A bicycle odometer (which measures distance travele)pk. hta /kjbp3g,v pk;cj7g 0z/ .djdd) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you us;gdk,c.jk.7vhk jp j3p dg/bz/ ap )0te it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doe 5knc00r 1qxris a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point h x5cnrq ki0r01ave radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described b.b7fe x2y0 d0 xuk/ufyy a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque7 ek(n idg- ./0 )tnkewcv tdj95tygk5 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 qp q /bd,tk2;4li/ t/ot0*yn+ ohhgyp sit-up with your hands behind your head than when your arms are stretched out in front of you?q y4l 2 *nqy,hk/+otdtpipb/hg o;t0 / A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whee 7-yp68govd1*fnm60g2w+ d rroa empxl and three at the pedal cranks. In which gear is it hard* mr 27m x-v8g+yeodp6agw p6onfrd10er 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 able to run fast have slender lower legs .blih ck1,tfpc5h69pe2s :w k with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dish:b lp p wh5ct.6e kc,91f2siktribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrymch6s:+v(m*o;ng1. v-jzu0 xacva pfv +)g hr a long, narrow beam?

If the net force on a system is zer) w.(b9: yaf/e;u xgij5b1ni sahk5h io, is the net torque also zero? If the net torque on a system is zero, is the net force ze19inie.uyk( ;/gh)aijfhb sw5a b5x:ro?

Two inclines have the same h8b464v;er9k;n y pgmw kaq k4qeight but make different angles with the horizontal. The same steel ball is rolled down each incline. On whqq4 yvk9;rne6pmwgak k44;8bich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from /b/lw/ 1/7htx4;q qj zpe.rqtz5 g7vtxi g5myrest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incl/17p5mvgt iw.;l4q7 /xgyj/qzqx te / t rbz5hine first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They stae ahg0wh5)lf zk/92agrt 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 ta/0eh9kl f )zg hg2w5ahe bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum artcsnvm 5iyje v d:1j(97 f06oye conserved. Yet most moving or rotating objects eventually slow down an9on(tfs 06 v7mydj y15cive:jd stop. Explain.

If there were a great migration of peopkpkppg.d; vdw1q 64a 5le toward the Earth’s equator, how would this affect the length of the dpkpvw6p51 ;dga .qk4day?

Can the diver of Fig. 8–zs0 6 z/pc;a.zct/puz29 do a somersault without having any initial rotation when she;zt czz06.puc/as/z p leaves the board?

The moment of inertia of a rotatinguvmet8vw dg 0,i7ww6lvm35 :w solid disk about an axis through its center odwlwv3v,u gmv7i6 8w0tewm :5f mass 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 9k: waq(eo+kduk;voy e m )5s2rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decre( 5dw eo:u+ )k2y9kmvq; aekosase, or stay the same? Explain.

Two spheres look identisko2 e9w o *kmzme972/g.u,zgobzx.ecal and have the same mass. However, one is hollow and the other is solid. Describe az/wx..zzkome k9m 2gu s, ob*792eoge n experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontanhae*)k0n gflk7 9hf- l axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear accelerat0n 7 hnh g*k)fk9f-laeion of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of ayta/dp( b3t qa97itv .crj,3c large freely rotating turntable. What happens if pt/ b,ry9vqc ac( 3 tjadi7t.3you walk toward the center?

A shortstop may leap into the air to cacd jqdxf 2)u bud0w483tch a ball and throw it quickly. As he throws the ball, thefb0du j2d3u) wdxc84q upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momk8( k6p-cyu fjentum, discuss why a helicopter must have jukf8p cy(6k -more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following an2 xpx:gl. jy0e8 )ixldgles 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, usi( os88wiy x-m,0cs7 cje xz /uu-y9nfvng the information inside the Front zwsv7,-yun 88mj/y(uoxs 9ec0c x i-fCover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direct t+ t)bkunx / -h+2teoqq.3mabed at the Moon, 380,000 km from Earth. The beam diverges at23)nq+/au q o+ bhmttt x-ekb. 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 rate of 6500 rpm. When the motor ie-ux/vijo2htwg x5-mh*o/ j, h9 pov s)5ckq:s turned off durc/m oo tk hv*eox,-)ijqxj95u2/ vp5sg-h :w hing 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 floor 3.5 m to another child. If thea w8eq-b4o7 m-03 x.ltechqx *s6zhzy ball makes 15.0 revolutey-e6qa h.hb*s lwtxqmxo0874- cz z3ions, what is its diameter?    m

  A bicycle with tires38nc9 r)m- u7 rcshwonox w6s; 68 cm in diameter travels 8.0 km. How many revolutions do thecuon6)9xcwr3s m swoh78-r; n wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotat: 813en8-h qr koapuyies at 2500 rpm. Calculate its angular vel18kh-e3 qua:8iprn oyocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolujrc/ o x44 a::axp5qz(n6u fsytion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a4yaqz( o:f rsc6p /aj: 4xu5xn 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 of0gkk( xg.zfn41 hwo-2 /vayr a*:lf6vjbk- s v 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 po+j3.hipuxyciov rzds32v) (be7 ,zr va2 * h6bint
(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 qoraa3,kve*: seth4 142f sby7.0 cm from the axis of rotation is to experience baey thr*efo43: a vk4,ssq 21an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpmar: f u3ohsaj+4u:/tx to 280 rpm in 4.0 s. Detjs/o : u fr:ut4ah3+xaermine
(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 2+ n)wosu a33ba(m3uh c3xe te$R_1$ 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, astronaucz*m ih0d+*( tgn xdr1ts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At th0 h 1tr+iz* d*gmcndx(e start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in p dh2* eyljo4m+k.fs-ae,r/b220 s. Through how many revolutions did it turn imj e2odl 4/fy-+.a he*sk bp,rn this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpmrw1+ y6f y wi3dleoy30ula;k59 j+ )zr(ircwy 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 pcg/y uwu15lq p 72phe(upt4 :highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its w c:ylp1/u54ug q7et(ppu2phfinal 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 accelvhd1ja *9 o e9s9wlg5gerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? gd la jgw9 v1o9*e59hs   m

  A cooling fan is turned off )fcqno5ofd-; o3.w2ytopm5bwhen it is running at 850rev/min It turns 1500 revolutions before).c ooo5 yb o f5tqn2w-d3mfp; it 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 reve07igeg 3 bliy-ae :bmj7y 8r:0 ov,xjolutions as the car reduces its speed uniformly from 95km/h to 45km/hib3 077:ey j8-g:aexbgeyr mo v0lji, 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 mak61 j26 rgtu8 r jl9g,rru t,dfmvqm5.te 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter mu1 gj jtrrdv6lrr9 2 ,q m58.guft,6tof 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 ped w3gj; *dtx7rwo3ms*obl1+ jfal when climbing a hill. The pedals rotate in a circle of radius 17 cm. ;oj7xw+1jl 3 f*o3*wdgbrtms
(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 w qdz3p/u, w:h/fbp5 anide. What is the magnitude of the torque if the force is exerzpu3p,d5 hq:b fw/a n/ted
(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 tor)v m5hhwbc0 2gque about the axle of the wheel shown in Fig. 8–39. Assume that a frc w)hm5 0hvgb2iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of massvscm ;2,z52 c (ofbdrqe2lm 0l 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 hoco(zlsd22 b ;mq25 e,cfvml r0rizontal 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 torque of 387n.8,xqizblm1 d 2zh q qhb . d,72q1zx inmz8l$m \cdot N$ 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 oj7bq xhgp1h /m 2od9+if a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cente h+ mb/hx7pi1 d2gjqo9r.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66s jvr 5rb,(va5k: xho1.7 cm in diameter. The rim and tire have a combined mass of 1.25 k5rx jor,1( k:sv a5hvbg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of b ob(gw8y h8u5a thin, light rod is rotated in a horizontal circle of (ybguo85h wb8 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 rotating atln80 iw)r -k( n4.pwxyshj*he constant angular speed (Fig. 8–42). The frictw0*8kn i(.h) se-j lrhxnpy4wion 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 array ofxgp7(235 xobdsuf a;e point objects shown in Fig. 8–43 about2x aoduxg;eb 57 (pf3s
(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 to .*ao67akzfnstal mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate,w gbpn62nc djg .zzgd)n2k m/,g6m19 a engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steadydb6g/ .nz29km)wajg 6dzcg,nnp 21gm 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 onbwg2 hu )y /j251vku(7kzoy8.50 cm and a mass of 0.580 kg. Czh/kou5u gokwj1(ny2yv) b 7 2alculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fro;ymo3o8pt6 p u +ylp5zm 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 merryz f2db.t,q)29r ro jn8qnh4mz-go-round and is able to acceleraq 8 bj4rmz9 o, tn2.nz2drqf)hte 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 produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broughtq+. g* pctb+n1(dfpm l uniformly to rest by a frictional torque of 1.2f+ct+b (qpgp1.n d*ml $m \cdot N$ 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 atkktuu8p2 8y3q l,jgq/ n5e3yh 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 9wbkdmpcg,s;wm 6-k8 of the forearm, which rotates about the elbow joint under the a, kp wc;ds98wmkb-g m6ction 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 blade can be consider.f,ei;6ue9dw r) mh wded a long thin rod, as shown in Fig. 8–4id6; erud,em ww.h9f)6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse2 fcg2dk gb+p;s, $m_1$ 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 withjg5 ml.7* sy tndhtf,/ p2p*r cq yhsq1q95ln:in four full turns (revolutions) and releases it at a7j/y5h,nc sn 9l fm.* t2*qq spp:5 tly1qgrhd 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 of y/r (pw.f nz8j*h xddit+ l e,bt1c+9r$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28njqilh 2wv6-7 0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg an3j ; tcnsea9e6d radius 9.0 cm rolls without slipping down a lane at 3.3ac3e6;9st j en $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energ(:kjh 9zu: 7m r 6suo,uiq0zeoy of the Earth with respect to the Sun as the sum of two tje koiu,(hu 7qu z:zsorm09:6erms,
(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 of 7.50h/jk dr9+k* zx m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? As r9d*/kxkh+ jzsume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts r 8(.)wmhj+bcf2dt.zwl , zw ffrom rest and rolls without slipping d r ,htdwbw8zw+ .f.jzc2f( ml)own 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 2o*xdda(d.m gkdz;d 3and its lower end does not slip. What will be the speed of the uppe(.; a*3doddgdk2 xdmzr end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular l rs:z7mv w (rs9w, c:tm6+wqlmomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m 7wst:lm l s, 9c:zq6+ (mwvrwrat an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical,hh 0u 9hgdgw o1+-zsd grinding wheel of radiu,-9 hgh0 uo gdzhdw1+ss 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ram,h; ga-y0 drcz;0)zr vc 8fjxte of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed )vagy 0;,xzm08r cf-;dzhc rjof 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 q36v aqcga/*0l( 3m.fd3 wo(yz voprhinertia by a far c63.a3y mvo /vh0 l3ozqgfw(q*a (pdctor 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 her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can inuho+2 2 pcte(bk;pztr0whx6 1crease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to ux26 pc whp(1t+ke r; t0zob2ha 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 its center atutzv4f2q;w :fgh +.by 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, appr4qb ;+gu w.tvzy :fh2foximately 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 momentum of a figure skater spinning at 36cz : .wwvwtxu d91 g*uwjmxg9sf;j :2.5 $rev/s$ 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 momen e/v ie3h+p2flsre ( 4i-l(zdgtum 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-um5+ k y3bxhn moment of inertia I is dropped onto an identical disk rotating at anx+bh3ymn-ku5 gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2jx:(- dqfybbc -kb0ks54 ohb7 .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-rpoyx * i5bn)t0ound platform of radius 3.0 m and moment of intn5bp*xy 0) oiertia 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 rotatm40y 0(pyvyp9vhow u8ing freely with an angular velocity of 0.8u9 yowpy0m0v y8(p vh4 $rad/s$ 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 9)b) up/w1 plf w3vosfinto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carrieswp3 lb 9u)1)wsvf o/pf away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involywadtb8dywzpf4etq;k -,ok :2 :q3u4 ve winds 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 h) x jj-x4i+wi$ 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 platforsdx lg / b)qw+w7ok+;fm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the+)gqodwfw;kl / b xs+7 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 edgr bghikx*)3v *r. 6jsoe of a 6.5-m diameter merry-go-round turnbr givhr.*xok)6*3 js table that is mounted on frictionless 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 lyig+3sz7u u a j v1g(akph1ra*.8wt+oqfng 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. T8hf (.* jp+ta1gu3 kao vu+zqsgwr7 a1he spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwayqp + b*rh4.ed rux3cf6lr/n(bs faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital anguurrfbl .pnbd er 6+*3x/4qc (hlar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fromvy (ahc7p89zc :z2p-odi llk. rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radiusjkd3 /at*vk 2xrom*j/ 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration.vxj ro kk/*/jd3m*ta2 Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eacc p(( h4uapk3mh 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 0.01(up34 mapk (ch0 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 spe krun7zrjm3 is32,hm .ed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotat ll 93fnvw,0p0pvmv(v ing 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? Assuw3nlp, pvv0 fvm(9vl0me 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 lowsbksdp *(x v;*-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywhe ;*d(skpb*sx vel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an ;w 4li(ap-.tchtv:g v $H_2O$ 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 spe.hl3hmwfxe yoyam9(4.zf(( n a ,ua6oed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igny kwolsl0tv2evh b(2 062wm 7eore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear accelerati(2w o2vw 2v0sm0ktey6eb l 7lhon 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 s 0+ta4wgitc5 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 r+4wg 0 taic5stests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tur ux3io45cl+ 0p he(vyhn with a radius The forces acting on the 0o hy4 lx(3 uhve5pci+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 lkez2h2 3e*yc +(qj0y kavr y/sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s.lkjqyrz0a2 k3y+2c/e *(vye h 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 kr4h2swabh g) 061oo) xe2:zl.yclib)ldwn; cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agai)w2x.n rigz4bdh: )0lc;)b s6 2 yehwo lal1oknst her body.   

  You are designing a clutch assembly which consists of two cylindrical platelr) x1c5shvzbq1bz;4tt ufo 8 :)e u)cs, ofc;fu1 5rhzus t)qct18l b )4vb)x:zoe mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig. 8–58.u (irskyl. 2(6oh8s6vix0c ss What is the h cx 0ivy.ru6s ss (o82l6ski(minimum value of the vertical height 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 assubgam2lezn. ozh 2m.s l.*yc:8 me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutiontr3 ii;k;09gyzb :frqs as the car reduces its speed uniformly from 90km/bi i3; ;kq zry9:f0gtrh to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s