A bicycle odometer (which measures distance traveled) is attached near the whe w7ck,s5er) q qiwj8r+el hub and is designed for 27-inch wheels. What happensrr)qsw78c +,i5w kqej if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulm 2i+*yvcw0r0wc o w-har velocity. Does a point on the rim have rw -+ ywovwcchmi0 0*2radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single va,/xo6a qgkn,f 1u.xjf lue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explbvl pmh+b* gdts4 7+s,ain.

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 youre2jmtyot, . l/ hands behind your head than when your arms are stretched out in o, 2/mytle. jtfront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rk k t4v9/ tkujxzc37z0ear wheel and three at the pedal cranks. In which gear is i/k9t 3u xv4z0k zk7jtct harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being abl:ddu5lequ,*x,p t2 pj e to run fast have slender lower legs with flesh a,de pxqjlupt,:d5 2*und muscle concentrated 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,azy4pgnlm+f :23 5 jgt narrow beam?

If the net force on a system is zero, is the net torque also zero? If ue. hr:o.v ulydd8 a:0the net torque on a systvd0 .eru.8ao d::lyh uem is zero, is the net force zero?

Two inclines have the same height but make diff 6a-h 4d+,njwiqh/bvao iuj0*erent angles with the horizontal. The same steel ball is rolled down each incline. On which incline wi*0 /j-jo4u w ahd+ibihqav n6,ll the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. jyc3 4ud)r3am3k8y myp1mm c6One 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 hayy1pm3a3 c3c yrdu mmk 6jm4)8s the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same u-hs ezk5ej)bwk7ky *6ub8k8radius and 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 has the greater total kinetic energy at thebuykj5)zhk*6e b8es8 w7 u-kk bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserveqe6m/:mq1s) 5dy aiw hd. Yet most moving or rotating objects eventually slow down and stop. Expa)e:w1 m5q 6qyhi smd/lain.

If there were a great migrygi ,f8 0,bsziq.t8bcation of people toward the Earth’s equator, how would this affect ts, izgfq8.bb8ic t0y ,he length of the day?

Can the diver of Fig. 8–29 do a somersault withou+ )okjdmafqi f0q7x0 )t having any initial rotation w7fq fd+0 ))0kiojxm qahen she leaves the board?

The moment of inertia ofnit; *oi7yj4 n a rotating solid disk about an axis through its center of ma4n7ni ;o* yitjss 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 09v tn6guimit 2dxk/6a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease26ti0gvdx n km6 t/9iu, or stay the same? Explain.

Two spheres look identical and have the savjpux9vb6 p6m*iv5tq(+.lez fq) y 7fme mass. However, one is hollow and the other is solid. Desyfu +fb7pvj(ve 56iz9p*) 6v.xqmtq lcribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west.sgi/*eck8dftm5 +uq xz4,xu8)y z a 0u In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the tt8 *4ic5+sad x)ufzg/m q,0xuue z8ky op of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on thesa0 1oqw:-lom, nx p/lli26rd edge of a large freely rotating turntable. What happens if you walk towar/msowr,2:l-xdlno1i 06ap l qd the center?

A shortstop may leap into the air to citn e09x)*sug5i mbh -atch a ball and throw it quickly. As he throws the ball, the upi t-s eu*5m 0ix)hn9bgper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation n mxe(oph74m/h+gi ojr **m1w of angular momentum, discuss why a helicopter must have more than p7r1m*w i * 4xnjhogmmh+e(/oone rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingsc)fu2o8trfz; *n*gy t l1 p7a angles in 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, using the12az/ 5lie2 t. :vtwso qyyqhjot 7zub0-99 np information inside the Front Cover, the angular diameters (in radians) of the Sun and th-tu2oj yi5:whyp9 9 t1sbzotqq/7a0lv 2.ne ze Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mllzomr am3rise26:t x s752yj: w r)5noon, 380,000 km from Earth. The beam diverges at an anglell22artsn : os)xm6:j357mrrz y5i ew $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mot eba0* g3obrjk 1e1v+hor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? g+rv1 01*hoeb k3 jaeb    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thetf etf/re vc+)f(-lbvdi1, ( x ball makes 15.0 x/( fl1evvtif( , b+e)d-f trcrevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions k3m( rt kjs6tqa4o89jdo ttk9t( 3k jqorsjm684 ahe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a.rjnj ut1 c:2b:zlo0 x:terf)(p,s 7sbbe my.ngular velocit1b j.2pbfete,xz0m uor::7ts (b:sln y)rj.cy in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-r,wxb8qb6jt/zvx q jym 5.ri4.ound makes one complete revolution in 4.0 s (Fig. 8–38). (a) Whatqb564rz 8x/v..ytjb wqjm,xi is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sunf ry6 s w,h88p/qqd4nn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe)idy4,k iaydvf3-+ u3v ym 9ufed 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 particle 7.0j*dc q3cz sky;3i 7/7bwkcbb( cm from the axis of rotation is to experience an acceleration of 100,0007jz3dki3c kbc ;*cw(bqb 7s /y $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its vbq5mo/awbn6 r+dc2-. g( klzcenter from 130 rpm to 280 rpm inm2.r zb/6 klawqn g vb+co-5(d 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius 2l*wg*l ;mmw :b 5 ni2pigm8b)iszq)l $R_1$ 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 space-b-ou8,pw:y qjlod l; craft 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 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder 8-ol obqdu; -jypl,w: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 acceler h5bov+e)xc:kates uniformly from rest to 15,000 rpm in 220 s. Through how mv5xh)c ke :+obany revolutions did it turn in this time?    $rev$

  An automobile engine slfi 63go ewe(vr ;t*r-gows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets ih9hqeaqj:/ph5 5el 9cn a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before heq5eqal 9hph95 j c/: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 acco i zm5hg5t0p;elerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of thh0ot;gi 5 5mzpe wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/rw x-h3ya28j)xw. xhxmin It turns 1500 revolutions before a)8wy.xwx r3x hjhx-2it 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 relih.cp: 4x2ykuzxl (:volutions as the car reduces its speed uniformly from 95km/h to 45km/h Th:p(2khu l xzy4.xil c:e 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 revolutions as the car reduces its speed unifo+pby j2w 9k*tormly from 95km/h to 452p+*t yk9bj wokm/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

  A 55-kg person riding a bike puts all her weight (qhxju4r* )40o lxbfgu.69ln8lp n lxon each pedal when climbing a hill. The pedals rotate in a circle of radius 1xl9ln(nblp )hu0 8*.4xr4l6fj o ugqx 7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. 2f.js 7vem kc zzzo.b8fik (:94(+c aqtwc,vlWhat is the magnitude of th f.( 4zm sco,8t9za+.( j wkf 2ci:7cbvkzqvele torque 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 abfl+9b8c tkn*hz9 ro1lout the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4 r nzlc9obkl+ 8*1h9ft$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attach88kxy0l; -*joi q6ik skt5y a8bcogv1ed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the ;l-t8x1 koyqiov0i*6y a5bc8gk8skjrod 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 tightening to a t rsxfz2il4h g+9k )a(worque of 38fagxkw h4zir (s)l9+2 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere ofr ih i /uk9zjwabiiu5p:e*./1 radius 0.648 m when the axis of uj *9.hib uiw/air1:k5p /ie zrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertitqu )p8 -5flpla of a bicycle wheel 66.7 cm in diameter. The rim and tire have a co)lp -fp 8uq5tlmbined mass of 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 rotated in aiwdlv2wz -td5tk: ywoy5 u v9)8r*kv4 horizontal circle of radius 1.2 m. Calculatrk* w45td kl:5odzv i wvu28wv9yyt)-e
(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 rota7e mcvldd71l/ting at constant angular speed (Fig7 lc edvd1/7lm. 8–42). The friction 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 iners/ t4;ld) yk,o znk*,vsg)s swtia of the array of point objects shown in Fig. 8–43 abl ;s*d)n4zytvkws, osgk),/ s out
(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 ato,upnh ha1hq40 ;7bi ljhblh:*ms 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 sppscjpv 2(k,;l4 z z2hkinning at the correct rate, engineers fihk2,lp2c zs4v;pkj z(re four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, 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 8.50 cm t+,i2j 7 ih 6*tcdyu sl+k dzlc;v6,eiand a mass of 0.580 y 2u hec vi+ic,+l 6,kd*ti6s7tdz;ljkg. 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 bc:7la*7tr/2lc-rn s-e6x y bu)v-+ nybf jtwbat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round ab ybyn/0kct2s1* 1jfit9h ek1nd 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. Calculate the torque required to producnkb2fyh0*1b tjic kt9 /es 1y1e the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10, xmkw: gaf 0ij+/526nqxpiji;300 rpm is shut off and is eventually brought uniformly to rest by a frictik/ iwja+50:6mp2nifxji gq;xonal 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 acceler4. do 0ag,o 5oyfccbw7ates 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 forearm, v oys7s8.tz 3p,.fg uewhich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from ret, .g7y 8sv .pfz3uosest 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 b3y: -frp7x qpgb;020k thqhn- bb p1n9ud bjb.e considered a long thin rod, as shown in Fig. 8–46.9qq0x-h:bh nd 3u -2fbjb b p1.7k bnp0;ytrpg
(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 consisvq0x5:. yba5k43 opmpppv c 8sts 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 hammer 5z8jan i0e yen52qry,tm- h9h from rest within four full turns (revolutions) and releases it t8jnh5neya mq 5irz-92eyh ,0 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 moment of inertia k ;bxl4lswrhsyhycp,p;(j) u 7:np89of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develi8(1zwbbf63gu /h - s2h uegltops 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 rol7ydyu(jba xvmp 5uttll)6(63ls without slipping down a lane at 3.3bj6l l()v6tup7amt3x y(u dy5 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of8pa +6sd(pc:j,uc wou two terswac c (o6ppu j8d:+,ums,
(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 mfv(:v/t:sy aand a radius of 7.50 m. How much net work is required to accelerate it from rest to asty:v(vm/ :af rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass nz+o x ,9,,zdbhddzh,8h mhd 01.80 kg starts from rest and rolls without slipping down a 30.0 h0nxdh,+8,,d zd,zhohd 9b mz $^{\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 itsoj8d9pzx w t9fpz,i,, tip. It starts to fall and its lower end does not slip. What will be the speed of the uppzj9otdxf,9p, ,8zpiwer end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum h eu(*;eh ouj3/ xp8penzq w,1of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed o8uxw1z o j3ee/ (hphne,u *;qpf 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momens4j7 wka xg7(itum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm gkixa s4wj(7 7when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotatinqs()ml+x ab st p,w8c5a7e 6+gb7yjwug at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 87y( gw+la8x 5c, 76atm bjupbq )ess+w–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 ineejy,iudl 5 23 z,;bshartia by a factor of about 3.5 when changing from the straight position to ths 5,;y3 z2li hj,eubade tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an iniqv.8pk8-ofswmc5-k 0dla ay xi2g,j0tial rate of 1.0 rev every 2.kxc gy a0vjs-k5,wp o2il 0daq-f8m.8 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 rota)0dbfzft p - 8gdq7d8hting 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 frequency of the wheel after the clay sticks to izpd8 df d th)fqg87-b0t?    $rev/s$

  (a) What is the angular momentum ofn .z7gfd47e7t8ac px g 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 the Earnd*r0aqt hray/ 3zr ez(7 n;n,th
(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 0):i8hjhvv gvwp )( ;hfs rltty(5)i mis dropped onto an identical disk rotating at angular speed ) iw) l(j) g(5ytr0v8ms;fvihvhph t:$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atrjf1l2vm4ruw 7w-*a u0az f7r 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round pl-tw0l,g6t: nw ( lfzjjatform of radius 3.0 z(t wt,6 jlg:lnj0w f-m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-rombusn ztb n5.w +*q*n7und is rotating freely with an angular velocity of 0.q*wb 7.+t5nbsmnz* nu 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, lo+ vxko nlg +6tw2/(cdhsing 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?(ro/++ndh vx (wl g26ocktund to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 1ye17j hl 6ey0u20 $km/h$ 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 ./gyo+( vndfr$ 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 c,o:2u0ymemywb r 3m ,jan rotate freely without friction. The moment of inertia of the person plus the platform is r3,:bmjm2muy 0y,eow$I_P$ 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--m8 8e/gdx jfhround turntable that is mounted on frictione/dx m8-gh8 jfless 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 theij2npr y(::916mvr8 tmomsgh.+pmt q 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.yhqivmmpoj6t+r(mp s18 mgt9: .r 2:n 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 always faces the Eart s6cwch3 or:+i/3 huiah. Determine the ratio of the u3:/h+i s6coa w3 irhcMoon’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 at a rate of 1 m rtq3jst,ca3-/$s^2$ 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 unxwu:5bb 9ntr0ci5exqv m3 4b( x 8b4yniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angwq4 xcbx 53yt95m(x r 0n4vbbei:8nubular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid c 4rplowd * 95qgbnha26ylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when 9o54r*gdn2aqplhb6 w 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 ofd4 k1a7ro* i(ir 8vdwc 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 ;8h0/to hedz smass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries t8 esoh/0hdz ;off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tornn( 5vtv c0tv3u)fa0 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 mass 240 kg, and should be able to travel 350 km without needing a flywheel “sp0an5vv(utcnrftv0) 3 inup.”
(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 iv; qigqv 8z2 .3keg+ban $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.i y4 sgxhfo7wf5462 sp3 $m/s$ 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 j(2nyh:v0 lqx L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fq0 x:yhnlj 2v(ig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius s z5j ;kx8pmt8R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The k js8tm p;5zx8step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makm w+jp*,sq yi8fjf8pm qu;:i/ ing a turn with a radius The forces acting on the cyclist and cyc:/p,p8+m w fsm8;qy*fiu ijj qle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins0p2.mi1a5:aogbsgo nx e2 u:t whirli amguxoea2sb1 np:tg0 i 5o.2:ng the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylindo uj7g6g,;f7cg nfi-xbf 05pjer and her arms as thin rods, making reasonable estimates foc-7b;o0juffg6f7gj5gxp , inr the dimensions. 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,ngg3vwc7avpz i m:ngt(3j 12 which consists of two cylindrical plates, of mav2 z,gvawn7jn p3 m(g1t cg:3iss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rough track of Figevhhcdda t+rno/8 pl0(l( ( n+. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the lola(v8e/n td +(ph+ncrdlho(0op without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do no p* b,9(gruusdt assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revp) xi )0pjie2v2l+cec olutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acc e0 c)2icvxjl )2ep+pieleration 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