A bicycle odometer (which measures dyj.e-(,w*uo.: yhn v ii xmfr+istance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheelsi.oiu wfj+r*h e,:. ymvn-x(y?

Suppose a disk rotates at constant angular velocity. 1xk2 rpng:u, bDoes a point on the rim have rad1burg kp:xn2, ial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be / 3j sdc;q .c3is:pgst(x4pwg5nc g(v described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu)olmnonqj )*81oyll0px- d0u: g dkx +e 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 wso4b3lvf6;t d orww7,ith your hands behind your head than when your arms are stretched out in front of yo ;7ls3,wf6wroto4dvb u? A diagram may help you to answer this.

A 21-speed bicycle has seve,yf,l+ f: j0mdnzh d0dn sprockets 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 pedal, a,d+mhndljz y0d0,f f: small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have 7ci n 3ma-znx*slender lower legs with flesh and muscle concentra*m -znnxai 73cted high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fio f; mek*u,eqmf(/2wxg. 8–35) carry a long, narrow beam?

If the net force on a sg xh5yude2j 1+5j ite;ystem is zero, is the net torque also zero? If the net torque on a system is zero, is the gh5i2tye1; 5ujj+ed x net force zero?

Two inclines have the same height buu. f).h:qven xwnn( .g7w)bg zt make different angles with the horizontal. The same steel ).g. e xnf(zub :.vnnw7gwqh)ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simoc-b vm6 -ty4+e1iwidultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the 6mv bi1d-eow+t y-4 icother. 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 hai1ia8r56s0+ a bxwd*sdf2f-) wmdidkve the same radius and the same mass. They start from rest at the top oa1ask2w- bmx d w 8f6*iid)if0d5+sr df an incline. Which reaches 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 angulnj2md3tnc+ b8ar momentum are conserved. Yet most moving or rotating objects eventually slow down c 8+nj3tb2dnmand stop. Explain.

If there were a great migration of peopleyi* d 8kdxa )u6d9ruo7 toward the Earth’s equator, how would this affect the uod7uxk aiyd8)r* 96dlength of the day?

Can the diver of Fig. 8–29 do a somersault ggve;gamc ,*i6: g-cmwithout having any initial rotation when she leaves the boarm*c - ,;c:amgveigg6g d?

The moment of inertia of a rotati:h6dwqh nty, g* ,npt2ng solid disk about an axis through its center of mass * pw,n2hn:tghdy qt,6is $\frac{1}{2}WR^2$ (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 outste. vd6 9,duotpretched hand. If you suddenly drop the masses, wiuopd t,9dv.e6 ll your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one 37xr,0fc nbmuis hollow and the other is solid. Describe mur 0,n xb7fc3an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal a8 ;0yy sp*wpsgxle points west. In what dir0pg ps8* swyy;ection 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:o0dy2tp5sr2kb-/l mik- oc w of a large freely rotating turntable. What happens if you walk toward the centedy-o-r / mitsp2l:kkocwb02 5r?

A shortstop may leap into the air to cb edmi (:6ecw6lz1i-cqd r6a3** bz ptatch a ball and throw it quickly. As he throws the ball, the upper part of his body rw d pm*66ectebd -cbl(i 1iq3 r6z*z:aotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wh4d2eww +l) 560.kspaf, ujtk lr(sez gy a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stablz 5 r ,)ds2w(.ea e6l4gup0j+stlwkfk e.

  Express the following angles in radians: (a) 30 ;,stdp+,:ybgo9g tv8xhatdjd 7e 1m4$^{\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. Calcu1m htlyu2k; g.:xz k9ylate, using the information inside the Front Covert uymh.k2 z;y9k1x g:l, the 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 Moon, 380,000 km fjp1x i71stl;s rom Earth. The beam diverges at an angle sp7ij 1;1lts x$\theta$ (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 oie nl*0.gq0pkff during operation, the blades slow to*p0in.k gql0 e 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 fksua-w7eb ad k p18w*) dbasyz+6 t3*rloor 3.5 m to another child. If the ball makes 15.0 revolutions, what is it6bp b 3s kzwwy1 dt-*)daaru7seka *+8s diameter?    m

  A bicycle with tires 68 cm in diameter travelsrv-j/(f rncg* 8.0 km. How many revolutions do t/f ngr(jvc*- rhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in dxmsy5 l63 jh6wiameter rotates at 2500 rpm. Calculate its angular velocyh3x5lm6s6 wjity 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. 8–38). (an 4 gudl3b5 yvm0(in/w) What is the linear speed of a child seated 1.2 m from the c43wg0d nbuyv5 nlmi(/enter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of theojzk-.lz;pk/fxo2 + s 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 point f vl*c() (kfqy
(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 r4u .osx(pybo +otate if a particle 7.0 cm from the axis of rotation is to experience an accelerp ox.us ob4+(yation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center f7rx 3rae9jsz1kh5* cprom 130 rpm to 280 rpm in 4.0 s. Determink1r7*9c5hzra espj x 3e
(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 radiussaw6-d37)iq0 vn ef mx $R_1$ 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 t kjof .o )/7msyeflg:)hemselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their)o.k:o7gj)fs fme /ly trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from res-,l)6j6km pj n alh/qlt to 15,000 rpm in 220 s. Through how many revolutions diln6p/lk j,ha ) m-j6lqd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcul2a: rmc1ot65 8vfuqfuate
(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 higg.e1fy1pi cww9f d 2x-hspeed jets in a whirling “human centrifuge,” whic .xpegw2d f1f1ci9-y wh 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 rpm to 360 rpm izcy+sp7n y/7m r+zz9 s4s3jiu7 -szcun 6.5 s. How far will a point on 4y+rsj- zcun/czs7+ smy9i7zp zu 7s3 the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is r,i c rbfp2/bo0t,)vaxs)q y5punning at 850rev/min It turns 1500 revolutions bef)pp)vr sa,c0,fbo/x y5b2 itqore 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 reduces its spe+w:jv)asw+kcolk ( na,7:ot )pw0y nged uniformly from 95km/h to 45km/h The tires have cna(nw:,+7akls wkj +ovy:wtop)0)g 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 dmpd7y:xk 7r2eu2 u:icqf;/ c5ss0i d 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamedyudi;57xm/rip cuq0:e 2ks2 : fcds7ter 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 weight6t 9gao zq(em, on each pedal when climbing a hill. The pedaz(o ema 9t6gq,ls 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 wide. tok4/ px*)eou(y kcq5What is the magnitude of the torque if the force is exer) oc(e*5yqxkt u4o/p kted
(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 ohk3 my(6wn k)dm*,w -rjz6fkgf the wheel shown in Fig. 8–39. Assume that a frictidr m,yh m)6-g*f(wknkj 63wkzon torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass mmt+4 a vzl0ik;, are attached to the ends of a massless rod which pivotkia z4m; t0+vls 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 system.

  The bolts on the cylinder head of ammnup. m5 98vnn engine require tightening to a torque of 38mnv m8.unmp59 $m \cdot N$ 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 radius)lqxgb 5- qi7dbr/ss+m 1mhr:kp +ql/ 0.648 m when the axis of rotation is thrpq)1 5 dmsl/qr+k b:gx+islh/m -r7bqough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The a1pdzd(4 3 e.gf98te 4oex lnnrim and tire have a combined mas z1tne.ea fol984xg4 dp dn(e3s of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,;af8rja ae 1ef4ku /1k light rod is rotated in a horizontal circle of radius 1. fka18jk aera/;eu4 f12 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angulap2 r 4pk2pfe,nr speed (Fig. 8pe nfk22r4,p p–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of pointgvxl (tgp+*r wu/k j37 objects shown in Fig. 8–43 about vrk7x/ 3gu(lg *wjt+p
(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 totaly5)wts 5bj087ls tezd 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 cylindurjeg* y*m.b6 q mra01rical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Figar1 bjrm* *u.m y6qge0. 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 ctsm,15 )b86oa3 tbh vc 6dtalwylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculadc,6avtt6wb ht5 l a1)mo8 bs3te
(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, acceleratinu-x4 boqfx i44n igv90g 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 tangenti v6az8zufl+f(ally on a small 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 edgu zl8f+z6vaf (e. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off.; 9lc97zw vy qu.ad3jcylb.o and is eventually brought uniformly to rest by a friva .qdyc7.ywu ;c9l l zob93j.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 3.6-kg ;b*e; pmdg-co k37tux myxm)2ball 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 theisa0 .fmt;i;) cue-x, xpik,y 0do7rt forearm, which rotates about the elbow joi7o0xf et;,i) icr-,aksitxp.0u y; mdnt under the action 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 Fi vy:in1k1; w y bq07lz ltzd3qkmh5-l8g. 8–46l:m7v1k;q1w ih508lbyd3k -zqy lztn.
(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 csdrx3.7.slk konsists 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 withivv( 5q)qleii f c;x* (ns4a0cun four full turns (revolutions) and releases it at a speed oi*(vqc)le5(4 xafuiq;0sn v cf 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)3qis;mwy0y cy9 vp:9fm gz(g 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 d8cl,oso)z+ hs 7:l-)z dlzylnevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm ro)fqu:pbmfv2 3) ayvk ,2: yksulls without slipping down a lane at 3.pya)f vfub2u2s mk3,qky:v:)3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enk.yqf 6 o tg3om((ki0* h6jhgpergy of the Earth with respect to the Sun as the sum of two oj*qfgi( 0ht(py36 kkgho6m.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 mass of 1640 kg and a radius of 7.50 m. How mudzk k-- w(k2btu0 2btfch net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8(-z bu2k2dktw 0t-fkb .00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k6efy4 -y9 -f8/irigrksn;hi2idz(-h xozw )og starts from rest and rolls without slf9-oy6k/wxzgi2- h; )h ifiers8r- odyz( 4ni ipping 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 fall and i :s3o- b-zdhrets lower end does not slip. What will be the speezh-3-e :srd bod 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 thiv,p k6 es69 kkpy1x2grn string in a circle of radius 1.10 m at an angular sp6kv kks e62xp,1yrpg 9eed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of 9altz ey k;ypd; y;;5oa 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when ro p y;k5e;yl d9oy;;atztating 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 w3quu-mpui; ,2 kp: fvrotating at a rate of 1.3rev/s If he raises his arms to a horizontal pos,i2vu :p; pwfu ukmq3-ition, 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 F4c-j2bs0do 1bz, *ro*sztyak ig. 8–29) can reduce her moment of inertia by a factorsctjzk14dor*s 2 b ba*o0y,z- of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from v)) myb fdi(d8an initial rate of 1.0 md(y d)ifb8 )vrev 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 vertiput3nvz4:um )pp 8m :bcal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the cpmp 8)mn::vu34ubzptlay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figurfx)rvevf-6/v 6iqo( k rcp32ke 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 m7 smf w nl s ++/2vtmq/hz-;qr1qv-fabomentum 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 is dwb1 s w8f,3sgb8c wj(lropped onto an identical disk rotating at angularw(f1bs8wsb c3 8j,wlg speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 h2e8jihfep01iyzib e7i , u;+$rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plbu,sc- kg7cd0atform of rd7c0 ,ks-ugc badius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angumeav/-ubd2,u6jb9ni lar velocity of 0mi,-n/bevbu6 d2 9au j.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losil:y b6t/a,f brng about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momen:rty 6/,f bbaltum, 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 cqm b,5vu;9-xa0x daf 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 yc)e9c k5/ ,rpt7e30ola;oub,q s iku$ 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, t ww2ecs m7v-x.l 1a xk7g7ae,yhat can rotate freely without friction. The mom- a7mayl1c7x72w wek ve,.sgxent of inertia of the person 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 y;h9 5k 5foi(nn 1 na1se ubr9i7/kulestands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a m9in5kie on7ulr19ufa1 kb;hyn/e s 5(oment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the endvt oi6-bj/9 bm 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 slipping. What length of rope unwinds fromi j9m6 bo-tvb/ the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the samed1it c (*o:fyd side always faces the Earth. Determine the ratio of the Moon’s spin angular mome*cto:f(y i1dd ntum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

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

  A 1.4-kg grindstone in t :z0r a::u,+vvub nr:ra4 hvkwhe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate ther+ak0 uvza v:bu:v::rw4 ,hrn torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, )hn+p h d25jgyeach of mass 0.050 kg and diameteydhg2 n)+hpj5 r 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 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 angug4epr: 97ahh hdbt(/bvd.ct: lar speed 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 stvt yug,6 dgmtvca .e9k-..t7ize of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 1 tck7.et.ta9m,.-g6dtv gvuy 2 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 automobile is kp:c m*9k,g 4clg r6bpfor 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 dcpp6,9r c4 :lgbk *gkmiameter 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 p x5l)6eft3) s gfmu: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) 7z+ ;9m8auvh7w: adnwb( j2ftoerqwvd-1 a4iis rolling 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., we ignore fr/)epe w47gct-*/alrdvr) mt wiction) 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. vre/dgrmwta/ * 7l ct) 4w-e)p[Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vert9ggz wwy.*bksp -,3(+ gmkhqi6h *grbically on the floor, and we want to exert a horizontal forces zg.q,-wbkkig9+mhy bg(6 3hp *rg*w 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 is making a turn w-e8 e(9dkijl x+2 h9evjyv n6with a radius The forchijd +x29k v6vee9-8lnw(ey jes acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and behu1h31f;qr evtwi5g/9v hzp j( w wm6;gins 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 required to achieve this feat, and where does ifq5g hw r(vw;wp31 muz691/ tejhh;v the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as t29ec kb)a7 xs/n7q+d fn8vwszhin rods, making reasonable estimat8sakscqv/+db7nz9)wx 2 f 7 enes 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 asvnd+x)zh l-+r w6(ob pabn37jsembly which consists of two cylindrical plates, of mas v3hznr pn)jdb++x(l 7oabw- 6s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rouyvqz78*1sp0)+9j9 ur g nadhyd:y mc ugh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to *)d0:7argny 1vy cdzpju yhq+ 99m 8sureach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass5a/xc e u/)f8l 6ugitmume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforv 4lonk1* u3 wrhyrv-7mly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration orw yv* o3urk1l-h4nv7f 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