A bicycle odometer (which measures dist.1t/vx:bcq0 p x ouk(gance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle withk:u0ogqv/xbpxc . (t1 24-inch wheels?

Suppose a disk rotates at l *.-ompvyev, :+/iqabu-lnhconstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of eib-ve :myl-/nolq u*h,+ipa .vther component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloueu ;trzu -q,5city $\omega$ Explain.

Can a small force ever exert a greater torque than a largerbl3ekvc as(2zi, 20szzj32i mub: w2x 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 diffiib,n (yds1m1 jcult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A dia1dm1(by sin,jgram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal )x7vc5x) ddcpcranks. 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 pedalc)d)5pdxx7cv , a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to runeatu y439vh)nays.j0c.v 2e/ld + qj z 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 distribution of mae 2qa0ueaj znv9 .ytl/)cdjy s34h+ .vss is advantageous.

Why do tightrope walkers (Fig. 8–35) carry i)3nka8gvs : pa1i:hna long, narrow beam?

If the net force on a system is 7fexfx5p (-4gh8 a wdmg yq.6vzero, is the net torque also zero? If the net torque on a system i7q.xd-pyg5f8g46 hexvamw(f s zero, is the net force zero?

Two inclines have the same height but make different angles withv 6dz7f5tg,qj the horizontal. The same steel ball is rolled z6t 75dqv,fgj down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inca,quylf: d1;m(w w y(yline. One sphere has twice the f,l;:yua1(q (ywymwd 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 and the same mass. They start fromtn5 t)u 4g2:tclk tjm, rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has , 2ttg)tucm : 54ntljkthe greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserve olz 9mfb sdi3x)-d8d:d. Yet most moving or rotats)f:xmb zd- 9d dl8oi3ing objects eventually slow down and stop. Explain.

If there were a great migration of people toward ex n/nx *8k+idthe Earth’s equator, how would this affect the lenki xe*nxd+8n/ gth of the day?

Can the diver of Fig. 8–29 do a somersauk0 ozin++hf *xlt without having any initial rotation when shz oh+ n0xkf*+ie leaves the board?

The moment of inertia of a rotating solid disk about an axis i -j((ihik+ps through its center of ( j+ii- (hpksimass 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 sittingsbydizz*i1-zwwtb6h) g8:c*qga4d f i0 vn3( on a rotating stool holding a 2-kg mass in each outstretched htw8f1 -z:zb0azgsqg iid *)v6 w bn(4i3hcy*dand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identiy- b/:lt;ztzc cal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determ;c bltt/z:y -zine which is which.

The angular velocity of a wheel rotating on a horizontal k*cz;;.eyyyl9j qw yfos8r26 axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acc;9yfw.z cjyo* 6sy;e2 rykql8eleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of awa x-c0pxyq;sp+or bc(/ r0.k large freely rotating turntable. What happens if you walk towar/ rsqwop 0c+r(b a0yp-xk.;xcd the center?

A shortstop may leap into th4jy4vww f2+ eae air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickja4y42wev+ wfly 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 9tj6g j0d)qba why a helicopter must have more than one rotor (or propeller). Discuss )a0gbdqtj 9j6one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rad/y r fbzcek 3:7hnjm/+ians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Cz4sv5qv yny; vim *:t,.j yu-dalculate, using the information inside the Fron yy4-yd, m jvuz;*n .tvsi5:qvt Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directedym: e rp,y3.mp7h f7da at the Moon, 380,000 km from Earth. The beam diverges at anrm p377a.h y fepdym:, 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 rot2ovjgrn7)f * wjf h/5gate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is v)wjh/*r fo7 jn52ggfthe angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anotak(nsjobxs* .0 t ad2-3)2 ywz9jp yokher child. If the ball makes 15.0 revolutions, what is its dos y*jb z) 0oktwpkjs-9(23xa dyan2.iameter?    m

  A bicycle with tiresuy 3 )b zx0 a)dz8s+emhyk)b7i 68 cm in diameter travels 8.0 km. How many revolutions do the wheels make)bdy+) 78 m3 z)z xikue0ahysb?    $rev$

  (a) A grinding wheel58up* 0fwx.wu5ty ( vj uyz.sw 0.35 m in diameter rotates at 2500 rpm. Calculate its angular veu55ypxs8wywu 0vf.jz(wt*. ulocity 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 co 74y6oq ye-vfgn/h1t jmplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from 1-fthg/ eyjqn7vy6 o4 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) f2j )xey/w)irzoc,gi 2x l8l,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 k25 zv;owu ow,
(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 7.0 cm from the axis s psczo-n 5/n ribt:q;o+rnvdhz/));of rotation is to experience an acceleration of q n) /t )s5drizscnz-+n/hop b;vro;:100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlynd+3xf+ zf s(x,h3r 5n9b ;xzt7qbusl about its center from 130 rpm to 280 rpm in 4.0 s. Determinebb,hx 7nsfd+ sf3 xt( xqu;z5zl9n3+r
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius zuqnyk e *fxw;-5 ws:5pg(po4$R_1$ 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 themselm )r(g8;nee rq wodc6,ves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceln;(ce6remwd 8g, r)oqerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,0000 ehda5vy bm29 rpm in 220 s. Through how many revolutions did it turn in tbv a2d e095yhmhis time?    $rev$

  An automobile engine slows down from 450dx+mtm. gix/jl/j 1 +i0 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 stresses of flying highspeed jets in a whirl-rr-yw 4d najqg 1yp47ing “humann-gyar4 yq7 1pjr4 dw- 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 accelerat0bwiic tw7u;qyqw8,+vl 3vol y) u. 6tes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point onbw iq ycuitl3;o)l+8w0uq,w v 7 vy.t6 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 ttt-fd fx v )9/9-rgiehurns 1500 revolutions before it cdtx9te/) -hfrgv9if -omes 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 speed uniformx nnt iuzy42 tg *yiaf(3bi6m( ).kr.wly from 95km/h to 45km/h The tires have a diameter of 0.80 .t(.m(y2 6yniuwi )r3*t z ixkafb4gnm.
(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 t/+wg )v(mqu om k;zm4-g9 wguaf3rl9hhe car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter o -z hgqlwfr )m4o(/+mvgukm uw99; ag3f 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 whenigb. +)vhb/f e climbing a hill. b+bg i.fvh e/)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 5uowvcuwj/)9jlr9 ;im*t . *op5 N on the end of a door 74 cm wide. What is the magnitude of the tortvjl9o/cour.p* iu wj)*w9;mque 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 toj/ d:1l39o(srjw;mtzse1 czhrque about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.413lw1(:tosjmrz ;9ezd /chjs $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to thef6t31 zn:12smxo /zh8 y7mg afwqyxm6 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 n16y1y 6tgm wqsf no/hxmx8mz:237azfet torque on this system.

  The bolts on the cylindzvass95(7x )dj) ep qder head of an engine require tightening to a torque of 38x aqs)dpv7) s 9ejd(5z $m \cdot N$ 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 o bzhj*zbi :3b f/qm/*qf a 10.8-kg sphere of radius 0.648 m when the axis of* h*zfqb /m:i3qz/b jb rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of iq29966xmw1sd uwq4r +qvmx da bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?4qrsm9 9dmwi qu+dvw x1 662xq).    $kg \cdot m^2$

  A small 650-gram ball onv y6xpxs: f3+d the end of a thin, light rod is rotated in a horizontal xsvy fp xd+36:circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a j0nx/onroox )(m/pg4) reo* zpotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay i opoo xo)n/rem nr()*x/gjz04s 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 )mw5knlel-e8z 83ujpj3y8ykb )2c q bpoint objects shown in Fig. 8–43 aboy3wb ejqk)p-lnec8b mu8zj8 325kly)ut
(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 td+-v vw8ra-azotal mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellor 7fw/e5 s2lxite 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 sa5 o/27 xrsfwletellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm ay1n 6 fd.vocn)nd a mass of 0.580 kg. C )cn6ofd v.n1yalculate
(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, accelerlql2bd ih*y6:vr ( oa6ating 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 ):s2b rk4dso 6; tyeahsmall 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, neglecting frictional torkr 4o6 eyd2 st);bhs:aque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating3 d;(1k.(iqo1x8n sty ql drgt at 10,300 rpm is shut off and is eventually brought unifor n(o3.l1 y1 qi( ;sqdrtdkgx8tmly 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 acceleratvuqjx6 4p4pk. 82fbxt:l d 5okes a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ba1yz62*szcfp8 w /ongs 8w,s1pada 0hvshr2b6ll is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest tog0swc n1opy2 b 1wf /h88zzrsh6av* a spd,62s 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 i6m.r ljkb0,ok wf (*lwn Fig. 8–46r.,okwk*m6wjlf( 0 bl .
(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 tw0nepo +;3zd hgo 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 within four full2oqiy z-f6 ayov:zf: 7ur7nj: turns (revolutions) and releases it at a speedu :7- 7yni:zo 2yzqavj6:rffo 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 u)m4mz)muc, x4j ) cu ll.ez*pinertia 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 torque ooaq6v)t9xd9p f 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 rollsp4s :j+ ceh;y mb+jkrk5mp;7c without slipping down a lan 547pksj mky;ph;bcre:+ mj +ce at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of:ra0g c(y jti0 two 0ca:y ij0 (rtgterms,
(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 o c tq1,ogu ;1t9d)vldad*q4 zkg 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 r tdgcoa*qlu9;)d1 4d1t, zqvo evolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sx9h*:wwbzz-2 bbs (0t cdy p1btarts from rest and rolls without slipping down a 30c bb2ys wbtp9zx:10 hd-z b*(w.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 balan w jjjc)zz/c60ced vertically on its tip. It starts to fall and its c/cj0 jw6zz)jlower end does not slip. What will be the speed 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 onn2m+ww;g1 m/bu t:ap z the end of a thin string in a circle of radius 1.10 m atnu;zw/ 1gw:t2mmab p+ an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg 4vqcqv)gm-z* uniform cylindrical grinding wheel of radius 18 cm when rotatiz)mc *-4gv vqqng 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 plat y6c 5xso6+fjz)eup 7x9rxq c0form 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 d c xq5rp7)yuzs 0+cf exo66xj9ecreases 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–eqk7(swuw+b r b.v(h529) can reduce her moment of inertia by a factor of about 3.5h(bu+ q7be(rw. k 5wsv 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 an initi/qeo yk nq7x8lf-v)p 8zph4(eal rate of 1.0 rev every 2.04-qxfp8q)y lo7he(z8/kpe v n s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cegid gr-j.lv( z3u l;38xbj/p fnter 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.fpg-j(lju;d /vx 8 zlrbg3i3, 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 3.5 1-ndb -cf:qizni3k1 ocd c-.l$rev/s$ 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 mom4nkrqw,1 f 6wm ov2o9z ugtk8e-ljl.* entum 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 inertia3w4d xrdgnkz1h h/79 s I is dropped onto an identical diszkghdn9 r4sd x3/ 1wh7k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atbly :f-:1yoaa 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg standssjbml: iatm5 w+ ha057 at the center of a rotating merry-go-round platform of radius 3.0 m and moment of i7atb lm:sh0jm 5a+iw5nertia 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 an ab b32 lh xcmw107esw1)+ohwvgular velocity of 0.87 ec wlhomx+b3s 1w)1bwha0v2 $rad/s$ 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 dh ,yjta bje5- egw9,c-warf, losing about half its mass-e ,cwejah,tjg y-59b in the process, and winding up 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 winmk 9q npk)vpj/ox xxutkp*9. 9-1(irf ds 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 je6r: )t w9 2x-ixbfjd$ 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 platfor 1ojdd * 6e:6 t4 mfyb9/q:ml4txxoznbm, initially at rest, that can rotate freely without m of4txz*1qb46tydjo9e l:d m b xn6:/friction. The moment 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 stands at the edge of al4 5ynyvox-a8 at5*pi 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momenx*54i-o pv5yynatl8 at of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyinc,gnff,omp 0 9 9hf.1zx lax*dg 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 slippinggc,hf ,xo.zm*lf9 0afx n1dp9. 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 sidtloe n0m.- 1lvojpca un hh5,4i k6):ze always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own 1.) 4: ,plozt0io 6c -mujnnhvk 5lehaaxis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest atnvn 3gvb*9 i7:2m avvf 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 s.9chdybylme2 y) 3 zd.hape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at con e).c dy2blzhy9.yd3mstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of macb,p 56wz1hod vn s83pss 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 thp1zdpvnwbc s,o5 3h86e 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 sn8m j w0/i9gtdlm ub/2peed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.drjn2w0 c ;sli*kir* /0 times as great, were rotating at a speed of 1.0 revolution edjn ilki ;r0/2r* *swcvery 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution au wt +5a 4cy9s mj;wrdb)6du:bftomobile is for it to use energy stored in a heavy rotating flywheel. Suppjsrdc)4m 6;9wa5b t:f d w+ybuose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a 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 pv2 nr: ccj((i5r j9qsau4 .on$H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.) hulg7t ri c4le)99zct*vz7 w4lkvl,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., 9 cr-x*l ,qopnwe 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-l,rxo qnp c9* 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 verticallwdi*ohu)ku :rkgh66 eqt6 .8ly on the floor, and we want to exert a horizontal forceo 8)hd urku66g:.kiw6lhqt* e 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 s5 a rgpln44n5making a turn with a radius The forces acting on the cyclist andg5n 4rl5 sa4pn 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 leqz7kd6y5.76f .ci p kd,pdrrpngth 1.5 m and begins whirling the rock 7d.kfcp57ridprkd6y. , qpz 6in 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 the torque come from?    $m \cdot N$

  Model a figure skater’s body biav/ gufbg y1+/l3 4jas a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinni/1ybgbiv3uf 4la/g+j ng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cons+5 ; 4r)m jnxy-os6nfy d-puzo(a s/plists of two cylindrical plates, of mass 4yaz-j/yf-onpnp usms o)6r(;+5d l x $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 theugkusf. bk491; 7z cuzbkyv +a x;q8n6 looped rough track of Fig. 8–58. What is the minimum value of the vunkqbs1+f;cv89 7a. uuk z;zg yx bk46ertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not axjjf1-j:-m peqv-)m qssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces it t6ryfa+ zt1go s3u;vewx02q3s speed uniformly from 90km/h to 60km/h The tires have 2 rw+u3 feqot;1x yt0 3zvgs6aa 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