A bicycle odometer (whilkjl* r((r. pzavo2/cch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if yora(cz2/l. v *lr(op kju use it on a bicycle with 24-inch wheels?

Suppose a disk rotateswizt33 + 4spy:cqnlqd;v h 38r at 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 hsl3; wq q34c8:ynp+ t zv3dirtangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloc,r vb w90d0fxcrkow1- )a fh tb5x)b0kity $\omega$ Explain.

Can a small force ever exert a greater torque thp; *xop5 ,g+q*yngybg an 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 your hands behind your head 2z)fsy; 2pl v.scae4v g82xg+zspg8 j than when your arms are stretched out in front of yous)p;f z zy.gcg8 +2xjvglasv2es p824 ? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockb;gw a64p(cgu hswm:9bywo94ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to by ;pm b4wo94g9 6(gwhsc uw:apedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run faspu ;*qxanx)f ;t have slender lower legs with flesh and muscle concentrated high, close to theu*p;fxx;a) qn body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lontc-vn( qtz9r3 g, narrow beam?

If the net force on a system is zero, is the net torqcnqz6 q*j( )*4or1pa3f jbi ahue also zero? If the net torque on a system is zcf3qnorz6jqbia* 4j*h1a )p(ero, is the net force zero?

Two inclines have the same height but my20 pi ch//5dp,md bakake different angles with the horizontal. The same steel ball is rolled down each incline. On which incly/dhmp0pi/a, 2k5 dbcine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously stop5;s+z0wu ,wy5w0iv *xd yv xart rolling (from rest) down an incline. One syi;u wosxwwpvyx * 5zv 5,0+d0phere 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 kinetic energy at the bottom?

A sphere and a cylinder have the same radius an p5c (, tktsr8l-dwr:3vugno 1d the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greav,tps5l8 d-( o3r wg kcrn:ut1ter total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are con jjdsaz0v k,dm5:)y4 pp7tp c9served. Yet most moving or rotating objects eventually slow down and 9 j5jm:y7k4t cdzp0,p adps )vstop. Explain.

If there were a great migration 8qh;.s2 n y4b v :qufrj1m+unvod,*mbof people toward the Earth’s equator, how would this affect the length of thv, udno ry;s.:* bqjmqf41+mb2vh 8u ne day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotationc 4v+ l+lvs c-nv qt2xj/n7b/g whe+bvq47 lg/vjcn t2/ xsc-n+vln she leaves the board?

The moment of inertia of a rotating solid disk about aegqm.l bz4 -hvjgj4)+n axis through its center of ma+z 44ev)jml-.gbhj gqss 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 sittinjf c+:cvxxqka .iqz d+c, 2j-;g on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop;2:.x+d axj cz-f,qik+cqvc j the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havbs9 xexq;( oyfo0:wiag t*0q t09:xgie the same mass. However, one is hollow and the other is solid. Describe an experiment to determine wh:x*g0bt9:0xa0 y (ogixq;f otwiq9 seich is which.

The angular velocity of a whes.bnf sy9/7ofu xd ,2iel rotating on a horizontal axle points west. In what directio 2duif s7bnyf.,s/9xon is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge ofqw k670npa ) q3dq5yik a large freely rotating turntable. What happens if )35qpk qydk7a n6 i0wqyou walk toward the center?

A shortstop may leap into the airtbzmr yc;ly-cb, gn;(7 -d8td to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips ant;,zdbl- yy7 c;dc(tnb 8 -mrgd legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular m3u dd esf:a qb2md78.u*)tfkbomentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more way a3tkfud*ed)bm :b7fu2s. qd8s the second propeller can operate to keep the helicopter stable.

  Express the following aef t+*bmowqjbos/c1+ 6et6i) ngles 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, usifbmvu*t)ad , g )l9 xn7ig6w+cng the information inside the Front Cover, the angular diameters (in rad9vcn * ,)6tlwg7xfb)gdia+um ians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 qv6e o /s3uyw *036g8qlafn2eaetvo) km from Earth. The beam diverges at an angle qavtque3y8fo*)2w6/o3 eg0n6 sa vle $\theta$ (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 th w k(6+phwkzts,7uo.qwjg .: ce motor is turned off during operation, th7jzou .wsc. kwkh +:t,p6 wg(qe 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 anothercv0zx 84f wk0q child. If the ball makes 15.0 revolutions, what is 00x48qf cz kwvits diameter?    m

  A bicycle with tires 68 cm in drm ppl;a7p2 gm6t0nm+ iameter travels 8.0 km. How many revolutions do the wheepp p mr0m7l;+na6g2 tmls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C0 ulhub 4a1*iqalculate its angular velbh laiu*04 uq1ocity 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. 8–38bsl d)(nupb86 p;z :va). (a) What is the linear speed ofv (zb 8nu ;6spab:pdl) a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in 8mar4sslnt-5 its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of agi nsu*i;vj,l(m z/l) 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) mugxx5kxg* kzx16u u x(cpn6q 8;st a centrifuge rotate if a particle 7.0 cm from the axis of rotation is x(5g66xx;1*qk8uz ug xnx kcp to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratombw7m9oo2 e*qmv(* bes uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determiv(2b*7 qboe9*wommm o ne
(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 radiuts ,-lx *d3kqeb2p5j qs $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themse5mb)mo(,u2npw vgc;2yx ge*k zgn 3w2 lves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceleran bmvywg3eg522*x u,2zw; )on pgm (kcted 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 15c;r-q lh8kk.;khyn,6bf )j nh,000 rpm in 220 s. Through how many revolutiq;k.;hb-n 6kkhj 8)lyfnhr,c ons did it turn in this time?    $rev$

  An automobile engine slows cpl7 k 3zjs6qy1ayz8*down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresserja:8u8d1 6c2nvg bkr s of flying highspeed jets in a whirling “humaugva 21n :jbd6rrc8k8n centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rh.hqoy /;t/u1lpa kgkb3f /5rpm to 360 rpm in 6.5 sh kk.o ;lfrag5/tqbh1u /py/3. How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It tu xrzp8wi7(x+ b3pgk6irns 1500 revolutionsbp8(3iik x +xgp7wr6 z before 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 revolutions as the car 9vtoy6 u,x /znreduces its speed uniformly from 95km/h to 45km/h The tires have a di/ n9 otyx6,vzuameter 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 it ajsy8-(:8*lswpawxm s speed uniformly from 95km/h to 45km/h The tires have a dia 8s8alwmj a-:y(*swxpmeter 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 whe6n6k)m(a yebeu3c tw :n climbing a hill. The pedals rotate tu)6c y ben wk(e:a3m6in 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 onq +be(uc* xw/h the end of a door 74 cm wide. What is the magnitude of t b+wqh(*xeu/c he 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 Frhmg.h7t 0/nz ig. 8–39. Assume that a h n.gztm7r/h0friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass )p cf s:p bpe1:gx:2jblpsra0e39p5j m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this syss3p f2 0 b pc:e:gbpe)9lpjxs5ap1:jrtem.

  The bolts on the cylinder hso9 bprfn) e4::4uem pead of an engine require tightening to a torque of 38r49)p 4p: fbnmoe:u es $m \cdot N$ 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 spherp5tkcn9kk f +7e of radius 0.648 m when the axis of rotation nk7f tkck9+5p is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The xg 6 *,f/c5 dnwp3eqqprim and tire have a combined mass of 1.25 kg. n,*xp q/gcwdp q35f e6The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ballvt0nvzk12 0 s9ctn;x e on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. 9 e zvvtt010;xn2nc skCalculate
(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 at constant angulwbhgk10 mkvycu.w/9 3ar speed (Fig. 8–42). The fricw/ kwc1h.9mg vyubk03tion 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 inerwfi zg* :o/,jl) caqe9tia of the array of point objects shown in Fig. 8–43 abgl/)qj9,wa cizoef : *out
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen .u-dd dy.*tq; fpc/ vfatoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, unifocdgkn6e8 tz2u3 vmf6/u0k:qwrm cylindrical 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 z8 vnmq0eu6k u2:kf63 d gctw/is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with ac:;v:f7oeb g s radius of 8.50 cm and a mass of 0.580 kg. Calculat:cg;b 7sfvo:ee
(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, acc3j0ky; dt-mbh elerating 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 smc2okl +5 rdo(rall hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the 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,r52c ( l+dkoor neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingl9zzi4xk8wb9 c3idk 9zozxfx- u 1(*a at 10,300 rpm is shut off and is eventually brought unil9kxiufa( z 8c x*idok1zzz9- w9b 3x4formly to rest by a frictional 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 3.6-kg;/;bgc wnnag0 ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm u7(f kfur;du2, which rotates about the elbow joint under the action; u7uudf2 f(kr 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, as9 lwtl; vr-0gj shown in Fig. 8–4l0-rv ;tw ljg96.
(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 massez/cd( f r vntj4f-v48fs, $m_1$ 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 within four fulk (h+1uk)* xuoqxf,ev l turns (revolutions) and releaskxqu+ u),ek* ho x1fv(es it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of 43w,- 3 lhglq9qy(t gi :odpac$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torquevz4p:c obsj vu ltd5-66(tcl: of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass sxlm95kci)8s 7.3 kg and radius 9.0 cm rolls without slipping down a lane at k 5cxilss89m)3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofzjhvi21-f j l9 the Earth with respect to the Sun as the sum of two terms, j21v- fzh9lij
(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. How m9gh 3(udb)qxsuch net work is required to accelebg9s3 )uq(d hxrate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withouto(w0r,nqao8r7qh v 4q slipn or(87qvar q,h4oqw0 ping 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 startsk4l;jyf)7r 0seqc p -s to fall and its lower end does not slip. What will be the ql k sr-0;cp7)e4jy fsspeed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin 5g*vzwwxolj8qs hx*m, f r 3/5string in a circle of radix/,jqwg*z h fvx55s* worm38lus 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of 3u7k uu03.pr+zge (oran7i t su h,kvg4w5s 0ca 2.8-kg uniform cylindrical grinding wheel o7u+7 o3(s0vgs u4h ak,pugrr w5n ict.eu3k0zf radius 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 rotatingvn,v yla /99:rgd sylmeh,:(o at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rom ,,v::ro9(ld /eygvy as hn9ltation 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 in: (vscmeat n9wg5a8 ; 93gocdxertia 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 is8 5mvn;:sc c3o9awt(exdagg9 her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of i7ldhij; q 1y/1.0 rev every 2.0 s 1i;y7h/diql jto 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 2s-dou6 :lipv 0mx0k trotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the v 6il2-0s ktox:0 upmdcenter 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 kc*(;kn hn- 2xw2gtrxa figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the j9 t(i:x1cvtp4uh no4n g8w2rq:9 e0wjhck.iEarth
(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 cylindri 0fjenguviq8/0i7g p 5cal disk of moment of inertia I is dropped onto an identical disk rotating at0up i0g/je8n 57vfgqi angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.*ztm/:ue( on s4 $rev/s$ 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 the4v u 8o/*ftgq ig0 :po*xem9ah;rm2fb center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 abv*0 o f/24m8qgo*p fgtmrh:9xie;u $kg \cdot m^2$ 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-rouv+hb+cwz b. 0kk yt4m a4m71yend is rotating freely with an angular velocity of 0.80vw+he7.ay m4bmt 41zkckyb+ $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing.sh5 sj-skx(q about half its mass in the process, and winding k -x(.s ssqhj5up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(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 wi6,u 0uyzeo)jrfa 0nry-; 0fgnnds 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 e;qzt4-.kjzkg2s 2a yid uzfleqg r667 ,i5;r$ 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, initiafqe(g0 b+ptwg:e +r2illy at rest, that can rotate freely without friction. The moment of inertia of the person plus+q2 pe i (0rg+wbgf:et 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 mppb l,-u:c 9fthe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertiac ,mlup-:f 9bp 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 gikt ,u4 auz,,+ig 9( xcfzzwe,5giwmf6 d 7-lmround with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipp9ix w(,+l7u6e dimi,m a4f-,zzzwg5, t fkug cing. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always facea 9alsc a/5 .9yhn7 kg9 (iyj/8bvgchis the Earth. Determine the ratio of the Moon’s spin angular momentum (about itsvli (g c99y 9sa.5gki/a7hj/bn8hc ay own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 5zphs 7j,q2o rr ;*ijorest 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 d3 xv;.4y ay0mj,nf founiform cylinder of radius 0.20 m acquires a rotational rate of ay,.jnm 03fyvx;d4 of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskgf7f frc 5 83fu*cumr.s, 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 g38*f5 ccf fu.7rufrmto calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed .*r9p9 eortxdkf*x(lu q0prw- y .ly4of 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 oks 2k+b9mj w7bur 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 times, and that the lost ma2mbk+k w 7sjb9ss carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energyr0k+hm*l0doz-5mg8itpp6/f:p ts jj stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a m:0mji6tgf+j/ * s5zr-pt 0 l dhpo8pkuniform 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 illustrate ihd. g20f+q.5b3vkc7gy8mlmmpf m+k s 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 rolling on a horhx,z r a3i8m3nxiq43vizontal 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 ftw503 :wl5mbhhkhi( z lu* sy 4v6ro4L can pivot freely (i.e., we ignore friction) about a hinge aymh*k:3(zo4w0rvh5u sl tf6ibhw l54ttached 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. 8–21g.]

A wheel of mass M has radius R. It is standing verticallyy *o)skhg+iqrn0a l2cyh34n , on the floor, and we want to exert a horizontayly4 3 qnhg*n+s0)ro2 iakhc,l force F at its axle so 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 isy)c i)z3: azil z8:-c3ja+duzdi/n xh making a turn with a radius The forces acting on the cyclist and cxzhzd cy3-)a+ii)8zdz::l c /iuaj n 3ycle 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 slingpbm*m ( e/qy.-+5 kcvzcm gt0h of length 1.5 m and begins whirling the rock in a nearly horizoh/v(ybtg-p+ mmz05 *ceck .m qntal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, ma*d hjl1 (eo8oxi eq 9qu*1qyj1king 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 against her l1i9d (x ey8qjhoo1q1e u*qj*body.   

  You are designing a clutch assembly which cyeeo *tyh / 9h:0cn,o.d,r okwonsists of two cylindrical plates, of masshwo/o.h,y0 t encek *r:,dyo9 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radi*zg)l yl6)c frmd7a7kus r rolls along the looped rough track of 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 loop without leaving the track? Assumgl*fd)ma7 z r76)ylcke $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n)fyee+xdq e6)ot assume $r\ll R$ h =    (R-r)

  The tires of a car make f13y7x-qrmz:w m s,0(nzigtm 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0,mwq:fsm (tz1 i3 70-nrmgx yz.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