;pizza. y/zc xxeacg d7((j7lkg of radius 0.648 m when the axis of rotatio.h az;gz+wl ;o30dkgzc( )elpx7i 7a(j dx /ycn is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyck9j 6-(dmls 7 ec(jh9dl)w jqlle wheel 66.7 cm in diameter. The rim and tire have alc(9whlkdjes -l( j79q 6mjd) combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatedz5q; b e5ip*fv/hqima 80pwh 5 in a horizontal circle of radius 1.2 m. Cal ipvmw/ihq5p0 z5eq*;8ha 5b fculate
(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 wheel rotating at constant angxh8ki) o+mi*tegn/ j 2k; qftu7;exo0ular speed (Fig. 8–42). The friction force between her hands ki*q;8)e;g7kxxjem0/ n+oi fh uot2t 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 point objects show2*/wlu jgc b/an in Fig. 8–43 ab2w*bc uj/a/ lgout
(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 ofp5,( bugje9n+yawn 5:jsa5:1puk x s z two oxygen atoms whose 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 satellit fqfdp9v-c y0)e spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force cp-yvfdq9)f 0 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 ur:z gt .o*bgr+kzr7w jg*lg81niform cylinder with a radius of 8.50 cm and a mass of 0.580 zb g8r rgr7jo g1k:. *wt*zl+gkg. 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 bat, a)4l9xl yw 9tytccelerating 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 r/9x2q d gk7fyeh ,0hvhand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm 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, neglecdr/ , kqg79yvfhxe0 h2ting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating aty z+ilu1a/ nnk .1e7dq 10,300 rpm is shut off and is eventually brought uniformly to rest by a feq nd1u7in1k./ay+ z lrictional torque 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-kg ba*on-vt y m1q)sll 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(oq( d3 wj eifw)ku4 ck ow/u4aa )z6vg/o/lf3 ball is thrown solely by the action of the forearm, which rotates about the elbow w4ovu 3(cld k//o(o/j6qu)i3fwaezf ak4w)gjoint 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 blahsvzzas3as*6 0yp /z) 86afatde can be considered a long thin rod, as shown in Fig. 8–46.3/0sva azy s af6ha6)p* t8zzs
(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 consistsf3iyk.qy )0os of two masses, $m_1$ 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 from rest within four full /ypfhfhoim 1 ias83+7 turns (revolutions31hhiopf7 +m/8sy afi) 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 has a momt1 b0t5am4j2 gc dblj*ent 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 zmcc+.y)- ei*i9ng ioa torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without b /q7k3r0 ssp8usq6qhx)9e4tugeon c c3+ b6dslipping down a lan0 pc96)se te6bh3o3qgqu7 n su x+r4csdbkq8/e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eq (;wfp(uotc b3o gt+/nergy of the Earth with respect to the Sun as the sum of twooog/ cpt3 w+ ;(tfqbu( terms,
(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 of 1640 kg and a radius of 7.50 m. How much net wor)h. onmazk7.e k isoz m. ).ahk7ne required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.86bi1wu brqv4k vb;p46g/xgukky3x 7()ptm y60 kg starts from rest and rolls without slipping down art ykqku1;6xwmg6p64b vy xu(b)7ig/b v p43k 30.0 $^{\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 starts to fall and itvh;qqy)+pwcq 2 j u-)ys lower end does not slip. What willq+y2 vy- hwq )c;qpju) be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rot5,- 9:srp 6 gvsuphix aa3)vqoating on the end of a thin string in a circlev5usia3:gv aops6x)h q,-p 9r of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniformea ldd)/6tm b4yuczih )xd8+3 cylindrical grinding wheel of radius 18 cm whcm bxd y +4/edd8u tal)36)izhen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side(v5rr3h:1un. t dd khe, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position,3e(d th .kvn1:duhrr5 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 reduce her moment of inertia ,)2xe*3shk7 u bkmka iby a factor of about 3.5 when changing from the straight position tuha,bxm )k*7ie3k s2ko 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 rat w :v6cfxu6r0(o j8waue of 1.0 rev every f86 c uj(r06:owuawxv 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 94mvaty6 ynny/v iq; (vmm/vcu2*3 rt around a vertical axis through its center at a frequency of 1.5rev/s Thvu6 *m;nyv2t 4nr( t/vm /y9vim3acqy e 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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning asi.30wqzc d2s t 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 the Earth4du 4mksikgthq; 0(n3
(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 inertielc56 sm v.8s* ;r up5ep.hdbt3sybd0a I is dropped onto an identical de s.8b phu5es;yrv0ts65*dbd mlc.3pisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns x2jdjcp x l6 ;-z(sef9at 2.4 $rev/s$ 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 sta5/:klrtr m: xqti4si5 nds at the center of a rotating merry-go-round platformmkt5:q:/risil t4rx 5 of radius 3.0 m and moment of inertia 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 rothxdx+aw5 s:9ie3tm )yating freely with an angular velocity of 0 :t9x ey+aswix) h3md5.8 $rad/s$ 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 eventually collapses into a white dwarf, losing abfs:g,h9d5oqt4 z*c7mxqe2q b i /juz;out half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest insq4 ju 29;o,g q fc57q:d /hxbti*ezzmteger)$\approx$    $rad/s$ (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 excess mz p86w1ri v.hof 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 x+ r x+yp4f9vvtq.jr7cmvt8 2$ 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 platform, i wjr2vvgy.18wnitially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform iw w.gv rj2v1y8s $I_P$ 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 the edge of a 6.5-m diameae(tvt:k(1v eter merry-go-round turntable that is mounted on frictionless bea(at vtk(:eve1 rings and has a moment 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 roy+nkbt (ii 0;ppe lying on 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 behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mi+ (tk0y;ni bpass move?

  The Moon orbits the Earth such that the same side always faces th :vudv.syqrh:2 * ,5 ulcuqy)pe Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) t,cyl hd :vq2)p*: vuy5suu .rqo its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest v g tu;i*y2 s18gvkiqx+)2vdo at a 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 a undc4u 0 dr+mn29nysyr: )zmul2 q0qfoaaee91 6iform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accel61 l+: 4q0yqer2fda2rnaumynu 0cs)9eo z9d meration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid etlw9i 59*kubg l4 6lqcylindrical 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 9 g6wqkeu 94tl*l5i lbcalculate the linear speed 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 iy c w*vhusfvs zz;p(*,0.lkaz+,gyx -s the angular speed 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 ma8 0a kpnh+3 .ai (ibpz9peaqf8un x:;bss 8.0 times as great, 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 tpkn (i8af h p:z ae3u.i0n9a8xpqb;+bhree-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all 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 ujo rucc2-2m -do41zoh5hfe3j se energy stored in z 21jrj45umd- ch-oc 3ho2f eoa heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, 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 illustrateani9 so/pa:n8e- (srpq gg5 z8s 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 horizontal surface at .a 9vwv13r2 vvdf0alg0hp fc .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 mass M and length L can pivot freely (i.e., we ignore frihr4n+ + gqbog.ction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontallrqg.b+n+4 o ghy 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 rod. 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. It is standing vertically on ttf+*qj uu( +o:*mq gtthe floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it r+ +mtg ( uq**:tjuftqoests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spee. qu cf;sr,t6oo2 2xubc2:mis d v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist andci:ur ;o26b x ,u2oc.2fss mqt cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 0.5gsxl8i3(sh4. p*4dpjwa , ot p0-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest tog3l8*pt i. ospdjxsa,h (4w4 p 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, mak-o2f xakc 3-iz;7gl , yw/w+o mcds-hping reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arm7;i-op 3o ,+-x2 dgcylac-kzwsh fm/ ws held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindr ix/gw8rt mooh3),1 qdical plates, of maxht)q1io ,r/mog 8 3wdss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 the looped gdagvx *z *13 b6ap0.c xtf9knrough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must f0x*ag96 p*vzkd a3.b gnc 1txdrop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do xb0l9uf/1ga tnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforml*u9zcco3n uj (7h*z hky from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ahuuc* 3n7 j(z9hczk o*ngular 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