A bicycle odometer (which meas5u 7y cl7 4/co3lpjghqures distance traveled) is attached near the wheel hub and is desigu4polcj l7y3hq g 5/7cned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotat3/(aw vb-ec.q ,ydp3-pkyo4ov, cbbqes at constant angular velocity. Does a point on the rim have radial and/or tan.q3v3dc,q k w( be /p-cobo b,4-apyyvgential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veed+sw3 0;zgad;:1b sqw gx-ijlocity $\omega$ Explain.

Can a small force ever exert)c v0 +js8p.xitsc tflgyn ):. 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 difficu bl9xg9w+ ) pduh6n3hzlt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to and+9gp9wb ) 3z6xhun hlswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the , y9xu (1ppsaotm( wn5pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which geay1p o (u9p(smwa,xt n5r is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend ;r6- wtb2d1gdpm awyo3v88 r pon being able to run fast have slender lower legs with flesh and muscle concentrated high, close d 1 rvb3tpy6;8p og2mr- aw8dwto the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkevz fn)h 30uz,qrs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, ism-m mi( flr,*r.k jg0 ia07obh the net torque also zero? If the net torque on a system is zero, is tho-0.hrl7rbmigfikj0 a (m*,m e net force zero?

Two inclines have the same height but make different angles with the horiqwhb:0hb2*cl zontal. The same steel ball is rolled down eb0qc wh2hl*b:ach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. O h2j8 c2eo5) s6z9vbbtedvhs2ne sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline firsthdt9os 2 v8beszj)v522bh 6ce? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder ri;l4fu1l 2,6v:wla k /*5dwmoq iwcbhave the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? 2rqwui mki4;w6f5dll*b, v:w/1 l caoWhich has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mjjs q-6l:htzncb73 ) joving or rotating objects eventually s7n)jl h3jzb6tcs-:j q low down and stop. Explain.

If there were a great migration of people toward the Earth’s8kl5b f ;5*qt/i +jofclex8cw equator, how would this affecttl8/*wf5;5qiocce8l+ f j xbk the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial r4id.35 -ou q8(zrpuiw,ydj7t js(d u lotation when she leaves the board?p5r du(q8, 7zu3wd4(tsjiy il-o .ujd

The moment of inertia of a rotating solid disk about an axis throhi,6mpjit )zu.+ l)o vugh its center of ma z,.il)topmu6hj) vi+ ss 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 s4k l-*ilssz ml8rq,v, tool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explqz 4 ,l,sl8ilks-* rvmain.

Two spheres look identical and have the same mass. However,v-ie c)0+ fhwo one is hollow and the other is solid. Describe+i we v0foc)h- an experiment to determine which is which.

The angular velocity of a wheel rotatinj o7 it6yi,o3jf72ifbnou) +e g on a horizontal axle points west. In what direction is the linear velb3fi7o,oiijuy 6+e tfo27 j)nocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge s mah zxhz* ;8dkfo2a8 sk9:2q0t l2ivof a large freely rotating turntable. What happens if you walk toward9m z: 8qsx h lds2akv0;toz 22ai*hf8k the center?

A shortstop may leap into the air to catch a ball and throw it quickly. vsc/:fudd9fk 030 yjdAs he throws the ball, the upper part of hid k9y:00d cjf sufv3d/s 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 o c4fd9nx+2v s:4jx0ygrdpaakcd *4w)fxq -2of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second k g4+ow xndcd4 9fs-j24 0 y)apvafq:*x xd2rcpropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) l0(kmev8wkp8gh(d19c (vxqs9isig( 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 ama 5tviw9-j; .w xjs7qujzing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radiix5juw7w. tjj9; vs-q ans) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 5o( twbgy1k2y km from Earth. The beam diverges at a (2o1gby5 kywtn 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 u ztl7j.kcacx-b1rmok :96 h5rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladxm .1t :lha9k6c5ju7bzk- rcoes slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another t*0 ttx.bjppg5fvp7 *child. If the ball makes 15.0 revolutions, whagftj7 pxb t.0*t 5p*vpt is its diameter?    m

  A bicycle with tires 68 cm isau fs*qv039o*xk)ul n diameter travels 8.0 km. How many revolutions do the s o)30 x alusk9*fv*quwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameteh2fo w:funv( 7e *5196hyk wr ynuxr-1ky+lwir rotates at 2500 rpm. Calculate its angular velocity r lwoiy whryvffu( h-+k5x*:kwyu21e1nn976in $rad/s$ $\omega$ =    $rad/sec$
(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 compl 2rd.q r op( d-ca6twp3c3, 7apz,devdete revolution in 4.0 s (Fig. 8–38). (a) What is tc6(ae d7r2azp,dd, poc. qwt-dpv 3r3he 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 ayoy ; tkiemc8 ,qo sn8hl3i4 m5ooh7,1round the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeddjeh uyi5 167 f6-ntrh9tb6t5yr)ez h 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 pan;n f qv4gsm f4(qa6u)rticle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 g4 f; )nuqmva6 sqfn(4$g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fr7 go 4.rexqyg)om 130 rpm to 280 rpm in yqo).g4 7e grx4.0 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 5 he)x8jxg+gv $R_1$ 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 obq 3ydocwaioo-/kh8t. + rdn n2586yspacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from nko8od hanny 8o2wbyi rqtc/ .5+36o-do 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 trg,; xb bkub) *zenh6)o 15,000 rpm in 220 s. Through how many revolutions did it turn in this ru , )bzn)b;he*xbk 6gtime?    $rev$

  An automobile engine slows down from 4500 rpm to 12 )s+4,njj;x.tzw f5tme ,ojgd 00 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 fl) 7 /bpnuz5tg q1ls*ecying highspeed jets in a whirling “hus5t u*7be)/ pq1lzgcnman centrifuge,” 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 dia vo-yh*+vnz i2meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travelen2vh+iz o*v y-d in this time?    m

  A cooling fan is turned off when it is running at 850rev/min ;i(b b*p sg1wgIt turns 1500 revolutions before it comes to a g;p 1(iwbbg* sstop.
(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 mazumlp,vs )bzr n ho(h +()f0;b d3d7jxke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h zs d fhzrjbv;xd7l)u((, +o)mhnbp 03The 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

  The tires of a car make 65 revz, evf+q )qa6t ay-b8dolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires havqt ady)b- 8,+fa6evqz e 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 pedal when clim zcc )im25)l jsvw3(sebing a hill. The pedals rotate in a circle of(z ec)53lwisc) vjm2s radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end os6 3x,wf)w3.dautha t f a door 74 cm wide. What is the magnitude of the thaw f.tws)x63t3a u, dorque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–1 0hqi v7 *vcskvp1-ne39. Assume that a friction torn1h0s17kvvpc eiqv* -que 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 masslesop 0.kmo5 erh i3v.qs*s rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the mi5os qo.vkrhm p*e .03agnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an end kw54ipi;x n.gine require tightening to a torque of 38 k i;p5d 4n.wxi$m \cdot N$ 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 ofs3niu-odcf; 93s ir6tb )rxl : inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its c:3b9 36 cous;ri )rsn-lix ftdenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheeajh9vqft-dx . q5y /d5l 66.7 cm in diameter. The rim and tire have a combined mass of 5qa- dx5/q9jyvdth.f 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotat7 btmfdd/j-d8 k1g mjghf)bj44.dz w;ed in a horizontal circle of rgdj8tfkd/ ;)jf.bm7mzwh1db 4 g j-4d adius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular sc xjrqj17jc+5 peed (Fig. 8–42). The friction force betwej rc71j+jc5qxen her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–4t0h 4-brcib +z/n0oys3 abou hn-zi+ b/otcb0r y0s4t
(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 i 5o*is-dv2r nhs $5.3 \times10^{ -26}$ 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 corre hrk)k1 v-u+6 rg:s7s+ u-denc cwpm)gct rate, engineers fire four tangential rockets asmrvuw) 7g+p: n -)6kug1r-cd ssh+ cke 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 cylinder with a8 el8 uffmqz ci)*v9 c-z,jb2 lo3ez-b radius of 8.50 cm and a mass of 0.ljm* ziqc3,)uzc-z8 e2b v8b-9lfoef580 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 53u lbbh4ze+l a bat, 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 s6a*i1ibjtc7 t9wgqm0 k9ax;ex )t .somall 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 opposite each other on the edge. i *j0. btx9mwae ;tsxqgk )9ct 6oa17iCalculate 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 of0mo6hpv.3 kae 0n4ph1(wk oejf and is eventually brought uniformly to rest by a frictional torque016v0 khmpwhn.k ope4ej3 oa ( 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 3.6-kg ball il ,,qm9os ay2at 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 forearm, whimjk- 87iw/-a+y zkb ;blzlkh 3ch rotates about the elbow joint under the action of the triceps muscle - bkymz7hl8bikw;lakz + -3j/, 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 bvgt1 xe v4gs5ejp u;78e considered a long thin rod, as shown in Fig. 8–46egj 7tx5vs18 ;ge4pvu.
(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 g2y6dfh;ndf/8wv90z z.y8 arb h+h p ctwo 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 hammer from rest within ffdd*y o*jf7b, our full turns (revolutions) and relb,fj f*7oddy* eases 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 moment4 i.1 c0erl ve ;f(u4topc2l8jra,gz g 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 d jsxq,:d4z0*-vs yqi zevelops a torque 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 slitvxmb 3equg.h-jy:6cy1voe m/l-6 e. pping down a lanv -: ux qm-e yove1.ghb/el6cmy6 tj.3e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with e 1ibguz)56 nvrespect to the Sun as the sum of twon5e u 1v6z)igb 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 iem (ph 4 hukm;x1 3engr(3bj vq-35nqof 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 guxnjekenq3b(5h mr- q1m4v3hi(; p3 it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80wx mvj1g2so cqfr7eec.(8ccc7 j ,1 j2 kg starts from rest and rolls without slipping87fj q oec2j v.c2j,c(rc g mws17e1cx down 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 vertically on its tip. It starts to fall and io4iifjq5) j j,khvgafa qk)q-4 +1k(mts lower end does not slip. What willjk j oqq5iqga+h )v(mk4,-ffki1) aj4 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 momentum of a 0.210-kg ball rotating on th97 t,zflmfra 34c)ekne end of a thin string in a circle of radius 1.10 m at an angular s9cmznke f l)3 atfr47,peed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angui45n5p rv,kq olar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whr5ip ,54kqno ven 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 side0hujdoj bh wag+1/.p/ , on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position,j+/hgdbp10.jwou/h a 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 shown in Fig. 8–29) can reduce her moment of inertia:vwc1r g:l9x n by a factor of about 3.5 whn1 lwrx:9:gvc en 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 rotatuja/a*oi/i9 h ion rate from an initial rate of 1.0 rev every 2.0o//9 a*jiau hi s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating arou aj:qsx3zo 1u8udxlbzgi 5/74nd 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 frubs/xlzgq354z17u o xi8:jda equency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angulalv,hiqt) u5p(r momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angularhs 1y*q2uz/bh2(we co 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 anjn (4edt b-a9etg5v*+f rzl q7 identical diskrtz + ndfteel(ga745b vq- j9* 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 2.4 i0gtnbt n2+u0 $rev/s$ 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 rotat* (ly9l9aivp ying merry-go-round platform of radius 3.0 m and moment of ine9*y(9ll vy ipartia 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 rotatingooai4 je0dyxxr: d. 6/ freely with an angular velocity of 0.8x/:a4.0re6d oiyjdx o $rad/s$ 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, l+bqyhsgma.s -7 2 g 35jjxm1tlosing about 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?(round to the nearest integer gbqhgt-1253 mxs7ml.y + jajs)$\approx$    $rad/s$ (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 oe rqj.4 o5a4s lnmx(bqy(e*7l f 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 xz dj-afx--n 2$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withoum8sz0wub qnxyw5+s4o6 xg(z-t friction. The moment of inertia of the peryszu+s(ox 4x 0w5b q gw-mn8z6son plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round twro 8(8h g8q *1/b e+je9cnagogsq;nvurntable that is mounted on fri qvsqgg;+1cge*8woo j8( en h9nb8a/rctionless 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 rope lying on tsb zp7mu8iv9pfp3d 3pzja4s-2 *qw-the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person w- tp-p3z*u dqps47swf9i28vmzpa 3jbithout 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 same side alwajruj+.lk7ld al.3r g0ys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) tj.lr37.gl+ l0jrudka o its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ra4v nr- v+ qxnwf1wi*n8te 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 uniformjbz4nnx; .is+y ;*l p0nzw; oa cylinder 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 the motornzps* l 0xwjn;ia+.boz y;n4;.    $m \cdot N$

  (a) A yo-yo is made cywk2 9 38s3 *on.rzh nfbcy*0ihl)tvof two 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 conserv.i)83h0ctlr*s2kfhzvn n 3yw yb9*c oation 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 speey-cbuq4kka;u03cp w: x2s m 1wd 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 ouv)+jz7 c-rfn ex,1to cr Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution e,x+ccnjt 7v-)1z f orevery 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile iu x52c.7zcuqaqm-9 q ts for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a totaquc acxq9z2m q5u-7t. l 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-ewdausb 9q ap*mp 1gxp) +3s1 an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surh ne41sc(hk ;pface 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 d12r sqd4(h)q vqhyv; ap s6g+and length L can pivot freely (i.e., we ignore friction) about a hinge ashd14 2+vs6vr a gy)dq; h(qpqttached 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 hase-8 74hzmxa odx6cys(z fv)2g radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its ex67csf x(gaz y2)m 4zvh8o-daxle 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 flaty37 4/tlgng 0*ytjzhgs h4jv e5* 0xmk road is making a turn with a radius The forces actingk*j04ty* etgm0xz shv 5g7jl43/ yh gn 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 intofdmz(53 o 9oyk a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his headz d3( mkooy59f, 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 cy8 uncuhgf( *c9linder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater g9nc f(u*ch8u with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which conse ;zzppeggv 6+(l.ip /ists of two cylindrical plates, of m egpvlpz;/pe i+( 6.zgass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 nu:pn.n1 hjpps *)2iaalong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the nhau1np* np: s2.jpi) marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assui gxi/kfoq-oers)/suvl/0 hj . t.3y 6me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces iz); yp:h6 htex4fjfv (ts speed uniformly from 90km/h to 60km/h The tjx4 ve(z6)h;fhtyp: f ires 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