A bicycle odometer (whit0m(pl qig3l /ch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens i(mlt30pli g q/f you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hq 9+1y-uxxf ,i nuinh1ave radial and/or tan1q ,f-xxnn9h ii+uu1ygential 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(5x 2c3x5f nqm n )z;z/xfmoid the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque gcf )p u6b4i+x of)i:-srdsed1 dfl28 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 y j27td(lt,ijv our hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this. 7it (,dj2tlvj

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p5 kmt0fezdj3/ne +r 86rer e-qedal 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 sr0e r /j+35efed8 zmen 6qk-trprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flesil.9 o o3urh3gh and muscle con.3orl3u h gio9centrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?da8d ssi 9;*lscd- *dk

If the net force on a system is zero, is theqgbeh )1(a 7hj net torque also zero? If the net torque on a h)ge7aq h(j 1bsystem is zero, is the net force zero?

Two inclines have the same h:7 vwpsd *tz5+xejf7n yj5-pyeight but make different angles with the horizontal. The same steel ball is rolled down each i:7 jet7 fzv+sp55-p jxyw*dynncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulttbtu:;b5 l) ahaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinatbh : ;lbu5t)etic energy at the bottom?

A sphere and a cylinder havg lto;.u8*hgv;t am2k e the same radius and the same mass. They start from rest at the tog8 tht.k; a2gu*ml ;vop 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 greater rotational KE?

We claim that momentum and angular momentum are conserved. Ye 6ht8yoxsuy +9t most moving or rotating objects eventually slow dowtxhs8yo9u 6y+ n and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woul6n5pqtynq(/ce7yyji 2umh:o +,9qmc d this affeuq9poq( :e5q hm, m yjnnc /72yyct+6ict the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any init)q9, ge( 71 iu widruz3ekur,zial rotation when she leaves the zqe1 )9i,k(7wzu gde iuu3,rrboard?

The moment of inertia of a rotating solid disk about an axis through its kzodi;m+ /o c7o5nd1z wkwh17 center of mass is /d75 1ko c wzh+on;wd1ozk7im$\frac{1}{2}WR^2$ (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 holdingpz k nb a1 .47lwip3sahv*),ew a 2-kg mass in each outstretched han74psbp)ik,zl .we ha* an3 w1vd. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the oxx5m u3p:0f:os yq8p h9fz t;same mass. However, one is hollow and the other is solid. Describe an experiment to dq5;:yx 3o:pmu fsoxpt0 z 98fhetermine which is which.

The angular velocity of a wheel rotating on a horizy 2afwb,+6vcmhzp 1) jontal axle points west. In what direction is the linear vbmc1fhzjyvaw6 2)+, pelocity 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 of a large freely rotating turntable.q;y d.ukrpe x ex1n-02 What happens if ex 1xk0py-u;qre2. ndyou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it q7u du2/rt8 yehj5,wenuickly. As he throws the ball, 2/7 58ee,uhwdy tnurjthe 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.ex4b* cn +c0re4poqx conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secoc4e o.*pb0+xc qrn xe4nd propeller can operate to keep the helicopter stable.

  Express the following angles i fmkls:uvd9dw :k m+,7n 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 amazing coincidence. Calculate, usky3k9r(bo1 f c6), htbn y/,h1fwtau iing the information inside the Front Cover, th oiu9yk3,b k fa6r/(h)tnwtyc,b1hf1 e angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,nybq-m8;ooqb w zg ;*z e8ktr -8/e6mc 380,000 km from Earth. The beam diverges atz-ne kbr*ebo- y8z8g o6mcq8q ;;m /wt an 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 rakoa.le57 z sc) s-8jse kma71ute of 6500 rpm. When the motor is turned off dur ea1s cmz 78jo.5)skakusl7e- ing operation, the blades slow to rest 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 flo olv1-fs1wu67wfq o4f or 3.5 m to another child. If the ball makes 15.0 revolutions, w1 vo 7-quwo4wf fls6f1hat is its diameter?    m

  A bicycle with tires 5/ j/xhf:-0x:m p vq:xifqkh0) ve w1ncc/abh68 cm in diameter travels 8.0 km. How many revolutions do the wheels mc vkhf5xicv/0w:hq/h/ f):-a ebp x:xj0n 1qmake?    $rev$

  (a) A grinding wheel 0.35 m in diametcl05d2m brc0m( q wt4ser rotates at 2500 rpm. Calculate its angular velocity (r4mtd mqws0bc0 5c2lin $rad/s$ $\omega$ =    $rad/sec$
(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 in 4.0 cap z.- zrz5m3z*i*lh s (Fig. 8–38). (a) What is the linear speed of a chil5azpr*3- chz i *lzm.zd 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 .ibfsq+7c9c evelocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pcww.k7 hv,ph kor/t63oint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from dy7t 1j5(ephbxqf m:( pr(t-1b xtsl4the axis of rotation is to experien( f(ydhttq (pb71rb15j 4lxpstm:e-xce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abou qzus2rj2o9byc ++zm.t,u q8d t its center from 130 rpm to 280 rpm in 4.0 82+bdrq,cutyzm s qu+ .92joz 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 radiunt70f9p p(nmd s $R_1$ 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 spacecraftl3x4c g1pqp13 fbw;yx put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0pb1xcx343 1 glwpy;fq 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 accelera7z2 l2(a 6oz,iihohnjtes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions d7o 62jz nzia,2hoh(li id it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 dk+l e ;e:-qn5;gunsh s, hrrdq 8(z9i5yeq-yrpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed l7v l5z/ie) uwjets in a whirling “human centrifuge,” which takes 1.0 mell w7iu5 /v)zin 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/ojnc9wws 2prg;1r f- 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel haves pf r1j w-2/wr;9cgno traveled in this time?    m

  A cooling fan is turned off wzrtw b9jl8)v5 g;q- hqhen it is running at 850rev/min It turns 1500 revolutions before it co qz 9lwbjghq-)r58 v;tmes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its sprr6 tz l(3wsw0eed uniformly from 95km/h to r 6wwtrz3l(0s45km/h 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

  The tires of a car makeg:e3 xvx x(9hs 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diax:hx e9v( 3xgsmeter 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 pedayqf0gt ;pm0(w l when climbing a hill. The pedals rotate in a circle of radius 17 cm.mgyt(w;f pq00
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. /v l6em-eb/p oWhat is the magnitude of the torque if the 6ve m eop-//lbforce 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 t6pxy1sxws1 0c9mmm9v he axle of the wheel shown in Fig. 8–39. Assume that a frixxmc06s 19w9mpm v1syction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mfng ,j /i,.ty6g,8dilt * ctojass 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. Calcun.yf,gj i,gdco, ltj t6*/ 8tilate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engikmfms;a 2) wc hwo8.w5 ta*.egne require tightening to a torque of 38 ;8)2w 5 se moakm*hwfc.w gt.a$m \cdot N$ 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 1;l 8emi;t9vx f0.8-kg sphere of radius 0.648 m when the axis;xmlvt 8i9e; f of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle w d3gx-vn 5.ft(,e0iymhlh c.a heel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the5cidn a.xe.0fm-hyht ,(3 gvl hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on txt)y ar-;i (kahe end of a thin, light rod is rotated in a horizontal circle rx;)t- i yk(aaof radius 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 rotatie)d,ul)- co.j3 c+xhiryy e 5lng at constant angular speed (Fig. 8–42). The 5ycioe e,-+ u ).hldrcl) jy3xfriction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the arre nkjets*1e 3bcewh 3zdt4-5,ay of point objects shown in Fig. 8–43ee13 jts4cbe*h, wt5dzk en 3- about
(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 oxq a *v(o3ek,m -*sloex w10wlvygen atoms whose 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 rn() h7mn.93jfif z)7gbgay fzate, 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 mn3f 9ff)7m. n h7gbjy)aigz(z, 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 8.50y dciq2/)alzdwcizvldv( 9 )mx. o* ;, cm and a mass of 0.580 kg. Calculat(i c9m );olid y) vzq2vwc.,x*zdda/le
(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 swib7k;j )h0m df0fk l7vwngs 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 ons(au5u1 h.:9fwvmkiz ysl(0b a small 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 merryvm(.1fi sa9 :yl5bu ( uwzsk0h-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating yew 41xt*q k.iun4w m:2y;;eai rsf0f at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frq2ekm0w afryy:;4un x f4s*1ii .e w;tictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates,( n2kx 6coqyx 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 foreabl. ylri k.9a2rm, which rotatesk. l2. a9ibylr about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor bla6a.7fwgz1m bxit1b;sde can be considered a long thin rod, as shown in Fig. 8–46fi1s7ab t6gx ;1.wzbm.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,j y,o . fgv3-gxr84m)prvjy)u $m_1$ 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 7wmh ry914 jf*7hhml k 7xo/u/mz1mfdthe hammer from rest within four full turns (revolutions) and releases it at a sp ym4x7/fmhh17k1/mh u zorl7jd*w fm9eed 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 moment of inert;/rc)txznqmx6y -,lz 5 m) wotia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2th ;. n tum-dbql43 gmm2/op9s80 $m \cdot N$ 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 withoutg,3 gk um69mhs slipping down a lane at 3.3 shugg 9k,mm3 6$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth 6kk4 ojqi8w 9iwith respect to the Sun as the sum of tw w8ik9ko46j iqo 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 a6g,hd )u8 doe kvk*h;ond 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? keho)koh*;dvgu8d , 6Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st7cspu/0exi g hf 2j1e(arts from rest and rolls without slipping downiu2p/ 0ex7ecs( gj1 fh 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 start4a t8e9oa+mk ys to fall and its lower end does not slip. What will be the speed of the upper end of the pole just befot +e4mkaa9y 8ore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.2vz bd.r5h *tf3snu m0210-kg ball rotating on the end of a thin str5vt.rnm0zh 2 sdfbu3*ing in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenml/zf:h 0k eqyx 7xg1,tum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating qye7xhfm1z l0,:/kg xat 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thatk f712tb bl 3pkbzk-mmp3d+)o is rotating at a rate of 1.3rev/s If he raises his arms to a hordbk k3-pltbz73 2+b)m1pf mokizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertb7b:ljttuu4 0+y3 jjqia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is heu7lbyt jq 4j+3tj u0b:r angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.al 9) ttf4q-t5u hl,hiz)4;d/fkilo e 0 rev every 2ltelhidu, l a/f 4io))q4t 5-hfk;9tz.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 rotating around a vertical axis through itc5s:xe4e(y7zzho5u ; boo0xue 6,hasv ez+ 8ts center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter8z05;e ,cvo(sye xuxsz5h h46ot7zeu +:eabo 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 momentu92w2yzse ,gb bm 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 ees4 vw3a c:vl *u6r vyayg;5,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 ofl yfa6-t3 u6ph7 b.2yjgn qc7l inertia I is dropped onto an identicalacl 6pf.nhl-t 73yujb g2yq76 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 2.biw f*il*tl mo29hx 3n4vn7r, 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 /,*njok8 34tbltm5*h dxw gyzof a rotating merry-go-round platform of radius 3.0 m and mome4/w8d 3,t*z h5lym tjkxon*gbnt of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0jzbvm(b c,.c r:lh a1:.8ja r1b,:l(c .zhvbc:m $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half v3jve)xc 3i2dq,4 49cp cgpmfits 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 rotatfcmep v c xq9 24vg3c),pi3jd4ion 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 wpu mdqs7x,4x n k(u76einds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass ijc/j+ya+ qdf),x8. jag8z0aog8 epe$ 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+ 1p mhzc:nz8f freely without friction. The moment of inertia of the peph f c1z:+nzm8rson 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 p/m f6: ;czyzyof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless beac pym:6 ;/zzfyrings 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 gr7ewj*5xmw w s*ound with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What lemww *j7*5se wxngth of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thacx14:f ypn 8)xw io 4xa)wcr7ut the same side always faces the Earth. Determia 41x4:)px wxcy8woni c) ur7fne 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest az6i-*cpuzfiu(dc,- pv)q5 pjt 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 shapea7 8 fh( g3g2tc+azhmc of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate +2 amca cgg83(z7htfhthe torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each ohn8r j0:yry , d/zxq1pf mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrijrz,8:nhq1 /0 rypdyxcal 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 of thm bq /vm 5den3uhwh9/4e 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, wep(n iwqbbomelu/24r7o a 4-j;nt2;zare rotating at a speed of 1.0 revolution every 12 days. If it were to undergo 7rnt 4jb2a4o/zmw; ;p bqa -uon (el2igravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored imqe :u*j9)y-fmzu8dbn a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywhe bjduqmmyz- e 9)8u:*fel “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:df(1 xmkg5vf 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 (ho s+; f-m j 7t6gmd3wyjcvrq)rptuu:6* op) is rolling 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 c e2io+pdw,a8p6 530wr(drc8jtthwq L can pivot 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 accelpwttdd+ cw856iqr e (2prca 8,howj0 3eration 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 radius R. It is standing vertically on. pc ixnl.rcm8lozys )86 -km2cil(i8 the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has hei)mcl8 p.k8 sl-(imon26 ry.ci8c xilz ght h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is maki5s/lvcbf(,gy 9l q wqgj6fi )s-n8 ;gmng a turn with a radius The forces aq ;5jcggss lngf -(qymilw 8,f/)v6b9cting 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 e zdpuv7/ov )ctqq(+; 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, v;tcevq)ou zd /7+q(paccelerating 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 rodsk 3)karu7ih5w n,5htz, making reasonable estimates for the dimenrza 53 kw 7)h,ntkhi5usions. Then 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 which consists of two cylindrical plates, oft9dn ot+0g 3wt mas d93totnw gt+0s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rou(weaozxf* :rs o 9lwlxu )-nh5-jx8 o)gh track of Fig. 8–58. What is the minimum value of the vertical heigow5wxxxs 9j: - ) l-eo8a(lnrfh)zu*o ht h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a r3k 5nfxx6v.qssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cyek-0 5xi+i tnar reduces its speed uniformly from 90km/h to 60km/h The tires have a-+xit0n 5iye k 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