A bicycle odometer (which measures distance t*z(n-8 k -(o)/fckzk (ywwvac j3(ojmlwj(fqraveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on azo(w)/vf-wj *f(c w 3-a(olkmn8zkjky j c( q( bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocyp:iltndae1*/ fok :-ity. 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 of either coktl1/:ia y fe-n: o*dpmponent of linear acceleration change?

Could a nonrigid body be described by a single value o o*o9 msap8(7j52 s9m x)1dnsjdvksrhf the angular velocity $\omega$ Explain.

Can a small force ever exertrwx w+ *+35.iyizc mwe 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 th qfo8i5 u,sah l.t.qm2an when your arms are stretched out in a8i.t,o. fqh52qsuml front of you? A diagram may help you to answer this.

A 21-speed bicycle has shm 9slxc1pf26/7h oiseven sprockets at the rear wheel and three at the pedal cranks. 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 lahfis7cm 1/hsp x2o9 l6rge front sprocket? Why?

Mammals that depend on being able to run fast have slenderkl24.n2c (bbgnkn x31esry,s lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributi.ebrn3,k(b1cgx 42n ks yl2n son of mass is advantageous.

Why do tightrope walkers (F(jmdfp a,g6pn9,sl:a ig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Iff txqj.54h 3q zu*8*8tzjg2-foidknz the net torque on a system is zero, is the net f-t4joft88 qz.jh5*3zfxg*udkzin q 2orce zero?

Two inclines have the same height but make 0 njvh emu 8y9h4(d,aidifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greateju 4vmd(89 yhna0hi e,r? Explain.

Two solid spheres simultaneously start rolling (from rest) dou b91 de.it5xe fw6(ncwn an incline. One sphere has twice the radius and twice the mass of the eci xnwe (915dtb.6ufother. 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 have the same radius and the same mass.m; mu jukw+(n9 They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greatn (km9wmu;+ujer rotational KE?

We claim that momentum and angular momentum are conserved. Yet most ( )h9wc1utxola h;fz+ moving or rotating objects eventux9tuo; 1a+)fzwc l (hhally slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would 617vk2 .wmy qxbvp6-d6ym htdthis affe-7 h6pyx 6tmd kmvqv bdw16.2yct the length of the day?

Can the diver of Fig pyl jj*ynr. xn7l+g)6. 8–29 do a somersault without having any initial rotation when she leaves the bg7 n +.ynxjlyl)j6pr *oard?

The moment of inertia of a rotating solid disk about an a.mv i 9gj71slfxis through its center of mass is mgv 17j9.lfis $\frac{1}{2}WR^2$ (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 outstr 3/2mk+ut6jx*c6 tph- hi z u;vou7shqetched hand. If you suddenly drop the masses, will y 6jhtm; v7pitcuu3ko*6s+u hh2/z- qxour angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holl0 wcr13v1dudkmr/4lyow and the other is solid. Dry lwv 03kru4c d11m/describe an experiment to determine which is which.

The angular velocity of a isg.w g hzj3 ow)w2;-vwheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the tos.; wgijv-go3)w2wzhp 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 ed. da2-hmc d2tvqok :xj7b4vt 4 whe:-bge of a large freely rotating turntable. What happens if you hw.x--e2cvk2t: 4 t7dv:4b bhda qmjo walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Ad tiokv5g2m i(q)p6u3 s he throwmq2u igp( i35kdo6)t vs the ball, 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 a3d, k n.ulqcz .y0pte;ngular momentum, discuss why a helicopter must have mor 3dl,ptc k.;.equny z0e than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles /aecbc)opkz n :laa:q8 s6e +c.8vw;xin radians: (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 amazbvy2jyg6 ees ;4jn .8king coincidence. Calculate, using the information inside the 4jee. svj6g; yy82 bknFront Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direct q7k+l2z+eb coed at the Moon, 380,000 km from Earth. The beam diverges at a oc+27e kb+qlzn 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. When the motori +pbi-:pkm* i is turned off during operation, the blades slow to reb* ik+-p:ip mist in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball *(ixwk 1 higu(ikkpmvvk0s1 rg; z086 makes 15.0 rx0 (8rwszg ui;g*1i (kk6pi vvhm10kkevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How man v2z(lzqhf )fl)z4,gp y revolutions do the lzp f )zl(f)2qg4hvz ,wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its zl,bo.to ppx4c/: v7fj2t es.angulaox 4 ,p2vo.7zc:ef jttlbp/s.r velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolutiafs jd4 g/qie 5pas7 4vnzi;,.on in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fro5isj;eagsi7.a4q4 vzfn ,pd/ m 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) f.l 9q fy j:)4plqd5gdin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear ssb1-dramg w3n3j46n cf,m/mx peed 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 centrifu(lg.yky6mb(re+* ctwalg( r 6ge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of l(* wg(g6k6etbl a(yy+rc.r m100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130+mox 4to ;8j3hqwlw.q rpm to 280 rpm4q+m; toqlow8.3hx w j in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius ev*k,yxe f ds)m9wi36$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themsdadyq24 (ym-j)i b( ivelves into a slow rotation td-4 i)dyqb2 a(vmji(y o 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. 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 un-w b 00ddnna*r/ j2xdeiformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it n2a/xb*d ed 00n-dr jwturn in this time?    $rev$

  An automobile engine slows down from+e9d e 84(cxf,9vgyfs3r aoyp 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed68 4.lt5qtf qkm1wscr jets in a whirling “human centrifu tq .wrstfq lk65m14c8ge,” which takes 1.0 min to turn through 20 complete 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 9npocer4mw-/vot3j/:h 5gs h 240 rpm to 360 rpm in 6.5 s. How far will a point on th/ve mw 4htn 3/9-p jocos:5hgre edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It tpxry(+1 9vyms*oist 4urns 1500 revolutions before it comes to a st9vymy r1 s*xp t4(is+oop.
(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 65y)q u +m/x1xydby6tkr: onp.1 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have yod1 y1 p)/ mqxbx. +yt6kun:ra 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 its speed uniformly frc2uvtb q8xg3x2cetp. mdz v1,+ 8i,akom 95km/h to 45km/he.g d3 2+vctxit88 pvmuc, bkz,1qa2 x The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each+,nf ;(pvmw0pnub:ia pedal when climbing a hill. The pedals rotate in v ;bw:+,um0(a pfpinn a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the endc gdwz yno;mg:4r4r/. of a door 74 cm wide. What is the magnitude of the torque if ;4cgnr/mg r zod.:w 4ythe 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. 79p blq4u+agb8–39. Assume that a friction torque of 07u bqlba4+9pg .4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mas(drq;jxb7 ;hts m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the nexbhq ;( rd;j7tt torque on this system.

  The bolts on the cylinder head of an engine require tightening to a toy/+oul e)rra. rque of 38yreuao +l). r/ $m \cdot N$ 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 rd g :mfqr:f)qmn f)q a8;i*vco(-tvc(adius 0.648 m when the axis of rotation is through itsm8cn ( g f;dv:of: vq) *iqmq-rfa)ct( center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in 6a ejm)xg(ci9diameter. The rim and tire have a combined mass of 1.25 kg. The mass of t)x6gmja e(9ci he hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balteemi ydb8ju 2/l.e qu n63g/-l on the end of a thin, light rod is rotated in a horizontal circle of radius b8me//unly q2uj. 6-e igdt3e 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 b;q hj 8(xad45rz50*lcbp js)zrf t3c at constant angular speed (Fig. 8–42). The friction force between j(5st;acb*f5p z3 dlczjx0 h)qrb84 r 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 aa 3:q/bwb7 oe0wci(nh+*c frof inertia of the array of point objects shown in Fig. 8–43 ah3(o/ bcwbaifa:w*0r7cqe+ n bout
(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 is t*(tmnts u7w) cq0j+s$5.3 \times10^{ -26}$ 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 sat5dcl :ad5h:opellite 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,pdahd o:l5:c5 what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8fh8.;9c3 u(0d jojwehn/bbsv .50 cm and a mass of 0.580 kgs c9 0j;83nhd.ovj/fwhbub e(. 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 bf)uynie324j k,p7 ndvat, 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-ro d a qsr/hnnz8,ft:3)g08awajund 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 radisg8f n:a)n,jawqa0t3drhz/8 us 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 andz 8ayvz1+ui.za0my:eh cw4s vk,f h52 is eventually brought uniformly to rest by a frictional to c.ymz z :1f 0u wv5ezas8yh+4,hv2kairque 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 accel .rgyvrdnd5+ 84, d v3itwbh)berates 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 foreabx-mrn3zd8 kgq 19y b/rm, which rotates about the elbow joint under the actiq 3zy- nbdrb8x1mk9/ gon 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 b7srls p9uu-ng xtpr633rvl0(hu11lu ew 9f q +lade can be considered a long thin rod, as shown in Fig. 8–46.gl6qs 1urn3tp9s+x9lrv1 rfh3 l uw up07e u-(
(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 ma/eea 1.0o: d5sgvr fdqsses, $m_1$ 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 tu2a;zego+k; v* rl h2dwrns (revolutions) and releases it at a speerav+elh2;zg;k* 2wod d 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 msa mox5: + ia0de9uuk j2*4yqcoment 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 tostt t)q n rs0s5,4*puwrque 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 slipping down a 2p y f6k7ymlr3ypyn9vmp85. k lane87mv5ppf ykyk 2m r3ynyl6. 9p at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asiv3/drxu;c.q1 t. k sc the sum of tw.cd t.rqxs/;u 1 k3ivco 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 attxmj u9wq8sdnw +5;/ radius of 7.50 m. How much net work is required to accelerate it from rest to a rotatt5 ;+wd u qnmjw89s/xtion 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 and r3d,je(ou gf a jvg+pp22cv.j4olls without slipping down a 30.0 4ou.efjj2+ dgv2,gaj 3(pvpc$^{\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 vem3qw51-0agu r* ssxtdrtically 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 beforesuam03 wd qrxt51g s-* it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.2q b:jl8b .* mw/c2kmr ka*.82id aalpk10-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an ankl*:a/ ba*dp 2mkl.ikm 8 acb.r28jqwgular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-s(y41j vttn)y-( d r:vgbdne )kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500(y- n4ed1 vgn(sttj )r:v)ybd 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 platbowo5 10ubjo.cn* 2b iform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreas2o5 noj0ic1*b.ubwbo es 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–n4)c a 4rdl t. tk6ku*;5aaaibj-ici)29) can reduce her moment of inertia by a factor of about 3.5 when )kdj*bac4ik;laa r . ct6i4n )-ia5ut changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial n+w4x;s d) uv3:m7e lmq t/7 tmpjozl4rate of 1.0 rev every 2.0 s to a final ra7mdt m xvqnejz7o:ts4) +;/ l3mp w4ulte 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 throy x5:pok; sl0pbm4,bz ugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and dils pop bmy 50xb4:,kz;ameter 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 momel0 / n:vez6d7lc1 (:gv dbcqdkntum 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 angular momentum of thel l33a 22lpnggaq))h u 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 dropp elvy57v(4-*jh ee mwged onto an identicghm5j*vv-yee(74e w lal disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at tfvc5df*qc5u29n /go 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 qyl(3e+ jk1 awmerry-go-round platform of radius 3.0 m and moment of inertia31+ej w q(kyla 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely 09lfb4qu 0q0dq /ylg pwith an angular velocity of 0 9lu40bfqqy0lq d/0gp .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 ah+qf +1nai t51 gaclayz79v5ibout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(rou anyavf1i iqla 91 thg+z+c755nd 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 -j+ qj5bo zvs)66isx*ssr.d cof 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 0uxb;jrr7,df0 k3bqd$ 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, inlp134.qn(j. egor:.iw v7mf o)eonazitially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platfp1jz oq( gei3. 4onwva)eno:.. m 7rlform 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 th6h8h( q5xlxllau;:oje edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia x8j6h(5uho l;qlx al: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 rope lyi(s46aht6tza (s -x txquuq1) kng 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 beq46tsk1h (x-( u)tzqs ux6a athind 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 thew.lc2y * v,qq7x2bmlh same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon27mw*y l,q2l h c.bxvq as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest n72xvkvlca:5cj p1jau7nkv8aceg52 12a( miat 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 uniform cylindera1r)a1f jy cx) of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by ) x a1c)jrf1yathe motor.    $m \cdot N$

  (a) A yo-yo is made of )5vea*0tjo1 /jotjw )xjo lu98 :frittwo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined 10jto5o xa*r)ut9ijjvfl eo)/ j8:wtby 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 ofub-du n+ 9pofk sf nr.i uyc4:.nhv+ :rtd5/+n 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 tim le apjqi(h7-hu wp5:yb)(mz8es as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angula)ph:h ymu((zwql 5j- ea87ipb r momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lx dn)/flvc0;zy(t ja t6nbb2o :65m,orjly 5h ow-pollution automobile 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 flb2l 5ntjoy/6hjc0n;bmr:,y( 5txfal ) 6odzvywheel 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 an h;e n sx3;l)fz6: prnj$H_2O$ 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 u i)6 rsyib.h-t::k(a g+zurdspeed 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 freeza-i aw,bea+9 p6ia3gly (i.e., we ignore friction) aboutaw9e izba-,a6+ i3pga 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. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiuse es, hyy*5z:p R. It is standing vertically on the floor, and we want to exert a horizont,hsy z:e5pye*al force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road isgex689xmn0gi ui1 mf( making a turn with ag(xi gfneiu m0m 186x9 radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.mq7kl;31u ydz;b c7z u5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from b7u;3qz zc 7mky ;uld1rest 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 vr6 lm .qqgxx1u-494iu ks),y7fsv ogarms as thin rods, making reasonable estimates for the dimensions. Then calculate the )q4x k9yi,lsggx. oqu1 vm ru47v-sf6ratio 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 consists of two cylindrical plates,00: (1fzlk rifznl7 dx of mass 0 fxl(zkdf l7i:1n0 rz$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsp84 49a+wazhtmu yj+m along the looped 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 loojaz4m+p4mu a h98y w+tp without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assa n*jhv(q7z9j k*ej+r4a7 if gj u3l 6gxep9d3ume $r\ll R$ h =    (R-r)

  The tires of a car makhnb*0dk6r.entdvm+- 9d o qi:e 85 revolutions as the car reduces its speed uniformly frvq9 h+ tdn k0md *e6in-b:d.room 90km/h to 60km/h The tires have 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