A bicycle odometer (which measures distance traveled) is attached near the w.h4qkt9,of pjdj y) rrm ob.:be v;86pheel hub and is desigtj kh bmp:.d;.o 9 bvpojq6,r ey4f8)rned 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 oxygszf 5lelc,ey5 ss4)12 /hvn the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangentia/y1ee4)cs5llh fvs5 g z,s2yxl acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descrip-h ir*d4ojy .bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torq+* 25b8xex to+uzoua nue than a larger force? Explain.

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 youe* )s3 e82 rfpb4 ;k.uv- f /6tlracteacmah8jr head than when your arms are v.*e;p fu)kh3trb8la6je -4ecf/s82 m ca rat stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pei4ih5 eq,mc (adal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it hai5ahmq 4,(ice rder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast h zbwv2)gc6; /rec/kqw4 rb v;bave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why th/ c;/b ;zrkvb4w2g be v)qrwc6is distribution of mass is advantageous.

Why do tightrope wal)x s7l25 .ajeb-gt gwo,;io mbkers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque s*aen l9 43mrok:)xc talso zero? If the net torque on a system is zero, is thcr 4sexm)3a*tkon9:l e net force zero?

Two inclines have the same height but make . fk0wt8i suc8different angles with the horizontal. The same steel b.s wcift0 u8k8all is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frz g b(595jlmqeom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed therel55e (9 gbqzmj? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have mjl oq oyh k;fif, nrs e3j)bv889a1t)wc148x the same radius and the same 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 the greater total kinetic energy at the bottom? Which jtckfix),1vmn8 ls y)ea 8b1ho3f84j; qro9w has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving shuwc r, 9)(1furr0l ne.y-i;8f eyq *uqxw x,or rotating objects eventually slow down andr*h-e ,( q9f1qisncw rx,wr8f ueu;l0.uxy )y stop. Explain.

If there were a great migration of people toward the Earth’s eq;0z-fhpk yveu;rh g ,7uator, how would this affeck;z fu0reh y;p7 -v,ght the length of the day?

Can the diver of Fig. 8–29 do a somersault withog 1m.*n+f zzah*ga:a yut having any initial rotation whe* :af*h+gagyz.n1a z mn she leaves the board?

The moment of inertiaa 1v7prd 5s44dgqlg9a of a rotating solid disk about an axis through its center of massd7 vgd5p9 lsg aq1r44a is $\frac{1}{2}WR^2$ (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 holndrs w0lk*x10bwyrs70r zo -/ fged92 ding a 2-kg mass in each outstretched hand. If you suzyx0norr0fb- w*rk2s0/7 9dw1 lsedg ddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the syuv(( rddcvhhklt6n :/mpdh 1*s 6*q,ame mass. However, one is hollow and the other is solid. Describe an experiment t 6*k/dq 1d6lvym c(sh t*( hud:nrh,pvo determine which is which.

The angular velocity of a wh zcr43a+b.ogtk h ;(ro/jw4qyeel rotating on a horizontal axle points west. In what direction is the linear veloc4 bo43ah/k cg +rwjq(z;.orytity 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 exh3 .9 i*j ddo/j8js e7mfxb)edge of a large freely rotating turntable. What happebfj 8dm)eio7x3ejs . /j*9h dxns if you walk toward the center?

A shortstop may leap*y8ukiig: w.o into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and le :.gw yiiuk8o*gs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law oft3 j8z9/a)8qwo z)mz maj3aiyy s-v50. ri qtx conservation of angular momentum, discuss why a helicopter must have more than one rotor (or prop08itazw)x-5i y.z )qajvo3/a9s mj3tz 8 mrq yeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 a ujgu b7v.(2z0z kw5t$^{\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 bzs:m* v5d zs5because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radian mz:5bs*zvd5s s) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00ng8cm z9be) s4w+ (owc/xo 6 wb-+bcnc0 km from Earth. The beam diverges at an anglbbn6 -wn(zbomc/c go++ xse8c4cw w9)e $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate t0in p1+x/r440y7by zqy;p bd6xvo cmat a rate of 6500 rpm. When the motor is turned off dur64ox 01d7y0 yqz pxmtbir4 p;/ bcnvy+ing operation, the blades slow 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 level floor 3.5 m to another child. I17atsko5 . bkee)h,wq.k( xrnf the ball makes 15.0 revolutions, what i (.x,on. k1khkeas b 5etrq7w)s its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How mavay kia6b(it( kg 0(4qny revolutions do the wheels m vkak6 (0iby((taqig4ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cal6f*sgm:w8e9s0mme t 9f c6kj iculate its angularc6mt:9ek fmgj 9 em0*iw6 sfs8 velocity in $rad/s$ $\omega$ =    $rad/sec$
(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 complete revolution in /k ) grk-1x0y3sh-6, bzcnx y9qrhaqf 4.0 s (Fig. 8–38). (a) What is the linear sp3,r)hs-f- g6ykhxz/k 190 qrbnqay cxeed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocityl+bedk,(ote.x qz25t 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 ofix dk; * ,wa;blbko.-a 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 c0xatgzc5 6k8 hm from the axis of rota05a 6ctxzghk 8tion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml z/glmx)0yz +eu(dt+cafb,7 ey about its center from 130 rpm to 280 rpm in d+ 7ex,+z0 /fbaeclzu(ygt) m4.0 s. Determine
(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 vmtr6 h)byy:buwpxcf/:+ v 4wc co 9-9$R_1$ 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 the Apollo spacecraft put themsel0j- 7i1-vs b9cdqyvlaves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated fr svyl-v- d9a b0jc17iqom 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 uniformly from rest /esef( ,am,b pto 15,000 rpm in 220 s. Through how fmpebs/, (e,amany revolutions did it turn in this time?    $rev$

  An automobile engine slid)b5qwmi* ;p a*eex 0 x*n/tbows 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 stresses of flying highspeed jets in a whindwm yx7r/od8 z x) c1xzz83g3rling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolmn8ogxcd 8r)zw 1x zyz7 /dx33utions 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 uniforedf/p8w4 3kcd,x rb b.mly from 240 rpm to 360 rpm in 6.5 s. How fabb3pfx .dd 4cwkr/e, 8r will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it isg(.c :boootu)z 1 ixin7iz ts(yl.g5 - running at 850rev/min It turns 1500 revol s5xoo .c )1g7ub-il yto(zz (tiig.n:utions before it 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 wf mc2 m:mnz,m 8h.mdz, s;umwrxz)., uniformly from 95km/h to 45km/h The tires have a di:f.zc mm) 2 uwzhs,.n,dz;8wr mmx,m mameter 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 x s:)ntnc5tvwc:qib 4;c9/stniformly from 95km/h to 45ki w : ;t9tn:scq5)t4cc /bsxvnm/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 her weight on each pedal when ,at09 z.bd*cb:/jb rj z k.amkclimbing a hill. The pedals rotate in a circlmat9 b*bjb0ckz/.. ad:, kjzre 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 Nx( 1hrwbtf m.6 on the end of a door 74 cm wide. What is the magnitude of the torque if the for1.xr tb( wmhf6ce 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 tjujd37 v1xpbe6cja) f1q lyy,3 -10qq i3zmsq he axle of the wheel shown in Fig. 8–39. Assume that a friction torjfm1 6bqyq1s0cd-iaqj3 u q j l,1y 3p3x7ze)vque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are t t ;g+y;f*zzwk(z9cgattached 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 az9tt;;cygz( kfg +w *znd direction of the net torque on this system.

  The bolts on the cylinder head of an engine requ3/eh,vokfbli9gaficc/0 9 )dy w-k+a ire tightening to a torque of 38 w/-ved9y9)f/kalhc+a , i3gbcf iok0 $m \cdot N$ 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 ih*/p 94b:r vjq ni,uua 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. 4v/r ubn*p, 9i:jiqhu   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ina xnj)rng i+cf e9f4kr:7az6ks*/) tw diameter. The rim and tire have6794w f) *frkjn:r+ z/t) xiaeaskgnc a 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 the3i20ub8/dlvzpa; r+nu l n k7r end of a thin, light rod is rotated in a horizontal circle of radius 1.ikn8ruznu 3 r/0 dlbl2a+p; 7v2 m. Calculate
(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 whe k l3grz3 vi-+g)drh-gel rotating at constant angular speed3 zrig3vdlgk+r-)- hg (Fig. 8–42). The friction force 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 point objects:ck u9bk 2bp*.uow-86 x 6igplad zm4g shown in Fig. 8–43:kd4pz*.g82o6au mx-6gki 9pbwbulc about
(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 whose 403);h6oaizl:d3 o f:r purmt 9zts pqtotal 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 rat*f.h g tq+n sn2;f /3whfml4tfe, 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 nttff2gs f;+/mf h hl3*nw.4q.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 j*p;l; e)b1ob k e/ibk(;fki/j/cdlbcm and a mass of 0.580 kg. Calcb/;ek;iojib(;/*cljpb/k 1 )lke f d bulate
(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 ab7 +snv 00tpo//4or tz h5qgsx bat, 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 ha*wj5l m. qb4/xv v-bxd;2 ngvqd4bms8nd-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 enb5/8. q24xsx*jm blvv - g;dqmwbv 4ddge. 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 10,300 rpm is shut off and is eventually brougcr30 4kh chdc:ht uniformly to rest by a frictional0kcr3cd hhc:4 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 3dzdir5pmic4i34v 5bzlzh2l,*q6vz a 3.6-kg 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 so:( byjo2uv+yrlely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 uyyj2( vbr: +o$m/s$ 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, as shown 0qd*n4mdrx ao0 -6plein Fig. 8–46xlpdmnd4 e6a r q*o-00.
(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 px hgl0 0zrr,; 3rku*gtwo 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 f.bn3pwbw/r3p4dm ecmbsi / 2w*4.cva rom rest within four full turns (revolutions) and releases it at 3b n .sc2 pwev 4/mabw.drim3/b* 4wpca 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 gkux) (nk1)t rmoment 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 develo fsh6vj(1h; st (7 kodni8* w7via:yeqps a 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 c yjwxv t .21hxabbt-,uvl(x/:m rolls without slipping down a lane at ( xv2 tj/au-hwxbvtx1.:y,b l 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eam61wvn5 ok zrd3c 8s q(3dqnn0rth with respect to the Sun as the sum of two tedd(3z nnq q3k5 o0ws c1v8mn6rrms,
(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 wo(+fxfr(favwv97 3bl ork is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it i7(ofa3b 9vxfr +lf(vw s a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without szhjj(5 dsv ) qo/5cln)l5z)hdv5q(n j /j s)lcoipping down a 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 tix lr ofa9 .;8s tfdv;4x1o,lilp. It starts to fall and its lower end does not slip. What will be the s lxov1la;x fo9 8.f li4s;,dtrpeed 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-k)cn s(x.- lewj 6unks)g ball rotating on the end of a thin string in a circle of radius 1.10 m at an angulan(. l )nc6we-jksx u)sr speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniformcha.e0ea*r 2e4g; rpi cylindrical grinding wheel of radius 18 cm 2*a 0r;e4pe.gca rhei when 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 ma x orufr6j3m-2r1a.sb rw-hje t))6at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms b r.) osr6jxtu 2ra1m we-h6)fj3amr-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 s x v5)xm6u6hjur4 w,lghown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 wu u6, mjl4vxxh65gr) 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 sp 2siqgo2d u7*npen 5;cs:ve3a in rotation rate from an initial rate of 1.0 rev evere2; guisd7cn * n52:ao3ev qspy 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 bh0(wh tkijwg/+ qo:(bq1:r99md e gxits 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. h0e gg9b+9ww h(tojdrkq i(m:: x/1bqWhat 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 atn wjy* l(;qq - e1g7fv9vh)tph 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 th p+9 mlsaqkh-3e 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 cylind*23 mmrj5b 4l i8hgw/wpwnk1kwssy15,4 vihz rical disk of moment of inertia I is dropped onto an identical disk rotating at a m,vh4w4yzg/jswsr w 3pk15m i n8l2ihw1k*5b ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns akaek5ov lg-w7js +(hf r;d1 m7t 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 stands at the center of a rotating merry-go-rounolv/s mmn +.chv /3. 1p;sfhorw*jgfo h w)25id platform of radius 3.0 m and m v1wn;hsfc)r5lo w m3gm.h .ioho+2 fjp/s/*voment 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 rotatinlq9 i62x;ds 0q milry3g freely with an angular velocity of 0.i0m3ysl2r lxi 9q q;d68 $rad/s$ 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, losfuo9n1)wf 2 +-w3mf* vnf golging about 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 woul 9fw ofwmf vg)n+l-*gon3 21ufd 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 a5+yjsi83 txn in excess 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 ucz m s:sq2vwu3gl1. *pzx90 f$ 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, initially at rest, that can rotate freely wh pkfdlow2su;hs,4 95 ithout friction. The moment ofs4 ;s l,9w pd5kuhofh2 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 persor7 d 6wkr:w)eap2nb:z n stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted or 2d7zbn ep6awwr:k):n frictionless bearings 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-ui:w 0nlocmc1 s8aq 2 with the end of the rope 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 sliic18lsa0:w-cm nou q2pping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thatrtzolr tsa oa/ (8428v the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbititvs/ a2roo8 r 48altz(ng the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest 6ft 4k5mh3crom76 hgbat 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 o rvh3. b(e(jpuniform cylinder of radius 0.20 m acquires a rr(bjhoe ( vp3.otational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mablc*v2.nnr e;ss 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 speed of the yo-yo when it reaches the end of itel;rnbn. *2 vcs 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 isrk1 pw nf npc)l-z.f+8 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 mass fx)dx4 t:kxipy 2(ogw(w;a1l 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it w oxy wdx1l(:gp( ta;wx24ki )fere 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 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 use energy v 6 ejyirwq/;rj;wor. f*dc hyjd03/:stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses; rdfrodwv hcwj.6y/0jjrye ;/ : i*3q 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 illustratesuwvb155ss+5 r*t3nzsi h y 9ndss+,ea 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) jx.hf.1 3:ce onrsmn 9is rolling on a horizontal 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 mass M and f7:8 j*( d:mwaybezculength 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 rod. Assume that the force of gravity acts at uc z j(ywa f7me:b*8:dthe center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius +* k0xos3 ozlvR. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it wilz3o0sx l *v+okl 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 f:i,zd h(ugry+.2vib hlat road is making a turn with a radius b,z.+: yhvd2iu rihg( The forces acting on the cyclist and 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 0.50-kg rock into a sling of length 1.5 m aqr1dl ;e::ey n8 gfvx+nd begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requirey+gvx;qfd81:n el: e rd 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 thi8j(c-zhps8 ag8m it 0ln rods, making reasonable estimates for the dimensions. Then calculate the ratio of the8i- s zgl0 8(mt8acjph angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plak v*1md;krbwb a/ v,:kf0w7j ntes, of bd;w7 0ajmbnkv/f*v k:w,k1r mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rough track of Fig3ps7un5 6jz*f mtr-*s,ypgau. 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 loopp7 u ng -jf3,t ps*rsam*uz5y6 without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b z9n 4pag: fu8rsvi.4i fr(l18wa:i emut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thm+ cz uoig *v86k*6mnke car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accel vc6o *i*g8m6nz +kkmueration 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