A bicycle odometer (which measures distanc+kxu;a /o78 (pxdfblcxro 22oe traveled) is attached near the wheel hub and is desig /xlafd; +8ocuop2 oxx (k7br2ned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at i (:-2lobi i kvtglq0o v*u6e)constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linea6*0v(qgivik)t lbe oi 2:ou-lr acceleration change?

Could a nonrigid body be described by a single value of the angular vn92ha5* 5;k.uysuoa:g(jt0 g j gptri elocity $\omega$ Explain.

Can a small force ever exer pf: 47i7rknnp;er5dlwqh85y t a greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with od02qb eoduu*++h+vu your hands behind your head than when your arms are stretched out in fro2+od be uhov++qdu*0unt of you? A diagram may help you to answer this.

A 21-speed bicycle hasat z4 y92y.)(n 0ws5wguep zya9tc.cf seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or . pctwuc9a 2 y59wze)zyf sa4g(0yt.na 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 sl - 7mo69os9h pq;fzkciender lower legs with flesh and muscle concentrated high, cloos; 7 qm o9zi6kf9hp-cse 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. 83s5smkxqfrgcap c ) j,cv701 a8wi6u5d83 hiy–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? I*kw 8ht7t4 8i w*gdv0vg ,e scgmjj)j2f the net torque on a system is zero, is the net fo,wjjd w ktv8g j7h0si*c*2)4egvt8 mgrce zero?

Two inclines have the same height but make different angles with the hopo/a :/zw okcdqy5)n65agpf +rizontal. The same steel ball is rolled down each incline. On which incline will the speed/pykn: fw5)q+p/ a6dg5oc zao of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rior ( lfkm/ooie 5l6 p)4rwb)hs1- rr5olling (from rest) down an incline. One sphere has tw) 1ool-4b fr)(e / iporrhmk5wslr6i5ice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder 1w abxoahez+:h tpgc4zp .)-u8 hhtd70g;g r9have the same radius and the same mass. They start from rest at the top of an incline. Which rea9 z;xg:huhg1h+-zoa7 gt0dr ptwe4ha.b) pc8 ches 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 a:ffi1kik- piqb, , uc2ngular momentum are conserved. Yet most moving or rotating objects eventually slow ,iu:,k- b fi2ikqpfc 1down and stop. Explain.

If there were a great migration of people toward r6zl,-gl:skbkl2, kg the Earth’s equator, how would this affect t krs,lg-, 6l lbzk:gk2he length of the day?

Can the diver of Fig. 8–29 do a somersault without having any in gm,hyg6k0acg;x9+d i d(f )slitial rotation when she leaves the bl;fd90s a +dgm6 xy,g()hgcikoard?

The moment of inertia of a ror7eox g xu/cterx+;30 tating solid disk about an axis through its center 3x tcr 7goexr;u+0e/xof 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 rotating stool holding a 2-kg mass in each)u4 k7adjxog n3m9s,pj1xq c) outstretched hand. If you suddenly drop thek7nmaoc q sxg,)xp)j3d 149ju masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollowsn( s+qeg,:rw and the other is solidqges,n+( s w:r. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating onj),(kfm1xer p cwy0 j2 a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential li(pm 2kf ,c 1e0jj)wryxnear 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 t*egxuc ;; x7ec;h i938mcq qkfreely rotating turntable. What happens 3q8 c;emcgx;7;ute ih 9*cqkxif you walk toward the center?

A shortstop may leap into the air to catch a ball and thr bdrtjgr.3i7 rd my0+; 4mb2muow it quickly. As he throws the ball, the upper part of his body rrmu7 gj+ .0i42mbtm bd3dyrr;otates. 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 con3* )qd-ytxyimo8g y)lservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the sq3l) *y doyiy8m)-xgt econd propeller can operate to keep the helicopter stable.

  Express the following angles in radd+nm1 kgvwyh: fwrn7rry v ;-r0c,8b ;ians: (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. Ca0vi8,b6u0i ksa2plfv uqn .;xlculate, using the information inside the Front Cover, thkfp s,bvv0; ax0l in8uiqu6. 2e 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 blalh3 .yz 8wk1rig (1Moon, 380,000 km from Earth. The beam diverges at aykb z1wr8gl1l(ha 3.i n 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 650 /5jkjru, qjd7:v1r s5vwa/l k0 rpm. When the motor is turned off during operation, the blades slow to rest in 3.dsa/kjv:l v7ju, rkrjw1/q5 50 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 thep: hl/ ad3l7dd ball makes 15.0 revoludl: 3/dhd pal7tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k8:rloyx yn4 9vm. How many revolutions do the wheelsy r9 v8xo:l4ny make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2v04vdt 2 ffegacjp20 1500 rpm. Calculate its angulart41vaff j 2p0cv2edg0 velocity 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 comsl lk trdz;1n/lm +w5*plete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from *l5 +lswzkd t/;rnm 1lthe 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 aroui;cwmgh/ lp);p2 cv- wnd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spie+ 2z.mf2 s5 nnxm8qgeed 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 particu( edme7m/bslykxj y5z (r0* 7le 7.0 cm from the axis of rotation is to experience an acceleration of 100,00yzs 75m0l r*b( 7/kx e(eyjmdu0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centerhn z; k,l,mu1j from 130 rpm to 280 rpm in 4.0 s. h kmj;,,unzl1 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 radiuskb)grir(f f; ch1x,c dl+pg:j05 z;mk $R_1$ 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. csho pn6,0ac, astronauts aboard the Apollo spacecraft 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 dursn ahpc60c .o,ing 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 220 s. Through )wiw3 v9x7idp y cr(r)hbd vt+/t+uc5how many revolutions did it turn in t x/iv7 d 5 c9)c+ub(wrtwd+p3 )vyrithhis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculatel-8-gd i v;l66krob ho
(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 stress0e-)rcld5xly55tn, 7t e xbnmes of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rearx7bnly l5nt)0- ct5d,meex5 ching 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 dia2unsd : 04vnormeter accelerates 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 thinn4 :ouds0rv 2s time?    m

  A cooling fan is turned off whj3a j ced0b)wo3bat ,/en it is running at 850rev/min It turns 1500 revolutions before it/dj3cbw 0 3,tjo)aaeb 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 653pwz,2o kai.zflv88y revolutions as the car reduces its speed uniformly from 95km/h toi,8 .o 2 pvzlk3zaw8fy 45km/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 make 65 revolutions as the car reduces i7k*-dwn8g/u egipru+ejxe : /ts speed uniformly from 95km/h to 48uu7wi:r / d/epxn +*jg gkee-5km/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 p4ku9sw6, qf vwpl g7hi .a0bk3uts all her weight on each pedal when climbing a hill. The pk4v30s7 ah.lwk ,ub69q fw gpiedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wideu fy +y+n e5a,shpy,z7pubpka/8mk8r/b )l* e. What is the magniea pa/kfu,h yz/ 5)b ,lb*++sp8 k u7ymyrpn8etude of the 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 about the axle of the wheel shown in Fcey)kl3e r s(q/1ns /uig. 8–39. Assume that a friction torque of sye1cske r/ln/(qu) 30.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends fh ;d nxe,0b6eof a massless rod which pivots as s0nx6b hde,;ef hown in Fig. 8–40. Initially the rod 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 torque of 9/n//n0dp/i suv czkf 38 f9kp/cnd 0 s/uvn/i z/$m \cdot N$ 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 gr* h:ascgr.-,i. m v ju5;uch woar-+10.8-kg sphere of radius 0.648 m when the axis of rotation,j- *-r+.r; h.s5haa:u uowrmcgiv cg is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in ex:/lq-1bc w0vnv0i u diameter. The rim and tire h: -n iucv00 xqew1vl/bave a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on vpu e,emub:g *8x1 ogt(t(5a9i6ib lithe end of a thin, light rod is rotated in a horizontal 1vig*ae8pxb u6u,t:i olg5 me9i(tb(circle of 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 potted,; 1k7, v/ege0bvymtelc/ tu r’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N 0e7 ;b1k gy/m,vuelt,t /dcv etotal.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of p-4y pv68d6kmf f.omgsni p63ae ji2d(oint objects shown in Fig. 8–43 abouonp4 8ydsjm.k3i e6 fp6 (fam6dig -v2t
(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 tota /b wbx;no 7zfzg6ny+u h*gj74l 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 thvic9 t)mv4* 4gjcgb, ge correct rate, engineers fire four4*t4g,ig9c bm jvc )gv 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 ofrl1c4mb3 uxs; 8.50 cm and a mass of 0.580 kg. Calculateb3xsc1l;4u m r
(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 s9jz 1pwj+qfs2 jkq6y.z zu6* 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-driv+k,:i 2)6 hfmrvlnbi.bn mn:hen merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, nebn, i+ .mhfnhbn:i6v l:r2k)mglecting 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 brou ht/8d77 yhrcbxp)5l8lvquy.6nm bc8t5+szi ght uniformly to rest by a p/7m8dun5 llsy5)x+qvhi86.yb8h z rtct b7cfrictional 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 a (n 8b o72 i*,ly7y6 (rtt/vt spndgywt3.6-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 n(k kj4 t os7ex::*jag gq+7erby the action of the forearm, which rotates about the elbow joint under the ace(:gtj kk7q :*ox4+7 n grjeastion 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 considered a long thin rod, as shown in Fig. 8–4sfova 5cbv 222n4zcq-5 2v2zscc-foaqv2 bn46.
(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 twg xoxfwcnh/luo; 4jg;0.9hc: o masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from dfn5ig ilf/9a g3ub/tg *el)my/pb+)rest within four full turns (revolutions) and releases it at a speed of 28 u9f t/5dy * lg+nbl)g p3gbi)e/afim/ $m/s$ 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 o5fhm)6e h q jva5yyh*8f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 w7w.mtr m;.z s$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipp.c rq b4 (dj6dokoow/4ing down a lane at 4b(oo 4/krow d6.jqcd3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as22rby ,ytet90osm nz *kky8:g the sum of two terms, s2902 megyy *,tr:b8ky tnzok
(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 7.50 m. Hqo 2k(a1h,a;yb4xyp m ow much net work is required to accelerate it from rest to a ro4;2ab1ahp,x m y(ko qytation 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 1.80 kg startsn vz,ba f+lm .exz;6,w from rest and rolls without s;,lz,b zax m +.fnewv6lipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fijngn /o9p n6/all and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the gro6//ngjino pn9und? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momenty m 47sp;mg/rsum of a 0.210-kg ball rotating on the end of a thin string in a circle of radiupsmyg 4/r7;ms s 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 5s1zzh mpx+p*tdv70a )x.1 aah b1ujj 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whenh0a5v1a * .sz1 j mpzj+7t1habpuxdx) 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, 1fytaxc5j98om(6* w-f wi d)d yte)dlon a platform that is rotating at a rate of 1.3rev/s If he raises ym)i lo1 6f)c w-dx5 j(twe*yafd td98his arms to a horizontal 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 reducegcg 59s7+gq3n lob te/ her moment of inertia 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 her angular/gn 97 5telc gq+3bgso speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an f mcht 0 c*/vvc:bl;,w:yw2uyinitial rate of 1.0 rev c: ;0 /utc w*yfcbyhwvvml2:,every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through it*jeau ,98vp qb5 5jyaxs 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 tb5j *p8 a,qxv au5yj9ehe frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skq *8dgjcmp-n 8ater 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 momentp.y*a8 h/6v cdz9p pg+m rwtu-um of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment o yi 8td,l z:hhr2qx-q a*9*dhaf inertia I is dropped onto an identical disk rotating aqy8*lz2di:at,d xa9q - h *hhrt angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atsr(w )qs eho-gcv( :r6db-i(3ehwi 2h 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-rou;ll;w:tg mc 9wnd platform of radius 3.0 m and moment of inerti9ct;gm :l l;wwa 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 rotk2l8*apev jl( a53 q0r(rwigcating freely with an angular velocity of 0 0vwg2al5rk plr*a i cj3((q8e.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 dwar,f5 16 s ac4wzq */xjf-vi8 m :n3pflhushj,fmf, losing 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 angularfcfnu w8 m 51sx,liz 3:s/4vq*jj-h6fhpm,f a 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 involbf9n 2c(q(u):d5+pmjv: 7 uz*z rp0n kyqve hive 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 6-rz-v3w tihh$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely without(*trcq1/ ecigbl- g r3 friction. The moment of inertia of the per q1rc/r(l3gci*tebg -son plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-roup/f;g6dq *rah nd turntable that is mounted on frictionless bearings and has a moment of inertia of 17 rq*h/gpaf;6d00 $kg \cdot m^2$ 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 thmk*v/r:xwnhd 0x/ooi7y zg7br : gm 04e 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 spo/mh:0 v*z7yxgr0 :g x4okwmbo/i7n drol 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 Ean 2dh+ lyk(yh/rth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital anguy/l hy+2kn(d hlar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest lu,*hwp6;hrz g5 ,cjp 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 sha3d;x; oona qo mfd);k-pe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at x madf3oodk-n)o;;;q constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050(3dlmtu6 o jk2 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 it reaches3 t6(mjlduk 2o 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 the rear whb8 d4jur31 tut v;/ptleel ($\omega_R$) 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 m8ts+effh80q 000glwj, o lxnw ass 8.0 times as great, were rotating at a spel j+wqhnx0fg0w 0 8t f,eo0ls8ed 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 off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution 7myb . 2lz;ojlautomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a;bljm.oy2z7l 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 “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 h0bbjmwx 0lbp,auc5 ,h /e*l +$H_2O$ 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 surfkog* /yqdmjut -h4z4pjbi:e u; -mzh 59hi37 pace at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can o p +lpgi (3qvnl(w3 (icsp/v3pivot 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, deter3 lqv l3sp/((+n vi3 ocpgpwi(mine (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 R. It is standincsf nr dxj2:t4eu. ) 1q,;y 9mxaltr/0h +wuyeg 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. 8r,xltcm2ant9+du/ h y0re) uw.q jys;1f:xe4–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat ro4gwtb;k;e q wr ;c(6b uqitq2)ad is making a turn with a radius The forces acting on the cb2 it; c t6b)rqu4ewqkq;;g w(yclist 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 1.5 m and /z- ;clbenha; 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. What is the torque r/;;nb lza-c heequired 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 arm.+,ld3cbgitt4 .tyn cs as thin rods, making reasonable estimates for the dimensions. Then calculate tidb gc,yt +.nt.3cl4t he 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 cylingo63 ryhg wg4:drical plates, ofggw4:y or3gh6 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 ftmm,3.t p9j:tj z5q eof Fig. 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 lz. t:t359em, qjf pmtjoop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do xjr j 1 oss6q)k5up;p7k.ue y-not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thegsbj4gin 7md)f,fp1 ) car reduces its speed uniformly from 90km/h to 60km/h The pgg,n7s )ibmf) 1jdf4 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