A bicycle odometer (which measures distance trhsxk/hq6,k0j9 eihrgod 2 .:qaveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens ih2sor069,jh qxq ig:k.khd/ef you use it on a bicycle with 24-inch wheels?

Suppose a disk rotateit63qc1f nc xi;9ur t7s at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velo3i6ft rctcxu q;i 71n9city 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 v,hi- y*q0xa q, vq9xnyxwl;p+alue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than m4a *mt 7 vn0z7r9c7*khmqo;dkql-toa 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-upivptb tgu *my088 w;y0 with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to a8pv 0iywg*ty80 u;mt bnswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedi0 m8a1n c(q+noa( rbk 6eo8ulal cranks. In which gear is it hardu8 knmorq 8o+l(an0ecib6(a1er 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 c ge;)rgk::o xbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribukgex )ro;g ::ction of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bei1n,d did. rmmi; pf9f:d )(evam?

If the net force on a system is zero, is the net torque also zero?. um sd4z( sboj.mh8o; 7w0gpg If the net torque on a system is zero, is the net force zpuow.mdsm (0zh g.j;b4og8s 7ero?

Two inclines have the same height but make different angles with the horizontal. 5r2t+qn xqgl+n2v/ye The same steel ball is rolled down each incline. On which incline will the speed of the ball gqrl+ 52 2+yen n/vxtqat the bottom be greater? Explain.

Two solid spheres simultaneously sm5u isf3;n ;ny yl2 ny*y*emr5tart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total ylnmyy 5 mye f2s3*nni ru5;;*kinetic energy at the bottom?

A sphere and a cylindert0co4/baz ek7kbq 1q (v .d(sx have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed 0q(qo1b xak47 cb/.kst(vdze at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mos cdyw*x3,8+ mb n;ucdbt moving or rotating objects eventuall d,cx*dbynbw83;+c m uy slow down and stop. Explain.

If there were a great migra+t 6nehl a)4tvtion of people toward the Earth’s equator, how would t4)vl tnth6e+a his affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any init2* vi-b f:zi)sz8zmyurd s6 3uial rotation when she leaves thei3 8uzv-2fu6 is rzszmy*:bd) board?

The moment of inertia of a rotating solid disk about an axis through its center0smh q3 yp7 vtsb 9kz(+.v:xpq of mass zh:7tp9pvqxv3 mqb.k0y +s (s 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 outstr5iolrb3; s98lwuyhl: t / np0letched hand. If you suddenly drop the masses, will your angular v y5:8 in9lulhwtr3l/ p0 bsol;elocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howev)uw h;p1t(k vn4o ub1ger, one is hollow and the ; go1(nwbhvp kt)1uu4other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rota6 ujmsm0reb*-;l n3 kuting on a horizontal axle points west. In what direction isuru6j nk3lms0b*m- ; e 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 top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large free arziz o0+c5;ply rotating turntable. What happens if you walk tozp+or0 5aiz; cward the center?

A shortstop may leap into the air to catch a ball and t:,xou bofcui)vs-y/;hrow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite directiono,xub;u :cv/)o y-sfi (Fig. 8–36). Explain.

On the basis of the law of conserpa( th awwq-n;o(pg+7ly58fd )g 6boivation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller p a(gnq lyoidw p-f;a b(+7tg)8 h65wocan operate to keep the helicopter stable.

  Express the followinq71so7.nhl ohhy 9g;t 8bsba)g 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. Calcu (s)*snfhh 7dulate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and th7hsuh)*nd( sf e Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverurr9p y2kmq,c.l+ si0u;sh 1 uges at anl rqi,+0s2. h k; uy9r1cspuum 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 is turned oglo:; g hkxm0* yhm)r-1-mbxe7pdrm5ff during operation, the blades y r -h -p;m17kmlg:)ex*5mmbd0hr goxslow 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 the ballh y ;sz0jm)8nvr /l*ap makes 15.0 revolutions, what is its diameter?rmj)y/0 pzh sn8;alv*    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do8fag7h sfp-)2abu9ag-u.r x i the wheels make? sparg uugfafh9.a7 ) x-8b-i2    $rev$

  (a) A grinding wheel 0.35 m inj ejuh096 kk2.r1 9oazc /mtbc diameter rotates at 2500 rpm. Calculate its angular velociku c .r109ko2jtaj/z6m9bhce ty 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 revolution in 4.0 s (Fig. 8–38d);i7+f*bcte n jhm .uq.*j 6e-r1c h*fvhrs s). (a) What is th r *t7+6un ebf).iv r.m;fhjcse d*h 1q-jh*sce 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 velociuue8iqt,(k(tk5f0aa c;fjazc+ 5m: jty of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi):jfapo5j 76z lr)tyg nt
(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.: bhjm +c tqi3s5gq3,vi 59ft8tyhg)x 0 cm from the axis of rotation is to experithf5h m9ti5bq3g+iq ,syc8v: tj)x3 gence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center f,zk ia:ge2 j.jrom 130 rpm to 280 rpm in 4z2g,jakj: e. i.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 radiu60shb f,m ee)fs $R_1$ 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 Apolng3rwjdj *f o3msg2( +5qt7 oilo 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 during a 12-min dgsq5 +fjoin r7w3gm3*j2ot (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 22bpusv2 t 2/iu-0 s. Through how many revolutions did it turn in this tiu iut22sbv -p/me?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ca0 uidycj+rjg 77: keekp/7+sn lculate
(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 in a whirling “hu9xz a un)s8je8man centrifuge,” which takes 1.0 min to turnz s9xj u8e)8an through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly wbsn 3s8+ow vz- d5ml1from 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? zl5wn1b+v d3ms8-ow s    m

  A cooling fan is turned off s:ubg 6nqrczh4(/8zn7dgv wt b3cli8wi81 2swhen it is running at 850rev/min It turns 1500 revolutions before it comes to g/8 z vc16n82gl3 sqnirb iws 4d8 h(:t7zbcwua stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frof;cd1u+1w su 4vrzny+ 8ef -com 95km/h to 45km/h The tires have a diameter o fe fovdcs;-uy+1z+r14w8uncf 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 reducesalr /bf qetd4 v;v+v38hct/7f its speed uniformly from 95km/h to 483lb vcrv / ;7v /a4tdqh+fetf5km/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 on each pedal when climbing f2y a 8m19b/tmuepd/qrim p,5 a hill9fydmur p5m/8 , pit/ 1ma2qeb. The pedals 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 ast hg2mi9 a7icau *s26 door 74 cm wide. What is the magnitude of is2hma9cta*g u 6i27s 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 Fw uu hhijs 8:rz8xf3(xy*soo 1(-jpa1 ig. 8–39. Assume that a friction torque of 0.(ifpus 1(*r1u hhyj3xsw j: o-ax8 o8z4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachev cm7n;.sg0 k nu)r5myd to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relm0rkycvu nsgm )n5;. 7eased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of 91u2/as ,:c ,ujrkwli+txool an engine require tightening to a torque of 38 wjx u: 1cu +2l,oaskilrt9 /,o$m \cdot N$ 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 of radg a;1bsmy. uh/kkws-(ius 0.648 m when the axis of rotation is throushm/sga k. (-1wub;kygh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 *ysw f3.ug0 .g.vkqd hcm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub s0*..wgfuqd hg. vky 3can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light yvgr+lxbl3-i l-,6ma9 p lepb1d)8orrod is rotated in a horizontal circle of )x la6-r9pl erdgpiom1+b ,l8y bv3l-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 6(f 4e1uka 89mm c ewg.ez5(lsh y)opopotter’s wheel rotating at constant angular speed (Fig. 8–42). The frihze.gcy u4l(kose5 19wem6 f8o( mp)action 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 in* j:*:sqy coxie8x5 i8xz xh0uertia of the array of point objects shown in Fig. 8–43x5 iz*uh8sjyci 8oq x x:x*0e: about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose 6sjv y2ni;th,(yhi6l 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 cylindric qdx92jmp:pja: we 3:eal satellite spinning at the correct rate, 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 steady force of each rocket if the satellite is to reach 32 rpm in 5.09w: pa e2:dpmqxj3j: e min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cgf)lz+ kj op;87k1) gzdcpvx*ylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cal ppz 81g+;)c lj7kdog x*fvzk)culate
(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 from rest too3 vcvdy,sy.p8k3s(,n4 wei wezq )s8 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 anh4r 5y4p 7yy-;p2h nj eos6mvek nb,i(d 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 requnhryip;hb- p ,7sejk6emyv52 (y 4o4 nired 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 shulg1xl+ag-t kk g /2*:qxoclzfd -q,c;t off and is eventually brought uniformly to rest by a fri:zkg ;g /oaxqkc-2dl*lq g,1l x fc+t-ctional 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 3.6ve6 g ;m/uem-6(;a1jp r.rdyzer5oou*oo8vr-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 thro)3 trpr,y s-bty an0.mwn solely by the action of the forearm, which rotates about the elbow joint under the actina bty pm-r0ytr .,)s3on 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 Figr w pbo9pl3fjl951 zmqd/*;y b. 8–46lbpl ojwfy bdr ;99pz3m5*q/ 1.
(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 consi-i+ 4wc; f*qscsmr 1roe2cyqe,aa.w-sts 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 from rest 2bnzkc4 9btw8within four full turns (revolutions) and releases it at a speed ofbk bz2n9tw8c4 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 hafop64z5 h v.jz sfji8xd -;sv+s a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tor:eo-q+d 5gv beque 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 radiuswhowxxc)t oz), bd nb2,4myz1*.ey 7w 9.0 cm rolls without slipping down a lan7ywtzw),h)c d. ym *z2wxo,n 1 ebobx4e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth r:)rbk o2;s6sse;hqv with respect to the Sun as the sum of tw; 2rv hs;rb 6kessoq:)o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mzdm-0m txm ,x)ass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 dz- xt0,mx)mm s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1l2/a-bn i ai,27zn5uh/hs fgn.80 kg starts from rest and rolls without slipping down a 3z 7/g2li 2nhnuf/sa,a-ihbn5 0.0 $^{\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 verj*u)p j bryv2fr6- .+zm0pihw tically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the j6ib-y )w0z r*.j upr+mf2pvhground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of 4w3l ;c;tlu84egnjg / 7tb nxia 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1/3;tj eix; c4 nwunblg4 l8g7t0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular molv5 /-.l:* sfajgzrgzmentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating z/lz* j: gafs lr-5v.gat 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, hands2lpv.)7uxgrjo v4yj t:vz:kn,; s x(v at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreasexgvjr s7jlv vp;2.x, 4)nu:: yt v(ozks 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 inertia2 2th/5au bqkr 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 po/a2 rk25qub thsition, 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 initx 5 e5cqs hg)0*3j fmu;ogxm0wial rate of 1.0 rev evj0u5ex m 0*h;3smo)xg 5fcw qgery 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 voumgojbo mf/7d65,m. ertical 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 radmjod.f7ob mm,o /gu65ius 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 nbr xwspfw0 9zy ,-3m;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 momeni+ gph(pkj y0/tum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I isz;tea(d/ tk x+ dropped onto an identical disk rotating at angular spx;d e+a/t zkt(eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2klpc.taw- 0m7 .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 rota xpqoo jtojpp7+ ,:-3ating merry-go-round platform of radius 3.0 m and moment of inertio:ap 7pj j3+ qo,pxot-a 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-b1zt ; nhf-lif (75utk8b4envgo-round is rotating freely with an angular velocity of 0l bnz; tt(5vb -u7fi1fh4ek8n.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 : 3zg:q a0.fhg -tavhwinto 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 carries away no angu0z:t.-gf a :vqg 3ahhwlar 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 involve winds in exc::a7.rte/xwqo59hgk w f ky 5less 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 ,pu fe6 f0+yta$ 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, inxf; kxltva:1* itially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform isa vfx*tl 1x;:k $I_P$ 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 2epwnvnf mv-t/7d5i, po 4wojd7x,- smerry-go-round turntable txf w477jowe t- ds,2,i /nov5v-pd npmhat 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 grotu.xy;ne67.pb:r ks r-q:x 8 b0h rqhlund with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holdinb nhlhq k6 0ubyt7rerx:spq;8r-:.x .g onto it, Fig. 8–50. The 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 Eart 2efa k.n9;x3fo 4abruh 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 angular momen3 9 ub. k4e;frxaonaf2tum. (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 mm plpxu /)b,t)/$s^2$ 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 radius x g+1ga.m0f+sk fo-xu hkoc wd:pm4ik(,,f l, 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cafkaxf,+pm(hck1f:+ks, ,-.4owix0dg lou mg lculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical d d67fa 1r/re4 jgcw(;pkrv qvn(6onh6isks, 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 it reaches1r66veg(a/rcor6d(qfkh ; n jnvp 74w 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 spev0 t /yl1gpcn9i/tcer ,bo2d1ed 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 o5g u/lxrmqg/-ur Sun, but with mass 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 lgug-/r x 5qm/times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollr ;eo zpdebxs)+ b)w03ution 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 diam3b;0)rsow e)bz+dexp eter 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 illustrate lm , yx7a3shqd4hv zky(.:pu0s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is roxv n3sr b 3a+1.qf-ut-v ,zh 5p 4eigzl)/kwfplling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., wed mb99hhy0q ;zkpii,0 ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular 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. 8–21 ziyikdh9 hq0,pmb0 9;g.]

A wheel of mass M has radiu7aqv1x y+k.p0 rej.)uj qgt3a s R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so) vk3qp yq+710urag .e jx.atj that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn lph3mi7s2v p* with a radius The forces acting on the cyclist and cycle are *3 vmhl2ppi7 sthe 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 qpso55 n *du*14gomir 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 i5 ndus **41org5miqop s the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin royz35 jhf9 uv4rds, making rer u5fjh9yv z34asonable estimates for 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 which consists of two cylindrical plates, offt,p-j i)rfb ffn d9 b.l*y((i ) f f .pfd(bb*i,f-ytnj( 9rlimass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 8–8 b gm8o h.k;twe8nw:hp*js6q58. What is the minimum value of the vokj8e hp*:sht868 n.wwb qg m;ertical 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 nk+7ukxps b8s li-/6 agot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions ast4g87 f37f fyeld8cv v the car reduces its speed uniformly from 90km/h tov4f3t yef7 c fl7dv88g 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