A bicycle odometer (which measures distance traveled.4qwh pqt3sf n. rz6*r) is attached near the wheel hub and is designw 4.3 shqq.6prfzrt*ned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vi;x *uw rb .3eq00agzzelocity. 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 eithegr0e0;xw.z*a b ziuq3r component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vej f6 1epbp+j,.ttlh(vlocity $\omega$ Explain.

Can a small force ever exert a greatereyrede9s4s.dqp. *+j m* pf.n 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 me4mkmzn/0w wf6q(q8 ihf) w m.a *x(ahead than when your arms are stretched out in front of you? A diagram may help you to answer thisfi80h(6k)(q. fwa /4nmx z mmmw*wqea.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal 6 : n9g. t3w:zosnon9 r(ktxltcranks. 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 frontwn:3rl 9x:not 6(gtn.kzts9 o sprocket? Why?

Mammals that depend on being able todrxa6jo0 2+byhv .nn:xzj3d 9 run fast have slender lower legs with flesh and muscle concentrated high, close to :ba9dxn3z d+ 20nxoj.r jy6v hthe 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) carr +rb fldi7p2a qu2,8cz j3cc /s*-qw9r/bqb ofy a long, narrow beam?

If the net force on a system is zero, is the net tfl29kl:rv s(rorque also zero? If the net torque o( r l2rvklfs9:n a system is zero, is the net force zero?

Two inclines have the same height but make different w-1.5 dnwvgc,tpi9 gtangles with the horizontal. The same steel ball is roll5. t d91wnp-iv t,wcgged down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollingalz. 7m.k9rza (from rest) down an incline. One sphere has tw7m 9z.ra zkl.aice 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 greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and o ipoxxq gqhj fh0z(q6*+/x -2n;y4zuho;lg5 the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which haoofu;z2xq( 6 5+z/h4hq;xigp q-hx0 noy*g ljs the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are uftj 2g8z 76zoconserved. Yet most moving or rotating obj8u z6 2goj7tzfects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earthgc*: tgqx1:dw’s equator, how would this affec x:g gtcw*qd1:t the length of the day?

Can the diver of Fig. 8–29 do a *73l +gndhae0e4o e6s1;h uu cil8 simsomersault without having any initial rotation when she leaves the boareo* l7cgie04elhuud;3 s 81ami snh+6d?

The moment of inertia of a rotating solid disk about an axis through its center)ws7 (f jjr jp/mfvy909 eegx9ofcj7/ of maso7 e)jvjmswcjyr7p(x/fg0f j9 e/f9 9s is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each o7t4 tm;w 4 m0eckycq5dutstretched hand. If you suddenly drop the masses, will your angular velocyqt;e5 cm7ck04w4mt d ity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and sy/p,r2mjo-2rvpn2c the other is solid. Descrinr/ov, - jmp2p y2s2crbe an experiment to determine which is which.

The angular velocity of a wheel,il,mwjq*a9;3+1cqmt 4y i2f p7wkwk (ehid g 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, d+yd k3l4q,i(p *gw 7mqeiic;92wkfa, wh1tm j escribe 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 large6.vd4t x.js*g,obfbur z9h j1 freely rotating turntable. What happensu.*g.v9 zsrobhxb ,6jt1 f4jd if you walk toward the center?

A shortstop may leap intdf(z9 s-(5smfcsra0 y o the air to catch a ball and throw it quickly. As he throws thm rd9-zfy a5sss f((0ce 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 conservale 1p.mvt;w.2b g:pedb (y 9r0 zrhkp.tion of angular momentum, discuss why a helicop9v y.;l0e:k.dbp1pm.2 r gtbeh( wrzpter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r,su5 9 xve84w.irz slx*p*czhadians: (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. Calculatey7ex,, t: fjjt, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seey etx7fj j:,,tn on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth/e/p, asy)dni. The beam diverges at an anpae,s/dny i) /gle $\theta$ (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 rwm 8z -(j nw.ofrnvnn1g4xwoofj*6:9pm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blv-*r(m8jfxon.zowfw4n njno :9 wg16ades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chid*8wy) c f5 .xjqd(ktqld. If the ball makes 15.0 revolutions, what is its diam8y5tjq)dqkwf .*d c(x eter?    m

  A bicycle with tires 68 cm in diameter travels 8s :ovji ,vtl+(.0 km. How many revolutions do the wheels mak:tv(ljvoi+ ,se?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at*t8,p lfwoq(6/hsq)v r / pgul 2500 rpm. Calculate its angular velocity i 8r ll ,/h*/sqwp( )fotv6guqpn $rad/s$ $\omega$ =    $rad/sec$
(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 revolution k :bt6 kee ndofh2 .hzc/ls)0;in 4.0 s (Fig. 8–38). (a) What is t 6c ke olskhze;20dh/. b)ft:nhe 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 v6r ,ppyfr2()sjp ; beEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe bf)wj;3,9gd5 q rjfcbed 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 centrifuge rotate if a part i)uq/ +57qv1idrld7 wzvy/rlicle 7.0 cm from the axis of rotation is to exper dvl) 7qrdz v5 yl1riui//7wq+ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accel2ie; nyss*2 i.s mfaz u.ih,(zerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. , 2m.iass*.znhuiy s (2;eifzDetermine
(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 (af yeq7lu8. z$R_1$ 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 themselg- ;(pst tnck:ves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought ofp(:t tkgsc; -n 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 15,000vrwgs n 7u9i)ay)f- g jkn, s1)onm4t+:.yrxs rpm in 220 s. Through how many re-gna+jr)y4t n9go7mksu)y:,1wnxsif vs r ). volutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcu:52wlp7 c e6ut ppya v8g).efflate
(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:kf5o *bv , i5xsms3qic3wx m9 stresses of flying highspeed jets in a whirling “human cent5qm9b: xsof km,3cv3wi x5s i*rifuge,” 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 240 rpm vu(5p pwxd(9zto 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in (x upp(wz59dvthis time?    m

  A cooling fan is turne-dv/c:jn3z n7os r(j4a/yijmd off when it is running at 850rev/min It turns 1500 revolutions before mcj(nz j/sio v:dn4ay7/3-jrit 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 fft:-lk xc* y*revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of:flxk y-ctf* * 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;+it;(wu tkb s as the car reduces its speed uniformly from 95km/h to 45km/h The + uit( b;tws;ktires 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 weight0f47s+jcud d ) u4mg;c cxnyl+fgp)hw: 8uf4e on each pedal when climbing a hill. The pedals)u pmscuy);j+gwfdu c g 4f0d x744c+:8ehnf l rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcevyxse,j);v:,p s-dk t of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if t j,v;: keyvtdsp,)sx-he 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 pkugo:b 03hj/pxx - 8vthe axle of the wheel shown in Fig. 8–39. Assume that a frb0x/: 8hpvk-x g3jpu oiction 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 ends28mksslilo751tad9 gcz1e6 l 9l:ea x of a massless rod which pivots as shown in Fig. 8–40. Initially the 8l l:9z52gla617aemlst k9x1i ed csorod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tig5ar ou:ets( 9dhtening to a torque of 38a (rdtu9eo:s5 $m \cdot N$ 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 spherpq;z3fdxflz ys3: p)4n)m7g le of radius 0.648 m when the axis of rotation is tpgmlf d s )plfy3q)7x:3nz4;zhrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rxlq. uc.p,k9o 8lnu1h im and tire have a combined mass of 1.25 kg. The mass of81ol hluk.9,cnq.xup the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th/3 2)z7,u( jlwbay zb5wosmoee end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m.yz,j wsoo wu/3 be7bzl2)m( 5a 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 ab2p hz6jp m8r ey;c9y- bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totayj rc-2hp ;m6 epy9z8bl.
(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 jmp /5-irpz*x1 rw.4h/dv jgyarray of point objects shown in Fig. 8–43 about pw /j z/gpjx-y.45m1dr vhri *
(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 whosll97ef m3wr.a -,mgaw e total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engnd6yf w+:1m4c xal+x t 7:newgineers 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 mcnewm+7t4:f x1+g l6yand: xw, 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 suitaqz* 0x )ciu)4q v3lq,6ta radius of 8.50 cm and a mass of 0.580)0tii z ctuv xual)*3qq4,6qs kg. 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 bat, accelerating it frhj*9vom-. c rfom 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-driim+wm93g/ mhiiuk;cn fa3+ l;ven 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-;i9wni+ufg ml3h/ ;kim c3m+a 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 and is evenl x36uq(0kpj ftually brought unqpx6jk( u3l f0iformly to rest 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. 8r 9av2q hmqsb pr) 9ia5(l+h.b–45 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 solely by the action of therm8vzt-zbr5i u5g9 c- forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleraugr zvrz 5mct98-ib-5ted 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 blaeiftle:r tsu 7.)2sdhca5, i*de can be considered a long thin rod, as shown in Fig. 8–4.5cd:hs7lse *t2tfeau rii ) ,6.
(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 cons6:w/ 8+6gfqko k/pkdc)k zfkiyp- c k3ists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the ms5o-zr) 95)zzd-tihbfu *kyhammer from rest within four full turns (revolutions) and releases it at a speed o r*y bmh-5)zt-zsfdzu ki95)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 aiohc;,s *;58 fwfyxqk moment 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 tor/.b36 uk)mj vf1-cu* kb boyguque 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 roi7h8rv1 u 3yrvpzj3y: k;ga3ills without slipping down a lane at ; ri8p7a1 3vg: y3zvirku yj3h3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of t+z7t zcmx .n53 xf+vpnyg 81r9:qey kfhe Earth with respect to the Sun as the sum of two g1n zzfx8y q+5+vcf: ty.k3exnmpr79 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 afi24fec x v*qd04z )po mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a s *)ff 0odeq 2v4zcx4ipolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withouoh, (tg7+jc9 m+/s(xs dsytuht slipping dowg,mtyxhjo+s cshd9(ut 7/ (+ sn a 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 v pj pzwl0+l1w0ertically on its tip. It starts to fall and its lower end does nozjw 0 +pll1p0wt slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentumw -z9i: qyjw1j of a 0.210-kg ball rotating on the end of a thin string in a circq-z:9jjy ww 1ile of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum2vgd- ;*emzic5m lu6e of a 2.8-kg uniform cylindrical grinding wheel o cl*v6meg- iu52m;edzf radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platformw 6t u5lq rr.jsxo3thh 0ofd+c;us9)+ 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 dec+c f9)tuowtu3shlh. 5rsd+j6q; xor0 reases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onp-f2 om5pa (lge shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the stp2mlog-5(f paraight 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 incre zkj9m6s q/c)zf j 7ubvr50qc,ase her spin rotation rate from an initial rate of 1.0 rev ecqj/rv ,cu96 07fkjz bzm5s)qvery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is ro ck: mh ;kecmtffr.60/f)+rjuq2b q9ztating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the :mtffuc6q k +eb/k90zrh fq.rm)c ;2jfrequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater sh ajb(bo 9 b,ylf:6u.epinning 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 angula e5p9h8)m4emq grcwy .r 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 dt dyv:6q3a59b awh ita-1x bv1ropped onto an identical disk rotating at ang bi t1a x qavyh9a-bvd3:w51t6ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 p wwo 9o6.15;bnray rt$rev/s$ 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 cerz*seuboft 75k0r( 5j nter of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 ro5bk(r7 u*5jzfs0e t$kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an ang4fd5r 9p: gc6avun gb(:-brzx ular velocity of 0bgv :nbrp46ud xr fz5-( gc9a:.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 intu,q1 2y u-zbc 96kzxfhq rp+7a0bps9qo a white dwarf, losing about half its mass in the process, and winding up withax p6cup yqb9k1 f+u,2 qsz0b9rzhq7 - 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 winds in exc5+(pze,x9viiw 3w zvf ess 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 9rrg8fbq:hjf)0::m. mqyf bv $ 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, iniqj:9vk h80bpktially at rest, that can rotate freely without friction. The moment of inertia of the person plusk89kph0:bv qj the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go :vhk0 ,69vsn hxm/i6s1 rnfcrx: )obt-round turntable that is mounted on frictionless bearings and has a momr)vc9x,s6r1k0:o6 hit/bnv :nfh mx s ent 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 rope lying on thevkl.1ru -c(j z top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of r-.z(vc ujk l1rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the slf f)phq/ra1zm 5 c;i)ame 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 Moon as a particle orbitin;mpal h1qc)r5)zi ff /g the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of,9(gt3z dgtp y 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 cylinderiu 9c-oxmi1t- of radius 0.20 m acquires a rotational rt-xcui-i 9mo 1ate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kx5f086 c7hlm:jx do skg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conserval76x8:mj ho0c 5x skdftion 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 a*ubgy ;. )ag wq2jkz9speed 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 great, wer1v ii2+7vsim0w vv /y5tiz u8pe rotating at a speed of 1.0 revolution every 1y2itv0pvuiz78/isw v1+v5i m 2 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 automobil*bb l 1wlm6yjldku z0)195d7xo t+y pge 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 )oy7bumgy9 6*5pl lj1zbl 0k1w+dx d tmass 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 illustratr,ko1,s ( kf2uji *3clak**5kmws mtgq* tlf1es 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 royqu:c6hp / n35my8 bnw.uq*dcww z1 g*lling 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 *i33lh l+d jdlpivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [H 3 l+d3llihj*dint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is s2jday:mr5cwe:cgrns 3r(3u, sq ;zc - tanding vertically on the floor, and we want to exert a hori25eu r w(scg:jrnacdry,;c3q3:z ms-zontal 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 is making a turn witudk,snitb4b 92 2wis1h a radius The forces acting on the n is tkbw 1bdu9s2,42icyclist 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 )vg x* 6lhw:uzbo:0 rdrock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his heb)6:u rdzgh*wovx:l0 ad, 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, ma,gcd;op z -.ttkr0 (dsking reasonable estimates for the dimensions. Then;cdkgdstot(z.0p -, r calculate 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 .ne:hd7 pq 92pos b2zpwhich consists of two cylindrical plates, of zpp.qb27o s p9:nd2ehmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 7nn-ao)+h8dm mjv4 m pthe looped rough track of Fig. 8–58. What is the minimum value of the vertical height h an+ mnop7m) m-vd4h8j 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 not4jtcr7hvll8lg5+,e * zgc5 k u assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca .0viu9g u8hair reduces its speed uniformly from 90km/h to 60km/h The tires have8 v iahg9iu.0u 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