A bicycle odometer (which measures distance traveled) is attached near8ynqp,f)0hsu qlvl t) (,:ul*3s lhqf :qn; ic the wheel hub and is designed for 27-inch wheels. What ha8)l:fuf3i:hutl, , s(y*qhn ;nqcqs p lq)0lvppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at convcn;c.rlt 7 f/yx 5fk3stant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude l. r3 ;nt7ff /vkcxyc5of either component of linear acceleration change?

Could a nonrigid body be dedggp 1)oup 9yv) s4ud0scribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exe+ljzeoeq6 g k3ivb63l/z.mg 6rt a greater torque 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 your head than ha:, lfnjgv7 5,oi7hut oew,a6oob6 + when your arms are stretched out in front of you? A diagram may help you to answer th,bw7nloa o+:6o7 giao he,u j6 tvf5h,is.

A 21-speed bicycle has seven sprocket1t mx.se zbj ah5vt ye:j31/0o2nifil*:k gy;s at the rear wheel and three at the pedal cranx2o.k:* j1 a m3 t5;:ibgsz/l vyeyehj0nfti1ks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hzfwyn j0m;2 .9nj9l0k 8(jra vjv2y mdave slender lower legs with fleshfaydvljv.8 ( n 0rj;mm2k90nwy9 zj 2j and muscle concentrated 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. 8–35) carry a long,iq*(qcn*p j p3de-xr: narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net t ;xl n02gmt dde.x5n 6oy:ohu:orque on a system is zero, is the net force ze:nyl2tg: u.0dnx; oho5 x6 emdro?

Two inclines have the same heyq kbtv 1 usq .a:yc73m*ud,+vight but make different angles with the horizontal.ku*q3. ,:bcu +7yatv1sm yd qv The 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 incli e34zli p-jh(p4muvc 4ne. One sphere has twice the radius and twice the4e pjcu3 4mvh4l-p (zi mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder*;n- 2uo r aejqy-xr*z have 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 spea* y qrr *-eoj2nz-;xued 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 moving or1 * kn-d;ufmjf rotatikm ;-1fd un*jfng objects eventually slow down and stop. Explain.

If there were a great migration of peoples-kzhh6427mwwyb8w v toward the Earth’s equator, how would this affect the length w h7v2k m6s zw4-bhy8wof the day?

Can the diver of Fig.ef 0f2jimk5bb mo- ;c8 8–29 do a somersault without having any initial rotation when she leaves the boa j2mb;cf-b e 5ofi08kmrd?

The moment of inertia of a rotating solid disk about an axis qh)m :wo.t,x ar9-yz fthrough its center t,.oq9 xh yr )za-wm:fof 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 eacy2k6ru;-q x)(vlfe/e 9flt gf h outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decr9l2ful;tqy- /f) xgveer(k6 f ease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howxcpc 1rt ussdag3aq*145gt oad6x7f l-*: lb0ever, one is hollow and the other is soli*spx1o g4b*lc :1atu6fsd ta57q rlcx-3g d0ad. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating hdx i)u4difg7lry)+:on a horizontal axle points west. In what direction is the linear velocity of a point on the tf7 dgy xdl+4 u:)rhii)op 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 turntable. Whovb7yvs;2j o -at happens if you walk toward t7-j2b y osvvo;he center?

A shortstop may leap into th0 tlu5x/dd2wk gqz 6.de 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 y/d 05g wl6kdxzu.d q2tou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angugc/ 4jq)(c*h lz)q ggnlar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one orqc))zl gg( q/ cg4*jnh more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30p:15we.-j- zvdrj uk y $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Cg6njotl:qke:8 2983ga ahi imalculate, using the informl6g okgie: 8h 3:9n iam8a2jtqation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directemcy 4:t5 sgexxp56:/+a h34bjesnut6 o+v gbkd at the Moon, 380,000 km from Earth. The beam diverges at an an4hx:t3 p utsemscv5gyk: a 46e x5on/+bgb j+6gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rol uxk7 .p0wu86owr2,rz )qz putate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the an)zw p,67ux2 lu rpoq.0ukz8wr gular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Ih sgt:*.uxdq* e, pom,f the ball makes 15.0 revox:um*p qg,e dt*ho,.slutions, what is its diameter?    m

  A bicycle with tires 68 cm in1 mk:1v4wt 5zl*xl 5rncfbic8 diameter travels 8.0 km. How many revolutions do the w5c f1bxc *81l5kw4znvl: rtimheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cal*u9sh.j r533ia ewb7jbg(kbnculate its angular velocity ijnkawe5 *3i3u r7 bbh(s .gb9jn $rad/s$ $\omega$ =    $rad/sec$
(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 comp nlne/-qs3q)vlete revolution in 4.0 s (Fig. 8–38). (a) What is the linear spee)3 nnev-q/q lsd 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 velocity of the Earth (a) in its orbit around the v;tzs/ngaxl+ 3c.rw, Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 5 v,h-)mws)g ijn bb5mpoint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from thex/b1 nfu 0jwa9 axis of rotation is to experience an acceleration of 100,000baw0xnj/u9 1 f $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cent(0u )w( s-w .dukwc/qhsi kbc5er from 130 rpm to 280 rpm in 4.0 s. Dete /iq wu-h(.bc0w sukw(5ksc)drmine
(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 /k j)cvgq; 4mr$R_1$ 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 spa af .3y*,gaik(r50 mtmg rn-zfa00vxocecraft 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-mi,m5f0t r vmr0zaao* g3gkfiny .x0a(-n 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 acceleratech j iin0mucd:/78ta ;s uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions didj7c0tcdih/:8 ; mnu ia it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calc22 mubggmhpsi( ly1eog/.1 w7 ulate
(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 flyin-bcxh/t dw d:7br35g ig highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 r xi7 :-5cddghw/bt3bcomplete 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 240 rpm7r s wv-y,hgab45f+bb to 360 rpm in 6.5 s. ba7h,gfv 5r yb+bw -4sHow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned )bts6c dty/z8*4r xxc off when it is running at 850rev/min It turns 1500 revolutions before it comes to a s6z*cxt)rc st8 4/bxydtop.
(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 redur cs6f+h/)w +2uzq qnfces its speed uniformly from 95km/h to 45km/h The tires hqqnr2w)s u+ ff hz/+6cave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces i/(hh wn0 vreq1* 2wzqxts speed uniformly from 95km/h to 45km/h The tires have ahw/ qve1q0xr z nhw*2( 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 onpaore jf+aar1 *y* t-* each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.j raf 1opae*t-a*+ry *
(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 thh4n8txe. dnvtt0 ,c2 xe end of a door 74 cm wide. What is the magnt0e4xhn tnc 2dx .tv,8itude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque aboj z5 ph;7cmtu1og /1qout the axle of the wheel shown in Fig. 8–39. Assume thm 7 jo/ctuz5qoph11;gat a friction 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 o6sk2mfkj5:(exn2te ;6 yrqm (iqrp j3f a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calcula ex s(6:mrm6(kqf2tq3;pkn e52j riyjte the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requir9 (re36ubv*hkai x,h4.ja ayykf i z/7e tightening to a torque of 38z ayi(v faah6kb ki.,ex/*u4r hy93j7 $m \cdot N$ 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 wvxtrbt.,* (a do wwb-i31j, pof inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throu(,tdwt*px a bvw r 1.o,jb3-iwgh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheeif(,l :xlf g8el 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can :i8fl(l exf ,gbe ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod isf,by,g2oxpb . rotated in a horizontal circle of radius 1.2 ,b 2yogb fxp.,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 whl) ls2ltda-4:4uis st ei 4o;.fm4wnoeel rotating at constant angular speed (Fig. 8–42). Thuel 4)fl-st dai;ms4i4ns lo:t2 4w.oe 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 shown in Fig. 8–4p.wx 7:1jev uf3 aboup:vjw 7feux .1t
(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 total mass1ne7)m 8fp ;c(vcyvb y is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellit 3m5x9( rohvrp/ l ncbz6eum7.e spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is en.r 5xm3c/br v( zu6mh ol7p9to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm vk9r + eqpb):e(bqzee3 d(-d tand a mass of 0.580 kg. Cal()r+(-9 q tedbvb:ee 3k qepzdculate
(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 rest to 3m1whss 1.3zto $rev/s$ 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-round and +-flqbijt lot7 i/7idaqxm3s(6) w x1is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. A da) s q+j/i3l1 76 q-o7lbxmf(wttiixssume 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 off apz(m9ij2 svb* nd is eventually brought uniformly to rei2j (9mbv*s pzst by a 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– phwp9+e ,5fz d5anwiz5j1r6 z45 accelerates 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 sol3x x*mn-qvaf 8ely by the action of the forearm, which rotates afqxxv3*na m8-bout the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thiub vs:; 9 oek/d wa7gz y).+vi5nj*zoyn rod, as shown in Fig. 8–46.i*5e/ d.oyv wzanj7s 9 k:uo +b)g;zyv
(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 massessd:nk.* axf; )hm1k 3s g yq57fctpa;k, $m_1$ 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 turnb xs:2hdc,vw6 ew7 jm)s (revolutions) 2 ew7hjcm ,dsx:6bw)v and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mof x6,xf gunw-7ment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torq5y p(a:/ v t ,wxvwnuxfb-5xxk+47qk tue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipplqhyb7a fx11u(j .k)ping down a lal qp ybu 1hja(xkf).71ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respecp4*vdn9 z:w:qm -tq c-n gsax2t to the Sun as the sum of two-*tc: n:v d -qsq xnag4p2wz9m terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wor i7))l2f 7xrrigbk 6ihk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a sogh ikb6 72i7xi)r f)lrlid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wi7hzus0.lx /g8: utjt 7v7 zaqxthout slipping down a 30. a.v8gzqlxthusj:7 x0 /7 tzu70 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and v wy/n+rh, c*1zi4xb o9ny,fmits lower end does not slip. What will be the speed of the upper end ofvrfn omw91y xh4n,,i/+zcb*y the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mome: otl95x;ldvvk *ce;w cens*hb u69 v/ntum of a 0.210-kg ball rotating on the end of a thin string ine5clhl9 v:k;tsd* uoc*/nb v;v9w6ex a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel 8y i9u0.2kv/g wd/*ylxbdcb gof radius 18 cm when rotai2gx.lubgy /8kb cd90dv/ wy*ting 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 p/ pry6hgytnmalixf *)nit5b2(: *j0is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases tj250lr/ai inp( gyntym)*b:t6xh * f po 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showntj 4a,g 4 oo)far-kfb- in Fig. 8–29) can reduce her 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 her tbgrk a4j -a4o,o-f)fangular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 cu:0 3bqmbcdogn 08:lrev every 2.0 s to a fibl:c :0g m8ubdnc0q3onal 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 center at a ff:-b qe/xlh8 r muta4 gm+jnoek, .k.;requency 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, appmt a4j :+,8; fekbn..xemu-rlk ogh/qroximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figfqror )f 4/91mvj5jzmfi-qb 1 ure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of txx* e ;hw)s4p q*px0flhe 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 droxu/p;vo) 0bfy70. ml - kn6(stpr rridpped onto an identical disk rotating at angul f 0p(k od0/vi s;b6t- 7rn.rmxp)ryluar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnszy(xl3zv(crl z9aa 00 at 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-round platforx 7uqn)rjeymo y9g 447m of )roy7 yg nejxm49qu7 4radius 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 rotatinyc45 bgm2egdm3 /;cvrg freely with an angular velocity of 0.c 2;354ygv gmedrcm/b8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half i yjdty . jm :bt4/-wy5+) edf)ee0daqmts mass in the process, and ))+4edddmjte /m-y ya.yqjt bfew0: 5 winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 winxemtq.ux +l-2 u05 /fg bci8wyswqz8.ds 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 k80u4e80giv3 pkgi50;1yboltm h juh$ 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 withouk +)j gl*u0w wcbg/3hy)+ecoqt friction. The moment of *ulcw) e+j b/q)y0+oc khwg3 ginertia 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 th7h(9qcp vd.hjd xo:2me edge of a 6.5-m diameter merry-go-round tur( cvo:pxjhddm7.hq 92ntable that is mounted on 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 gr99xr+dm)ak; ni-ou bcound 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 lengtk+dx;bo n-) urai m9c9h of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces thu 7p+ jr(rej5ve 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 orbiti uvre+ p57(jjrng the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at yjd,s.0-4(ytriihc pr9 xnu-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 uu8u x;a.orl7i yw18xrniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculw ;uu7o x.ax8 8y1lrriate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each ofk sjunuds3vb5 6src)iy c)-00 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,3r cj5c bid6nu0v su-s) 0y)sk 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 of thelqc-2- v al:uu 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 g50,w lk)9t2l/ gy-iux.idm)wdgxkb of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 rwtlg m0 x.i)w,g5b9i2)kk -gud xy/ldevolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobilea)c /vog r/*j fay:uz, 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 y/z/uj g)rcfaoa*v :,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 illustrau2h87hij7 e6jobk6 qb4u ay-eos*ax dx*yu: ;tes 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=31hcjsy .tez7,3qny ayl,qs .8 .3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.op ep4fc94 pps.u:wk8 e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held hor 9sp fu:ce4pp 84.owkpizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has c.( uws5tvr mn1ykr:9 h/mhw; radius R. It is standing vertically on the floor, and we want to exert a horizontal mrkmn u.h5c s(;1htry w: /9wvforce 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 traveling with 0s. c ufq80fulspeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the0.l0uqf8 sfc u 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 begins whirlfz kmv4iy t2l u.:hrzc.0m8w8ing the rock in a nearly horizontal circle above his h f.t iyml8r :mh.zcwukz0842vead, 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 rods,i;pj3mnydnvbd m 9g n8r(e1,0 making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms ( bv rn,38dy m1pm9ni;gndj e0held tightly against her body.   

  You are designing a clutqk f:w-mt;vr+esh )0ach assembly which consists of two cylindrical plates, of masm r:)-;f +ek0vt ahwqss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollp1e q1gsvi 8n8s along the looped rough track of Fig. 8–58. What is the minimum value oq81 npes v8gi1f 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 assume+zx7b fnxj;pll5tf 41 $r\ll R$ h =    (R-r)

  The tires of a car ml+0kswf u -c7sake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires hav- sl 0wc+uks7fe a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s