A bicycle odometer (which measures dista /yzdkwgn8wewa6 /o .1nce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it o ea kg61y ww8wd/o.z/nn a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a po,-/a2qd0 e z0ehlu:m ,ff,squkq h0qvint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point afhzhes, ,l q-qfq:dqem k v200,u0/uhave radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid bodyp,7( j8 ajy2 rrr3 +b5 s4ryaoims2iqk be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever5w/fe:;q te jy:1tdbu71k*cnoxh i3 g exert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a s h 0ka2f/fxx,ait-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to anf,/ aka20fxhx swer this.

A 21-speed bicycle has seven sprockets at the rear wheepm54zzr2b qwaa:p y* ,l and three at the pedal crara z:25qz wapp,by4m*nks. 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 small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legoq 9mu/ nrox0jx */rs(s with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mas u x09or*q/(x/mosrj ns is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowmfzrh0 su+5 m:zvh,7 t beam?

If the net force on a system iwzrwn7 +wyy,a7;gq ;p s zero, is the net torque also zero? If the net torque on a system is ze;qg77yn;wz,a y+ prwwro, is the net force zero?

Two inclines have the same height but make diff hf2og3m4wr,eerent angles with the horizontal. The same steel ball is rolled down each incline. On which ing3 hw42ormf ,ecline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (9eh6tmip3tr 2vz p0 e*from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the botv9zt0h3ipeer 62p *tm tom?

A sphere and a cylinder have the same radius5 .gr 8h 65gwewtr*sbt and 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 greater totat * w5reb. shg5t8gwr6l kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are)e(mb/ )*u mg8s e kgdej0.zca conserved. Yet most moving or rotating objects eventually slow down and stop. Explain8eg/z) k mcdm(.)bgsj0*eaeu.

If there were a great mtpz,kd2aiee c)ci 1b3 z5(mv,yp,j-cigration of people toward the Earth’s equator, how would this affect the len1jypez5 eid taimv,c) ,bc 3-(, pczk2gth of the day?

Can the diver of Fig. 8–29 do a somer 6tm yzmc(/f*psault without having any initial rotation whenc *6m pzy(t/fm she leaves the board?

The moment of inertia of a rotating solid disk about an axis th0(tt 2gw cz1pk65vie 96 gnm uy8xsld.rough its center of mass isgns1yli u(pg8vwx c.k2d 0 6t9 et56zm $\frac{1}{2}WR^2$ (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 stofsi-0z0+p l)vr uf)7gaxy ki0 ol holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, d) i7 -+l)uszir0k0f0a fvxgypecrease, or stay the same? Explain.

Two spheres look identical and have the same z 0 l h3,a5m:psjtbm(gmass. However, one is hollow and the other is solid:h tl0mzb5j(pa 3sg,m. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In weetqp.nz:o b, ffu7wb2 ge3z7l5zv0-ucwj (4 hat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tetfbcu27 3 b7 ewu4z,q.f5 wzp- n:lzo(0 vegjangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntableje2hdodf 5 vbag+;7l1os0 ry5 . What happens if you walk toward the cvjdg fdh 25 eylra0 7oo5s1;b+enter?

A shortstop may leap into tvs *1tr3vcjn, he air 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 w3rjsv t *,1vcnill notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angula8;t lf y+y)xrss9p2f jvc3b 7qr momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second py82x+rtpbfvf y7) 9l;csq 3js ropeller can operate to keep the helicopter stable.

  Express the following angles in rad*fgd vu6 ;gql6ians: (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 b4tt1fzdg. g6;etvt,yl 5p s)zecause of an amazing coincidence. Calculate, using the informati dgs 61z)tt4l;f.tv gy5z et,pon inside the Front Cover, 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 from Earzs -23zk 3-afnla ;pmnth. The beam diverges at a -f2a-33nzsnkplz; amn 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 blend-+hu m/gr(euo er rotate 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 the angular accele/ou gr+ehm (-uration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the bu-ktuwuof sny)8u(uo 4 /p7k 4all makes 15.0 revolutions, what is its diak (uwuu4tso kun/8yo)4p7uf-meter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How ma+ 12r pcj1oucony revolutions do the w+cr jc 21puo1oheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. rjsk .3x+mpqw+7b6 :q d.sprx Calculate its angulx qmjp.7rq w b +3sdk.ps6rx:+ar velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Figk;y y9lxs-xpchg,3y z13 c2p q. 8–38). (a) What is the linear speed of a child sel;yx1yh ygkcx- p,9sp 23q3czated 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 its orbit around the hvq ck ; 4e:/vjj5qiw j:u+qz5Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a a pax+n.qtq00point
(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 jm.8if6umj ul t r9 7jw/h**fqyyzu;8 cm from the axis of rotation i.j*l*9wr/ y zi6 7thm8uju ;jmqf8y ufs to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to-otjcapf, k-g0s 6sn0as ).km 280 rpm in 4.0 s. Det0 a-sjo-af6 ,sm ksp t)0.gcknermine
(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 - p-*+blhzj lf$R_1$ 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 t7;:x 1fb :wmgd-gdnvdhe Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceler; gb-v1xmw:d7dn fdg:ated 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 unddnrjx)uvaw3 8s f-u1p/:,zk iformly from rest to 15,000 rpm in 220 s. Through how many revolutions dv,-xuzpj/wn)k:a u3 f18 s ddrid it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpnh+mapx+ nf0 /m 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 ivy+*9x*2h nnshhld; zn a whirling “human centrifuge,” which takes 1.0 min to turn thxd+2hhz yls*;n*9 nh vrough 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 irzn )-o 9uykbg+ om-n5nj1t6 mn 6.5 s. How far will a point on the edge of the wh okmn)zn6 + 1o- y5nmjg-9rbtueel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns e6)pgsxg16 pr jl.v5y.1u jto1500 revolutions before it c5)x66osvyj eg.plr11 pg .tjuomes 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 redui nft .w+tchd n-b.-oo ek/h29ces its speed uniformly from 95km/h to 45km/h The tires have adhec.t w/.bo-hit+-onn9k 2 f diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions aszk.u) 1qths;terv.w*m59via the car reduces its speed uniformly from 95km/;a*hkiv .e.9)1tz rs5qwvm t uh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when fcib vt) io3 fy8zy-:gp-yu6,climbing a hill. The pedals rotate i ugyit6 --fzyoc)v:8yf p,3ibn 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 for 2hfqz:2 4p ;:lleubxfce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force zef:lqf p4xluh;b 2 2: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. q9lzsx2xx :um0;duu the wheel shown in Fig. 8–39. Assume that a frict9 2x;l usu:mdzu0xxq.ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachetrkd v-omh:;xc5+nb- ; -xrybd to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal positio;rrb xmyv5 oh -c+k-t;x:nd- bn and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine r*6b-e dn 6dwm7nvlpb- equire tightening to a torque of 38 -e wpv 6nblm*dd n7b6-$m \cdot N$ 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/,wc( *qp bqvq of a 10.8-kg sphere of radius 0.648 m when the axis of rotation i/bp qw qq,*c(vs through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamet()owojeiby*e3 c 6;ecer. The rim and tire have a combined mass of 1.25 kg. Thc ij3yo (w*);ecoebe6e mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on r hx)6tiw,oodoa;1 6u the end of a thin, light rod is rotated in a horizontal circle of rit6ar16o;,h )d xoowuadius 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 potter’s wheel rotating at constaq3 ml0el,d w-pnt angular speed (Fig. 8–42). The friction force betweedl0ewp,l3q m -n 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 momentr , 8tbx4gck)qsl +uy9 of inertia of the array of point objects shown in Fig. 8–43 agbqy8xrs+,4l ku) c9tbout
(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 consog+meoue1u32j0.3*d; 6jb e2w sqta*zjfg qfists of two oxygen atoms 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, uniform cylindrical satellite spinnim1 zixf.:yct;ng at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If x zmc i1;f:t.ythe 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 a8 kf q ck jdy76w6f+n4orj*de8 uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatek86jdco4r+jy78e*df6wq nkf
(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, accoh ut1 *col62gelerating 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 ao 9 9a2 dhl- zs2zf(afua5.lfg 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 edge. Calculate the torque required to produce the acceleration, neglecting frictional t2suffz lagf-2. hlz a5(d99ao orque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off anhbu c*+rj q4n-w0t:prow a7v.d is eventually brought u0.4 a7hqwu*jwv t-:+ocbnr r pniformly 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–d h:h: 3ekvj;f45 accelerates 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 ball is thrown solely by the action of the forea8ys.5oubkz f7 +pauf6rm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ballf 57of apu.y6ub 8sz+k 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 aqkegm 7uuhq zh37/ a(. long thin rod, as shown in Fig. 8–46 eu a37hkg(q hum/z.q7.
(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 qux/;u o7di,n0 zcyf. 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 withini4k6egao rt fe.g*3f/nps6 .h four full turns (revolutions) andos64t.e/ffi* .r3p6ne gkah g releases 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 :1g tqg9(g4k 2 jixktlof inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 k eth6ja1pe7n64ez(k $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of masl -n:tbk hw5imy6al (2s 7.3 kg and radius 9.0 cm rolls without slipping down a lan l(t ky5n 2lb6h-:waime at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth witzvlbami584 ade/e8 5 b f5uff3frs07th respect to the Sun as the sum of twilemfzu0f8 /rf55bb8d54as7 ftv3 ae o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much nedcc yt.p2 cldf4o:lyl4 i+,) nt work is required to aclcf+y)ltod24p lc4cyid,n:. celerate 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 wi+53tylo8p;s 3qtkg vbthout slipping down a 30.3+gt3tvo5syq8l;p k b 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 balvn io1ak ui g-+0o0c.banced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it h-ai0gu01vibo+ cn . okits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end ofji m-tct 52 2ecnp o) o6ngj4ewt2zz/* a thin string in a circle of radius 1.10 m a jc 6n5 2ozotcwgjntit242z )/ *epem-t an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mome9a v:di,k52gwd+avvfmk5j) 8 fiay8yntum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when,m+95 wfi yva 8jgkd5 a :vfd)kiavy28 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 sidet+orpv :1 eusmm+ gdet6(y:x8, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a hy1dt8:p+ u6e(xmrv:mte+o gsorizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reducqix/xfbp c-wr5fw y)p:(f t xc 5/7x.ye her moment of inertia by a factor of about 3.5 when changing from the straight position to t. xtq:7cx () ppfwybf/yx5iwx5 -r f/che 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 ratpmbd9x eb*jry/9;ib 3e from an initial rate of 1.0 rev every 2.0 b ;j9/ pbxi3ybr*de9ms 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 itlv),ldpl1dm 8 s 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 throw,1lld)l 8 vmpds 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 clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skate;f37+1- .xxa s;phd-dwgj bet;w siscr 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 tv2d y:e6;b engsuod 5*he 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 disk0tv tr2f+ osych1n//c of moment of inertia I is dropped onto an identical disk rotating at angular spof +2 s0r/tcn cvy/1theed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ,ki4/ u3 dujgq2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mer,*kev:bndz-itq3kvqhz /(y-3 +v tth ry-go-round platform of radius 3.0 m and d*nqve-vy z3t z,3hkt-kh: +i (vbqt/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 dm*lq,o t zmsqgu)4w/t qe;4 3freely with an angular velocity of 0.8tlozu tqq*sqg4)3;/ m dwme ,4 $rad/s$ 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 aboum7dbo40w/ii* wdq lk2v z; -h16 xhszst half its mass in the izvd*2dbq o lw/;sx1shk-i4 z7mh 0w6process, 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 winds in excess of 1hsj q5hr;(ns euf86( * vaknwe6wi e3020 $km/h$ 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 xjv;jjw/8 0 0j5gz xhy$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate seik9 :t-5waydm6 u,d freely without friction. The moment of inertia of the person plus the plu-:, y9 5idedakmts6watform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round tuy y z:/bcljz5(rc /yz: blyjz(5ntable that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope htys sj0yx6sa0o-xk*8pzq - 8rolls on the ground 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 yxp80z-tjy 06hssxak 8osq *-–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such ths3 r,evr7,a hm4p1 vspat the same side always faces the Earth. Determine the ratio of th4rr7 s3pph1vvms,e, a e Moon’s spin angular momentum (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 fr8jzo :z gjq*/l15yewdom 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 the shape of a uniform cylindy saszwtq g rgyyei-2) xjm5,*5(y,*aer of radius 0.20 m acquires a rotational rate of from rest over 5y(agjst 25*z)sx,yywi-ryea*g q , ma 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 cylin qml(x y2kegf-,96tzndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear spxl2e qn-f9 , y(tzgkm6eed 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 gl q*;wu,y)wgg21g eke g6ia2 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 8.0 timev+qn2mf,dyu ) s 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 ofdf u)vm2y, qn+ 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 for it to5*; 9elic bzox 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 diameter 1.50 m abc x 95io;*zelnd 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 illustra ;r7x05bdebu45 gx7/a sves jqtes 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 h+ y l, +yhejiifz q+r5;wlfg1)orizontal 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.13 f.6mxfu3cdcxt f*ne., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the c6cf.m*dux fn13 xf 3tlinear 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 ;d ey0bf*x sb2m7fi,ok ;cw)kis standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, w)2bc,f;fw x*mb ioy7skk 0e;dhere h

  A bicyclist traveling with speed v=4.2m/s on a flat road 8ia 7b)m)uctm is making a turn with a radius The forces acting on the cyclist and cycle are the normaluc)8mi) 7tam b 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 e pj;4i.b xq97a z8txlsling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to 9z b7xe.l;i p8j qt4axachieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyludqzdj+. 5y7i8ja 5hdinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the anguud5ai .jj+z587ydd qh lar speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which con8rc8t a3,ujxc u*r- susists of two cylindrical plates, ofs u*tuc a-c8r8ju rx3, mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped r,ata w9yxfn3dn w9c j8u* kvifi869( fough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble mustn9 9w 89 i*8 j, kacxtfufnwidaf(yv63 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 h0q fl,(6bbhnds x 4c.not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions am 8b6z4.:h lgglsn,bt s the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accel:.4zllg8 b btsg6m ,hneration 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