A bicycle odometer (which measures distanc ssd6el ,ko1y db,0s7xe traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happex6s1 7lssdoybd e,,k0ns if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pjp-htsmihx, 77ifyy.6a r :7djy5 2wgoint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incrx7hi7ytym r f 5pdjy,si -wa.jg7h6 :2eases 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,pc a+y*k il ; 3o3es1dotc,ma dybm71 described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque gn68h z/q sc0lthan 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 sitw0frggz/a k c( :feio/+t4bi(-up with your hands behind your head than when your arms are stretched out in front of you? A diagram ma rbf0+zk ow 4eag((/ic/:tifg y help you to answer this.

A 21-speed bicycle has seven sprockets at the rumu9v 3fva,c8ear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Wha3v9mfuu v8,c y? 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 fas5nv+ppv/o1v)w6+/cs uv mijzt 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 this distribution of mas pvujvc v i+z/)o1sv /+p6m5nws is advantageous.

Why do tightrope wal4hj9vhbrw qblsn ssj0 ) /;q6-kers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net tmcd;d1f15u b7wqj5 qlt* qmv(k7te( horque also zero? If the net torhtmb;*dwj tqdvu f 1k7((751 cmq5 qelque on a system is zero, is the net force zero?

Two inclines have the same height but make different angles ko:i/8c 4juq oo;hv.v with the horizontal. Thov4;.jcqooihu v8: k/e same steel ball 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 (from rest) down an incline. . xf g-axcpkp.8sw5o- One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the-.so. -g p8xwp5cf axk 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 same mass. They st0cn azsuz5( lj4a o-+part from rest at the top of an incline. Which reaches the bottom first?+s clnaj50 u4poz-z(a Which 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 and angulapjf9u 9fs( +o oygd*:lr momentum are conserved. Yet most moving or rotatin:*syo9 u(o+ ffdl j9pgg objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, x r9d hx8.47ttcqng) jhow would this affect t q)8x ndh. t4crjgx9t7he length of the day?

Can the diver of Fig. 8–29 do a somersault without havi7 /8immc-y7mp gu :h 0vrsk6ndng any initial rotation when she le6 mvg c78psr0/- k7iyhmun :mdaves the board?

The moment of inertia of a rotating solid disk about an axis through 7h i+5krjc8m.szgp;.to i bg6its center ofgo8i+ p mc;sj.7zkg h.5ib6rt mass 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 holding a 2-kg mass in e vupo.zaay(srgvs439 h9s 3q5ach outstretched hand. If you suddenly drop the massess rg54qosh9pu3vyza9 s. v3(a, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same masspeb.tdxok6/ : . However, one is hollow and the other is solid. Describe an experiment to determine which6o.e dt:xkbp/ is which.

The angular velocity of a wh.uufd 7jg6 c 9k.f8,boh-uxe oeel rotating on a horizontal axle points west. In what direction iscgu of,.u -jxe df68hk7.9u bo 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 freely rotating turntablm3ymviz- ,y4ia k/1tqe. What happeiz,mt v yq3i-km4/y1a ns if you walk toward the center?

A shortstop may leap into the air to catch a ba c xoerg(2)uka) +nyb9ll 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 legs rotate in the opposirxy2)9nab()ocue +gkte direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discusm( inj2v,tff. s why a helicopter must have more than one rotor (or propelljff 2v(mt .ni,er). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30+nu0ba/3 r:ow56u zvepgsil , $^{\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 Earby)d)s;s mlb 3 f eqtxc,z70p*th because of an amazing coincidence. Calculate, using the information inside the Front Cover, tb)ptx)ydl,fz; 3 eq*7b0 smcshe 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 fr23bhgr; kv we*j;8rkzom Earth. The beam diverges akhw*grr8;j bek2; 3v zt an 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 rotate at a rate of xv*- nglzw5t3fxhud0. 8k z2m6500 rpm. When the motor is turned o-vxhnm2 kzf* lw.zx50tug3d 8ff during 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. If 4hrv7 zk k 5*3ohg6wmethe ball makes 15.0 revolutions, r4kh 6z*o e gwm7vk53hwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How ma-ayn3flxst0 -gr3 eyt7i y.8 g e6jgyu* (.oaqny revolutions do the wheesgyqlej-yu x(a.- .tet*g7368a0y n gyrfo 3ils make?    $rev$

  (a) A grinding wheel 0.35 mz1aws+wr/ 0 kz in diameter rotates at 2500 rpm. Calculate its angular velocity+w /1za 0skwzr 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 se m/:;j yt/wmu.rh/) nmntq8gyn4 q8 makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chim)hme j/qw;y: yu88nttm//g nsn .rq4ld 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 velocity of the Earth3io6arw91sf u ;pbj g. (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,mn929q grig)z/ptc)rwf qps.( z ehm/:7rgrnt
(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) mug/et ;-bn *9 -jzenfpxst a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accel-9xn*ftz gp;b/ je ne-eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates k5 e2fod-vhx )*1bhiuuniformly about its center from 130 rpm to 280 rpm in *52)uhhx1fdb k- iove 4.0 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 radiuui,i :kt5b+1r tasm::rfehy* s $R_1$ 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 spacecrafts l,mn:y3rf ew8h .ffrih 6+w8vd8gc7 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-min ti l :h.cw68fdn3g+mrhy fei v ,8sr78wfme 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 b*x- xppzjnkx*) wy70to 15,000 rpm in 220 s. Through how many revolutions did it turn in thix 7y*k xwjp*-0)n bzxps time?    $rev$

  An automobile engine slows down from 4500 rs v dfnx:;gbmh.*i7*1 fi epu6btxk(2 pm 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 stresmbyv)0 v +c;3f9 tn zvbl(g:yjses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cogv+vz mv:0y;bfn9 3jltb( )ycmplete 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 24p (9ks5*afy yx 28y.ol3e pbqr0 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel havbe*(kls 532.yap y9 qfoyrp8x e traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rjun) )koa6* 1vxgy+ha evolutions before it j*a+y1 u)xn ka6vhog)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 fru(zhap4d0 ;ke+vlhj) 2y pblt,3o2y aom 95km/h to 45km/h The tireslp(hk)u+l2vp ojaye2 abz4 d y t3;0,h 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 ybh+6zz *lvky/8hd/u:a bkr*revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have akbhyz/z*v/:a +* 6 l8drh ykub 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 climb5ui6fs.cwas x4 zy/9j8x )7zxk v c4ihing a hill. The pedals r6wk/ xx7z8ivy59)cfzisha 4s 4.cuxjotate 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 9wsv ) yzde; qoyqfj di,4,/e0 rj/h7mN on the end of a door 74 cm wide. What is the magnitude of the torque if the for; eewq,yjdjv/ dq y0r/f)mh7i,9 z4 soce 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 th yve5ywkw45s;q5p(hh,sq-i j e wheel shown in Fig. 8–39. Assume that a fri pijw5(5 ehws5;,s4yqkvh-y qction 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 the ends of a massless rod whicq9rxw;yi +d)e1ee 9uggho7 0hh pivots as shown in Fig. 8–40. Initially the r9;7gi9 r0eex hyeu hdqo1w+g)od is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cyli ixw1uy(+ke(np l). .i qlfkw*nder head of an engine require tightening to a torque of 38 *qi.fliylp (x1u)k ew.n(k w +$m \cdot N$ 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 aygjmwh mpwyy; 4)f.y8h4t c 98 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its c. hm8494wpg hyt) fj8ycyywm;enter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim 0m)m z6en b*w-2b baud u)k nj)up*d.zand tire have a md p)*2 j6 ud*z-za.)m uneb ku)b0wnbcombined 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 tj2mzls b pa0u ;dl) ((yjxy8jvp.h9g ,hin, light rod is rotated in a horizontal circle of )gzljy(2bju.90ma;yh p (x d,psv8l jradius 1.2 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 wheel rotating at con;+hq -/l frfajstant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N r+-f; hqa/j fltotal.
(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 om; fpro: wb r+)m/s.vmf the array of point objects shown in Fig. 8–43 aboprfmvmw)/brsm :o . +;ut
(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 oxygj0m- uh3onkl;en 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 satellite spinning at 36laihazuk;pd,t4 k o3r.8 bdjc:g7y3vuv :m 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 of each rocket if the satellite is to reach 32 ri z:ouh33pjmavut68dc ,v3 ya :k; dr7gl4.b kpm 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 +6fiki h.sosrk97 9ed mass of 0.580 kg. Cal9es69. kfr+i oh7sidkculate
(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, accelerating it from rest8 7r80hqw mvox 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 hanmb 0*0ggyz -qroi)9nx d-driven merry-go-round and is able to accelerate it from resto b0rz xq*g)-gym9 i0n 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, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is suhjc/z( f t9l4hut off and is eventually brought uniformly to rest by a frictional (tj9lz/cfuh4torque 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 acceler w+gprs ;l67i tiv1 qqgvt(+zwi97nyd pf,03 jates 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 solely by the action of txvghf-hi79 z -he forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45 v9zgh-- 7hfxi. 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, as shown in 1vqp:06fitw0wn jh89 m v*rst Fig. 8–46.8r 0wts0m6vvhj w*: 9pn1qitf
(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 twon(op8uvp kh mep(54 ,c 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 turns (revbmqzs)gy yji 6+ 2i0f3olutions) and f m2qyi +i z3ys)j06gbreleases 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 moment of inerticjaop, 0r5zpdxngb k98g hp.j:7 y;ja;t;vm(a 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 9 xcid.tf,r2oof 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 slippiypgngw/a *pd q i.484c9 ve4ayjuu)c2ng down a lane at 3.3 )qujg9 py 4a24cc*/pa v.gn4du wyi8e$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as l5e -9z n5ebw-tsrcb/ the sum of two tb5wce/z be9 tsl-5rn- erms,
(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 wo1ja. psfjr 1iu.sptuuyk0d ) +*q. vt-rk is required to accelerqs pyk dj-itt1)u.1su*. vjrfp+ .a u0ate 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 1.8see32f edg.-a0 kg starts from rest and rolls without slipping down a .2 d-geasee 3f30.0 $^{\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 d i;wo949 e:crqiqe-h m bd*o4o2r s6ltip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conserer9w iq:-ecr4qd2hmbd69;ol i oo* s4vation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the twhb1: r ;fb9gend of a thin string in a circle of ra g91whbb: ;rftdius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m,vcpmaxdlozcy)8d) 2ncv c0kd4 kj 9972z sr+ omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500,vrpv)zl29 m82nkojcx z y0cs 7c ddk9d4+)a c 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, on a plr(*xuci b1i w/atform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreasiwiu xc /r1*(bes 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 4k, u7je 2ns-y* wdbumher moment of inertia by a 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 he*e7b,nkm2duj w-4yus r angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial ragkow n r odx +s4;.owus*).y7dte of 1.0 rev every 2.n 7wwgodys.ous *) ;ox+drk4 .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 rotau jcxi8* v2wj4ting around a vertical axis through its 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 cent iw2v4ujc*x8j er 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 su3:5-xzzp s, ;7 nxcai y:zjrcpinning 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 momn4hr0a4 2wxd:khkcq5entum of 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 isd*5xv9cm dsug sg,m .9 dropped onto an identica*m gvmcsd. x9u9d 5gs,l disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ( v*uo -8zctu65k f.zhnkra y/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 rotatingar7s: dgml :u5 merry-go-round platform of radu:rgm:l s7 5daius 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 rotating freely with an angular velocity o98;/kyo:d,8tdxa dwcq w l7 g8rh)v cff 0.8 d/ylqhfc9xra87wgk d o,:);cv8 td w 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 event/rnqea+1ag x5ually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its ex aer+5a1 g/qnxisting radius. Assuming 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 exc:cs(rd k:dk1la. np.xess 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 cjo6b7twvd(* $ 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 a gq +t9.qps)pyt rest, that can rotate freely without friction. The moment of inertia of the perqstgpyq. 9p+)son 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 the edge of a 6.5i ipz : hwd 6r0lf ;j/r3r)zz4y0th3ci-m diameter merry-go-round turntable that is mounted on frii z03r/zhfc)jtzd lrri piw3y0:46 ;h ctionless 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 grogv1/bk3j -3o w3tngxkund with the end of the rope lying on the top edge of the spool. A person grabs the end of the rokgbt-13wx3/k g3 von jpe 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 mass move?

  The Moon orbits the Earth such that the same side always fac7vp r8*z4u9io /gjll des the Earth. Determine the ratio of the Moon’s spin angular momentum (abpo 9/*dllj8u4g 7ivrzout 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 a rat4; m bzpvv 19+3 rssao(p3ffw2 hju4wae 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 unif3o; cr*h.v ;zcicq6el mi3/jcorm cylinder of radius 0.20 m acquires a ro3 * z6o/i;i; rcqle.h3jcvcmctational 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 z josa/nzuu -uj-48 . mmk7yv+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 speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it/ajk87z4 yu n+. u-vujmosm-z is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of thert*dx1 him2ua/ yvnadx3 lr(x;0w s :) 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 ai 0s,tb2fc 9rk+ 9zhuas 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 three-quarters of its mass in the process, what would its rotar 9c0sz,k+ f92hatiu btion 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 autom xp+/xs gb/oj6obile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindricb/6 gjso +p/xxal 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 illustrah,e5scd b ) tt8k*qj;ctes 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 speed v=3.3 e(zv0o khnxtc,o(s q/89,mv2i(tq rp$m/s$ 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 k8oc ykmi+ njq8::coe),w qu1 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 thenc i w:um )c8k,8oeqjy kq1o+: 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 ver12u:o;ae*own/ymo dn xc2w; n tically on the floor, and we waoa1n2n / n*d omc :wo;u2;wyxent to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelin9zampo mzx+s i)t )8i2g with speed v=4.2m/s on a flat road is making a turn with a radius The fo)xostm 2 89+iz mpa)izrces 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-kgw ;w1g* x hk bh+s0izguh wh:,v*s6p1y rock into a sling of length 1.5 m and begins whirling the rock in ahgivs0 pbgk,szw ;uwy+hx1w1:h 6h** nearly horizontal circle above his head, accelerating 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 rodsh(lw6ja-hc/( r:we 6jkq ln )s, making 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(njqwel -:(6 caw6j)hhlk/rs .   

  You are designing a clutch assembly which consists oo ck :and:;v;y do5gf,mu 4p8ef two cylindrical plates, of mas8: gd nfmo5ad : okpv4,u;;ceys $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 hr+(+wdwx5t u*j:e tc of Fig. 8–58. What is t*j5+ (+urx cwdtwh:te he minimum value of the vertical height h that the marble must drop 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 not assum er: nda.cmvt 4bb1jlhaw.n5+n )y7r ;e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformbrbn e)wq7+:zly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acce7z )n:w+ bbqerleration 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