A bicycle odometer (which measures distance traveled) is attached nearu 0d, 4uigv 4rjckz(0v:wv5qu the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle w:vqz u0w (k0v4dvcru5ij4,ugith 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the3wrvhj1t5t5r / ( efjn rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tange n fttr5j wh(5v13ej/rntial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desclcrh4 rtx5i+ a)47nrgribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater tor/a 7c npilw1,hque 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 headz2kh z (wv0-/fg7mqqk87r dot than when yk2m dohf/t 078(kz qq-r7wvzgour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three avb6-l/1*wd 7auu1ot e oicj:ut the pedal cranks. In1u iuawdcb1t u v*7 :leo/-o6j 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 have slecmm w* :ux- p3xv-n-uw)g5yu ender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis oun5 v-wxyu: ecp)m--w*xg u3mf rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 85r ,zfuoq fr53–35) carry a long, narrow beam?

If the net force on a system is zero, is the neydm u/72llr88 wyjc )nt torque also zero? If the net torque on a system is zero, is the net/2crnydy) wjlul m788 force zero?

Two inclines have the same height but make different angles with t4pa rv 5q 4iap.xja8y4he horizontal. The same steel ball is rolled down each incline. On which inclpj58 p4r4aviy 4.aaxqine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frous-mu.h vgu);m rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greateuu mv;hs- g.u)r total kinetic energy at the bottom?

A sphere and a cylinder02mrrn;k uop;8iff(xz y6 x32+rnfqv 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 speed r fu 8vxxzr+0rm(o 3p2n f;k62fiqn;yat the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angulacgqa 9u5 i(:op4ne ar6r momentum are conserved. Yet most moving or rotating objects eventua5erq(o:u aag64nip9 c lly slow down and stop. Explain.

If there were a great migratqw n183 8 ywsysz)4zyzion of people toward the Earth’s equator, how would this affect the length of the dyw8 8n)y4z1wzyszs 3 qay?

Can the diver of Fig. 8–29 do a somersault withk2r(*ghef)e 0ffkqt j 4kv j2*out having any initial rotation when she leaves the v2retkegjkfh2)*jk*q 0f f4( board?

The moment of inertia of a rotating solid disk about an axis throughls5(/qsly mq1 i7- ibq its center of mass is 1sb(mq i-/yqi5ls7l q$\frac{1}{2}WR^2$ (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 sittinv0lji9yd8+ b5 kj6we sg on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incre9 y6k+5v il0jbsd 8ewjase, decrease, or stay the same? Explain.

Two spheres look identical and have the sam * 2aae) l sw.skwovrxtx6902au h+3gge mass. However, one is hollow and the other is solid. Describe an experiment to determine wu as2sx)hxgv6.oa*+9e0gw2atk l rw3hich is which.

The angular velocity of a whee0jr7,l r b*hm)loej6x4hin *y l rotating on 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 accelerrhj7xn ,4irm0 )lby h*jle6*oation of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standingbwh /r 73rea1i on the edge of a large freely rotating turntable. What happens if37i/w re rb1ah you walk toward the center?

A shortstop may leap into the air to catch ax2ix 0-y9a9 ;m/n(x *rncrn6b dbco im ball and throw it quickly. As he throws the b2mnb9 rxbyx(rioi*6cc9-/ ;a0nn m xdall, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a xzrq nl*:x 1j3 brme+1t,zj :zz+sl)d helicopter must have more than one rotor (or propelle) ,j1xzxr:j3le 1szd*z++ztln mb rq:r). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianstram,cz8egh ,f qcb.ob y *909: (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 of an amazing coincidence. Calcucr) nitsvi.6(p vyqv:wh3 :z.late, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as see. c3z tvisqhn6ivyr ):v(:p.wn on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fromk0uql1hhoh;odfhnf( :q ;ow- z1d:(y Earth. The beam diverges at l ;( z1(1hkdho-hudq:o q0; :hon ffwyan 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 o c1.(i*d 3lw)lk v,v. tztzz 8dcjde/gf 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slo tldgk1/.(w*3vd zv cl).cz,8e tijdzw down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m tya:a,e/bk w hx* xcmy d.w:.*bo another child. If the ball makes 15.0 revolutions, what is its diabxwd x /e**w cbamk,: ya.hy:.meter?    m

  A bicycle with tires 68vg:5 ro9luo 0eezn40 o cm in diameter travels 8.0 km. How many revolutions do the wheelsv on0lg eo04r95u oz:e make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculn5;u2 eaq32 dty:do qsate its angular veqqntyoae:d;5d su3 22locity 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-rimj4z2 w-vg4*fw lr+ kound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is tjzmw42v+r4 - k* gilwfhe linear speed 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 th j(xk63y/(g iaz+xljl e Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speelc obr:acws./s0k0n *eol w 4y. xg)a:d of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centr9;jl8w5,hd8e5li drq e9v*di ,bb vgkifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1d8d;j*h8 9bl5 ,e dgiki qbev9w5vr ,l00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpmus 6 h)3 6 sjmiv4a*vx2ccvth( to 280 rpm in 4.0 s. Deter smv6 4 xv6 )32*uchjshitv(acmine
(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 radiuscw- 8.r:itz rw $R_1$ 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,j*.asmtergkcu8jv t 7 1+ma23 astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, thkm.2*ju+ eca1j gm strv37 8atey accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 1 zxyy .ry;: a6m0-xaqp5,000 rpm in 220 s. Through how many revolutions did it turn in this ti:mxa-0;y. y6qpxzy a rme?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 r()5p8j8)nmjbol bbs wpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whir wy4yp pbzk e5r095jivw; ed73ling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final 7yj5 repw 3kb9i5;e pdzwvy04speed.
(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 rpm tozf g -cvq*mr:3 360 rpm in 6.5 s. How far will a point qcgrmzv-3 *f :on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runniol )j: : lr7*d7z nxqvbv(gxh,ng at 850rev/min It turns 1500 revolutions before itld,(x zhn *voq7lj:7v r)bgx: 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 caroy*4r+38 o 6vt epp/ytdfa/t o reduces its speed uniformly from 95km/h to 45km/h The tires have a diamet3at64*8 p /dreyyt op /+otvofer of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its suk04r5dzee;aw * zc o:x o/m0cpeed uniformly from 95km/h to 45km/h The tires have a diameter oza uk00rxce/;c:4mez *d wo5of 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 q.z/tb q:m ia7climbing a hill. The pedals rotate in a circle of 7.b:mq/z q tiaradius 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 door 74 cm wide. What i*yq1 wzdgdf;3qo qt*zrn+4tw (.f* hk s the magnitude of the torque .*zdg 4 rwyz1 d*wft*h n3k;o q(qqtf+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 iw ,99wijffuu0n Fig. 8–39. Assume tha,wwj0 9iff uu9t 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 of a masslej wu4epntuxmlpk39uk7/ :d;-3wg - lpss rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and xu-n uw/djpk mlt39e 7 lkgp;:p4uw-3then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening;ts+p jk ah4d9 to a torque of 38ksjp a9d4+h ;t $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radxhow4x -dx;jgai;.a :i5 ytm 8ius 0.648 m when the axis of rotation is through d yhj4:gioxit8am;5;.a wxx- its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim 6p3 5 ),blbxx(gswytqand tire have a combined mass of 1.25 kg. The mass of the hub can be ignoredx)wbyqb3(ptxg6s l, 5 (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, l :qgqr.x b4ta;ight rod is rotated in a horizontal circle of radius 1.2 m. Ca.qxb q:4;ragt lculate
(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 potvk2):.uqy rqnb+ p;gfter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.n:kbq fqr.u+gp y; 2v)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 ob)coea ;oo/jjaen , 3q+jects shown in Fig. 8–43 abou,+anoa/jc q) ;e ojo3et
(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 mass,b)jwh xy yuw qu4i-,-21iv wb 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 +qudt ns91y86e io9kr satellite spinning at the correct rate, engineers fire four tangential r69ety k9d i+r1nquo8sockets 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 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylu ec ;v:; n-cz1yf 5t2b2gaxruinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcula -trngb:c;15v u;xcz a 2e2fuyte
(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 resti-iyk 9 b8) lepxt3 ,zjy7oyd( 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 smo*(4 r/ pawqdc9hjuimh(/zi4all hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposit9 i( /z(pw4h o *dhijau4rqcm/e 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 and is even bpd9amk.iwfyu7( k1f8 g6nn +tually brought uniformly to 6 9.b1nknyff gmiu(a+pdk 87 wrest 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. 8–45 accelerates a 3 z1- /)zt,sso up.)rakwp+l o em+vk/v.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 (/ hq7zxm*xvdthe action of the forearm, which rotates about the elbow joint under the action of the7vmd x*qh z/(x 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 consi cjm*v-xu .6qzcii47 ydered a long thin rod, as shown in Fig. 8–4c q4x i6y* .-uzmvci7j6.
(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,qz wt 1p tz2yq,xk4zdv.u xpf0dpu-jy9 3b/4+ $m_1$ 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 accelerar8)c vi7 ;yulc.t8li rtes the hammer from rest within four full turns (revolutions) and release ) lic;8y7r.ltcr vi8us 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 moc8wp o5,i y0jxment 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 torque of 2;q 2drd.xccr .80 $m \cdot N$ 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 b(c3j*cnv,2yr .8wxu oduf6t rolls without slipping down a lajwd2cbnf* 8u3.v oy r6tu,(cx ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect1liw 8pp)hbo 7;l s9eqyv.rj) to the Sun as the sum of t1rewvs. h8)l )op;j79pbyliq wo 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. 2azul* 5yaztxw4x r jyk.)7-gsjae. 6 How much net work is required to acceleraz*x. 2z)yja ga.aexwk 5-ury6tjs7l4 te 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 1.80 kg starts from rest andovv5ym4/ cyp5( 1p( qp/qt2raf hu3w l rolls without slipping down am pw2 pfqyryv3(4th5puo 1(ac5l/qv/ 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 verticatuwk1d8ck n/0 ai u53qlly on its tip. 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 conservation o/c0i n5at8qudu3k1w k f energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end ofr0db 6a 9zj*;zf sam6kyrk1t2 a thin string in a circle of radiusad9k;1zbt2a0r jr6ykm 6 zf* s 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angul )hf,-a*nw)sefj tz2war momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when2)wzfj hew,-sta)n f * 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 rotating at a rat/ so 28m x;y,u8u d9l.h- bh)bqdrehece of 1.3rev/s If he raises his arms to a horizonta-bo b8drums8)x ,e;ucheh9l.h y/2 dql position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showdi05zxd6g3xxn z 4llep nf r92.we6*ln 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 angular sd6z9 gixx .34lw2ne5dernl6 l xzpf*0peed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate fromz,5, aus ttok5pad:p8 an initial rate of 1.0 rev every 2.0 zdo, k5u:sa,a tp85tps 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 center k83(m4bm crkch -jn*tade 6c2at a frequency of 1.5rev/s The wheel can be cona6cedkcmt 2 *j4mn b8hk rc3(-sidered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figuejk- y63glaz6re 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 ic*thajwa;/w9pt7 lq:hsu(4 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 identicxl k g74z.knrx-r,bxarc 6, ue/u7 d4ial d,6z lkaie., 74 b-/gcn rxxuxrd7k 4urisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnssof8 d;ve+ anb q* iwx-,n0r/a 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 pl8gssnr. * v-kwatform of radius 3..8ws* r-v sgnk0 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 rotatinavx9ocewjy ;9 k1v4. 6oyxf;u g freely with an angular velocity of 0. eox 9ofj6;v4 yv w;uay9kxc1.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing abou /0f:4ggowcbi t half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away4biw g0fc :go/ 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 wily2 8l4;*9qws s fxspw7imm/nn.d 6 ybnds 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 a 3u0vxixqi (*$ 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 plavthq2( foy 8:utform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform i( 2t:vuho8 yqfs $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round tby*bw nj kphv;xo (dld;.o1d sh87(x5urntabl*ykd ;vd o n5swl dxx8;ph(oj(.1bh7be 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 ground with the end of the ropukpdo 0hw04. 9khkut+e lying on the top edge of the spool. A person grabs the end of the rope and walks a90 ukw+tpohhu0dkk 4. 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 sa uz9a dtm.n.,bme side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentu ,n.dzbut m.9am. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest ,zm 44(zid pia2-cwu9s 6ip lhat 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 u(*7*xuse h;q wy ;bphnlpg . a f8v1kp;d7yh(cniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interva7ge;8p* b1y kh7(cwh*nqxpf(y;; avp.hus dll at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of t, b1rd:p b a9ylxqku9c4jl -9h(jifd9wo solid cylindrical 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 oc y, 9x4lrh (-k9 99apd1ujdbjbiq:lf f 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 sp7qzq1wg3pilc d o 7x p..u8uh)eed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as g-3stdnllyb+-+ )*xafa)fg:zrbo0g x reat, were rotating at a speed of 1.0 revolution every 12 a-s3br -0gz) lb)y+:x noa+*lf gdtxf 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 automobile is for it to usg0- b: q.gjwsf dp9mkod3az0v * ;dxn.e energy stored in a heavy rotating flywheel. Suppose such a dkddn.p:0q;30*.gjgowz a-s xvfb m9car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates aeem .qu(r,, (nknrsq0n $H_2O$ 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 tf)abwra b 5cnxwyjvg+a4; c2 z1( s+3rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot f* ec3ywql;z+dd5nw3)f: riyfd r- x1 ereely (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 gra dydex3+3 )* w 5irzlrfn1-yfe:;dqwcvity 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 wegp.f *5p0 yxis standing vertically on the floor, 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 step has height h, where h*pfgyp e 05w.x

  A bicyclist traveling with speed v=4.2m/s on a flat r*skbiw) *tgeo 1ss,9* t/ba faoad is making a turn with a radius The forces acting on the cyclist and cycle are the normat/a bts*ioswf, egs1bk 9)**al 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 /cnm- c3zzg(q into a sling of length 1.5 m and begins whirling the rock in a nearly horizo cz- nmz(/g3qcntal 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 rods, maki - lf: )sdyefp3qpq1,oz c2/cwng reasonable estimates for the dimensions. Then calc:c-f/ qof,2 q1wdlsypzp )3eculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which 3k u g7m dan3/(5x2b9ncn)ytlk :cpnoconsists of two cylindrical plates, of mass mpl g39x3n5(k)yn:k /tbn cdo7 uca2n $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loop dypkbd).y(k * g npsx33g7vn1ed rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving thp)13syykngkv b.p*d7d x3( g ne track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doaskc)mz1u a9 e 2,ed9e not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revo3aj 6y .mpd ur*ra8xg.lutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) 3mr*ypdj .x .ur8ga6aWhat 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