A bicycle odometer (which measures distance travwl 36qd;8c p j+6uzibln3i: :xhu)ndfeled) is attached near the wheel hub and is di3wjhpludu8c: n6l x: q 3dnif6;+ )zbesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant anp aovv6 ilaec .to5r9g-:.)rvhr5xv3 gular velocity. Does a point on the rim have radial and/or tangentiall rgc5v6vr vat e5-)pxo .h.3iavor:9 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 described by a single value of the angula+:)0bosefx 4hikh :a*qrz*c u r velocity $\omega$ Explain.

Can a small force ever exert a greater torque hsg3 dv isaxmt*3lp,a ,7 f vi96r2h)ksmj70hthan 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 sit80uqse :c k8ncoo (egu v x.b2uc/pg9)-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer t2cco9v)gu0u.:8pck/ bu 8 o( q xesegnhis.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedazd6g qpa 9e+ 1wba2hf5l cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In 2bd 9z+ 6ephg1waf5qa 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 of ;-nry+(yws fast have slender lower legs with flesh and muscle concent;+y wr os(ynf-rated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–imvc v:cp,v0b6xdh2 :35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? :i n2we2df4jub2ky78x p6 mhrzz1 w xuv2z s-;If the net tory s1z48u2i2zr;pwdhx kb 6en2j-wuf2v: x 7zmque on a system is zero, is the net force zero?

Two inclines have the same height ;u lh-bdvt dr9ea 4:tb+*m7w)kz3rok but make different angles with the horizontal. The same steel ball is rolled down each incline. On whic 3u)*oarvbbh k9erdw-t :dz;7l km+t4h incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. 7. o98h hjsioxOne sphere has twice the radius and twice the mass of the other. Which reaches the bo.9xhho s87i jottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They sta)4 wv5fnsz5pv7tf l9u rt from rest at the top of an incline. Which reaches the bottom first? Which has thetw75p)z u v4n9 lvff5s 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 angular momentum are conserved. Yet most movi: c/vtq q940(,pol whs(.b tond aw*q3tz +evwng or rotatttd39a0 zq *ebw(+scq w(tn,vvhqwl:poo .4/ ing objects eventually slow down and stop. Explain.

If there were a great migra qano- vp/ck(-x:n)omtion of people toward the Earth’s equator, how would this afaoo q:mnk)v /xp --(cnfect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatf (jt2i1tv8cpion when s vp f1tj2tci8(he leaves the board?

The moment of inertia of a rotating solid disk about an axis 7e94uva;t up d/z,flc through its center of mass9 ue uva ,z/f;d7tpc4l 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 each outstret,x2da j6svm hkdw6r/7ched hand. If you sdvxj7 wa26,dsk /r6m huddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeveibo g6qwzkk8k4aw):( r, one is hollow and the other iswgok(k 4)q8:zb k6w ia solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating onmyh q snce6p.n:(yw 2./f3wg 8vkxrn* a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point agfq2ye. 6yc* nskwhpv/8m:nnr (.3x wt the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a larg4sd9:xig a ,gte freely rotating turntable. What happens if you w9d tx4g,s :gaialk toward the center?

A shortstop may leap into the air to catch a bk,0(ego7b93 l8n ovrahapa d,g:qol 3all 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 thoea,o rgbh ql k7 9g:a3d 8(0,pnao3vle opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wjg6n,5u pmkyct 05hb 5hy a helicopter must haveu h555n60 typmjkg ,bc more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianshn6m)d+xcrm: h*j4s+ fiap ;u : (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because ofw*dx6 ve (k7s3k7nbh5tm o1zm an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seennxmk b56(s3kv1me*7h dtz wo 7 on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed45dt4u )redsy5m4oi y at the Moon, 380,000 km from Earth. The beam diverges at ane5)u do54y 4s4rymdit 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 6500 rpm. ;q60 xf. t,vz:2jtk2s/jk chn-qxoiiWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the ,ijk x;-c jtfz/kno0hts q i62vx.: 2qblades slow down?    $rad/s^2$

  A child rolls a ball on a lev.ww1o ;r(qst b-/6w1zi sgwsp el floor 3.5 m to another child. If the ball makes 15.0 revolutiosq(twsw1si-wp 6. o/z; wbgr1 ns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 kz4zkxf, fj8ta twi.+ ,m. How many revolutions do the wheels ma,.ifaf+,txz wt4z k8jke?    $rev$

  (a) A grinding wheel3y9zwub o :d.wlf t8xk.q)w5b 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velocity in 5w.oytqud3x:8z9bw wf .b )lk$rad/s$ $\omega$ =    $rad/sec$
(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 com .t( 4l21wi,mfy bjb6wlj0rnsplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centerliw24tm1r.n b yw0 bslfj(6 ,j?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) i)eykbh;1438jv n wur ohj.hf7 n its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poit(tug7e/nn5a lvjh:yh 32* ownt
(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- ty3yv64dm8tryertz882e2 g if0klw a particle 7.0 cm from the axis of rotfym3iere04k6tzygtt2 l8r82vdy w - 8ation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleraten b(q)qej k34arel3tru ,;za 6s uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ nj3rlq;ek3q(b,t 6u)za 4 areine
(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 radiuq :vj (rn7qn8: mapnn4s $R_1$ 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 aboardb jv2yvxk1-1vgiuo /3 the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determineib1jovg 2 3yu-x1 /kvv
(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 ): bintp4 t*nmto 15,000 rpm in 220 s. Through how many revom) inp4 :t*nbtlutions did it turn in this time?    $rev$

  An automobile engine slows yx9 r.prz.ii4cf yzzx5a 51+t 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 stressesmk1t;,h ezly/ of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching ,hyl1zme/t;k 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 240 rpe4o phfkg0f09 m to 360 rpm in 6.5 s. How far will a point on the edge o pof4kf 09h0egf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It tus x u ;trzf*xly-) .4e8jpgq5mrns 1500 revolutions before it comes to a j4eq. l* ;z)sfx5gx p8y mr-tustop.
(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 th0h:vs/vgh;xby2 czs8ztc/hp0 -;i yhe car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of hgc ys2t -yv /xi s:z0h;c0zhh;/ 8vpb0.80 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 m5*j) nyr2m xlas the car reduces its speed uniformly from 95km/h to 45km/h The tirenr m5j2*xy )lms 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 aer h8ct3we/w+ bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of wcetw+r3 e8h /radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a doorl;jzh w sb5d/ay* +5rx 74 cm wide. What is the magnitude of tjzy b /;sxlwah+5*d r5he 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. 8–39. z ,;tq;moe;,f*3knq s)zxh htAssume that a friction x h;,3o zh; ,en*;fzqqts tk)mtorque 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 massless5r4mqo. bc9dx rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal 5.bqdr 49xmocposition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine reqa yoq3y ,w8- vo)*-q.mf bhsoal 3fs;wuire tightening to a torque of 38 afo8al;3 .f w,svq-)oy*wsy -mo3qhb$m \cdot N$ 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-ufm q;j xdvh279l+wkh c/q8q(kg sphere of radius 0.648 m when the axis of rotation is through its c;j 9 qqhvldfu (h8w+cq/kx2m7 enter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter w9 hmbz)/pixm*0 ehpc+r/r ;i. The rim and tire haveb0mr*cx +zep/ ;pr)h/m iw9ih 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 the end of a thin, light rod is rotated i8in;q::za s n*/+jvugqfor3gg;tq . z uz9u7un a horizontal circlefgg3uu nqa:nu i;8jz ut*q. sqz g;v 9+:zr7o/ of radius 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 constant angula4rz) 5coj/ k -od0grzmr speed (Fig. 8–42). The friction f-kjo/z 4dr0)z5gc mroorce 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 the1tc g1w *bz x2g;:wgr:r8 ouci array of point objects shown in Fig. 8–43 abr1 uo ggr1 w;2cwcbix8t::*zgout
(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 at elq(c 1; r(y+spb;tc9ei5o fkoms 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 7k.m4jqlwz;g- iy2.vryv -wnc * k:hssatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sy -v*h k7-. nyz:m.2srcwgkj4 ivlw; qatellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rg -3vq-ry y 9*n4z8 /.mimjy/tm6. eysuimgqwadius of 8.50 cm and a mass of 0.580 kg 3y y9y/it/qq8yj us ig*gm.r v- wm6.mnez4-m. 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 :( tak0 3k)r+ wyxiwophv p;u7bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-xb* psi;m*n ;pround and is able to accelerate it from rest to a frequency of 15 rpm in;sx**b;p ipmn 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 shut jxjxt u9d*l9sc )6nz+ off and is eventually brought uniformly to rest by +u* djsc t96nxlz)9 xja frictional 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 acceler,2 fbm1bfo5cfu7fdi8 b( :v.mo tov(oates 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 the foreu+b5ul7hffof xi1e*(46ywo tg87j hd arm, which rotates about the elbow joint under the action of the triceps muscle, Ff8 eo4uyb*uf6o7h i xld(5g jw1+7hftig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, afl0o6 ag8aqyp6a 2 bvfb a.:x5s shown in Fig. 8–46 gby:fla62.foa vpqx0a8ab56.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,i7mkd (w h xi1l:;iat* $m_1$ 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 fu8i ozu1x89at /za7twjf 2 xjjwm-1m:all turns (revolutions) and releases it at a speed ofwjaxf1:m88jox/m 17zj ztaut9w-2 i a 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 inert(2bw4e k/ bzl ykg37dqx.lh5 iia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280a-jka2n5pkj p6e j-p( $m \cdot N$ 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 slipping my 3gvgl2s 3hg:2uvc9ea- 6 lkdown a lane at 3.muv9:l23-6g v 2 eh scga3ygkl3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the n(u35 dxaqks :lo eq*8Earth with respect to the Sun as the sum of two soxk:ad(e8q3*luqn5 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 l y/p iu1axl931up ee4work is required to accelerate it from rest to a rotation ralu 1yaepi41epl39/ uxte of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest a;ask z-7x .caqnd rolls without slipping down . s7z xaqak-;ca 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 tiw8lqilz:(:)kv 0 - xwakboh6vp. It starts to fall and its lower end does not slip. What will be the speed ofwb0 lzvi)k(x 6-:hkl:aq ow 8v 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 wq))e bzv00q/xqq ;4owznjq 0,uwmo)ball rotating on the end of a thin string in a circle of radq /)n00 , qxo)mqz0wwq uvwe)b;q4j ozius 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 uniform cylinju1u.5mjwiw/m4 d.g,hj lr 6 bdrical grinding wheel of radius 18gu5.rmj wulwjm,64ji/ d.h1b cm 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 at his side, on a platform that is rotatingeo bk5/lb)d uyn26 8pju(my3n at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the nbelm(65/yy3pb8 u2kn) j oudspeed 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) c rys 9,+8t8xd ahxfc/ian reduce her moment of inertia by a factor of about 3.5 when changing from the straight position tf98x xty i8hs+/,a dcro 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 spins5:-)xqox ;k .iosygf rotation rate from an initial rate of 1.0 rev every 2.0 if x5.;x :ossygqo-k )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 its cenf 193v0djv bnfter 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 b9d3nf1vfv0j 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 angulnk po8.xb-,+)r4d z hl quot1mar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angularmzgtsn6buj,d8* y 2.fycd -f 6 momentum 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 is dropped onto an identi/ illb (xd8w)bcal disk rotating bx)b( 8d wli/lat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns u -tz6ol h3fo/:5 /wa:hmmrfnat 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 sjsju/r lq m2bt g bm v941q91hn:uy5x2tands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of ine uxsqbmy5 u4/1tr1 njhgqvb92j9 l2:mrtia 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-roundz6uf lp:f4n upfnh 13u+f z+)z is rotating freely with an angular velocity of 0.pl+f41hf n fpuz zuzu+: )3n6f8 $rad/s$ 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 whitsj h4y)4limvoz(m +*te dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass j*mz)vml4hiyt 4s(+ocarries 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 excess of 12yn634m i fsm:r0 $km/h$ 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 lt;v:s(znq;icz3gz3r02sj ppq38b2 tn u(gs$ 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 kxrr w,v1m;ymkj9 rz27 )z8 blrest, that can rotate freely without frictivrkr )j18bmry, l ;zx2z9m7kwon. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at t+it )rxn.x y4 bkv*ri5he edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlessv ir.*+inkb)4yx 5xrt 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 grou/yyvp,ez4l c 2l.yvy4nd 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 slipping. What length of rope unwinds from the spool? How far does the spool’s cl4vy cevy/l y4.yp 2,zenter of mass move?

  The Moon orbits the Earth such that8d2yb w ddo0ox*2)xn k the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own 8ywx 2kb*0d )2ondxodaxis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromby lp-y 7j-ewq(l84qx rest 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 shap + /*1yr w(a8koupwdmhe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at c/yuhdww8ar + 1p( *omkonstant 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 mas*y/xvi c 8kr/ss 0.050 kg and diameter 0.075 m, 8cv/y*xs ri k/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 is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed w384m;z hpwrkof the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, we5f (t g wgkud0d*f ifxy261zl,re rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gd zfd,k0txw 1y62il(f5fg* u 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 wrdqh/ ;fa;j,4v j/ xbautomobile 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 cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup/hqajrwj,b ;f4xd v;/.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an u2czm0q g4.lp$H_2O$ 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 at8(brz;lhn2/3t tce mo 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 pbpk 97wpdffi0ki+ ;t ;ivot 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 the center of mass of the rod, as shown. [Hin0b i;k9kp; p7idtw+ fft: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floeqn bpcx(uy xc, v:9)4or, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The sc)q :nu xe xpc9,4(vybtep has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making p wy/7trinme ++ic(bd7bdb3+ a turn with a radius The forces acting on the cyclist and cycle are 7w 7+/md3dcybeb tr n++(pbii 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 and begins8hw+-z9kofs61yc )y2xivn v u whirling the ry+x 62f-)yk8s9u1vc nh owizvock 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 required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a s os(m35cw:chml, rtmg1*b- fuolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for bo3u fm*(lrg1s mmwt-h,c5c: 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 w,t* *of3x dmkplates, of ma,xtd**3o f wmkss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trackl.p snwp )i;ps s)d5(rfuvl) 8 of Fig. 8–58. What is the minimum value of the vertical height h that the marble musns)sr lpl ppf;uvi5 8sw.()d)t 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, b(sf 9ee.o2j1c)iq-t d zg6uo hut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the chfwf4f2 6+xjkvd g;.oar reduces its speed uniformly from 90km/h to 60km/h The tires have a dia;ffw hxd2+gv f46k oj.meter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s