A bicycle odometer (whlqa ui g(;0jj,q,kp q8ew5 g,qich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wh q, qigqque;g0l j5kj pa(,w,8eels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at vdmr07w6vrz 3 constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have 7 rvzw md603vrradial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vel;bwz(m9nkd e 7ocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger foxmrxwj9q6sw : xrzz/oc:);g5 rce? 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 witaqh 93 fr4*/ho1wx iwah your hands behind your head than when your arms are ao*1/wqrw 9ah4i3hf xstretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sk;kdal h(*z.kfi pi 96prockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a sma;fk (liza9*6k p id.hkll 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 ( khrs89 ;8ftpoyq,ztu:me i4 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 mass is advantageou:(8oeq; hm,z8ritkp4 9usy t fs.

Why do tightrope walkers (Fig. (5o cdooy/bc )8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If tma)e t,ve/c4(ubr os :he net torsre: /o )ebu(4tcmv ,aque on a system is zero, is the net force zero?

Two inclines have the same height but make different angle8 n5fzgf2rqdhsm, r-7 +llak(s with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explai mrkf8rz7dl h5 agqs+nf- 2(l,n.

Two solid spheres simultaneously stm *..sg/f kigkart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at thk*fg / s.m.gkie bottom?

A sphere and a cylinder have the same radius and the szh-xql8z9 5pb )bkx k1ame mass. They start from rest atzqx8b- p9)blk z hk15x the top of 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 angular momentum are conserved. Yet most moving7l+w:e sn.c5e l *vc5hdyw+.hyia;oy or rotating objects eventually slow down and st7vewlec; a.+c y5ihslwydn* : 5h +.yoop. Explain.

If there were a great migration of people toward the Eak3)9edt 8ahda rth’s equator, how would 8h3d a9a)et kdthis affect the length of the day?

Can the diver of Fig.;/eox nfx(;me 8–29 do a somersault without having any initial rotation wh;;(oe xm enf/xen she leaves the board?

The moment of inertia of a rotating solid-6hcnro layadk0/0tswydb 0x sxa7a43 n5 h.6 disk about an axis through its center of yhxca -dro0h6da 3lswtsa b.a0 k5n7x 046ny/mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotat +cka 6gfj.ye1+2j5)qfcbnpting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop tk152jycf ge.j) ptbq 6+f+nac he masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sam6b bup )o(2)w8lpqdmhe mass. However, one is hollow and the othq8omd) bh(u2l6bp wp )er is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rk- n vz9;1lj(x h/rde);s yg-l,ddn vzotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration pointsz1kh l--rj 9 v,gd;dx nd)ynz(e vs;/l 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 standvn /*ck0ld*w1 c d)a dl8sbl.ding on the edge of a large freely rotating turntable. What happens if you walk toward th d*)wcb ldn *0/kl1scl8avdd. e center?

A shortstop may leap into the air to catch a ball and throw;a bpvo ;o17v)vt(tu*b beu (0put 2mx it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs; ab1 ( 7tb)e0vpxv;um( ptut *vobou2 rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of anguir sqi 5vc5b8 lc204tcz-v 7gzlar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the sg-civb4tz 5 cv78isqz05c r2lecond propeller can operate to keep the helicopter stable.

  Express the following angles 1a m8jx nie(4..fm fb8 x*vlnuin radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coin hgykh1lfbqh +k;)91 ccidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sg9lq 1+bf;hkhh1 yk)c un and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 *80)stq+tha puhga1tkm from Earth. The beam diverges at an 8t1p) 0asu *taqh gt+hangle $\theta$ (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 7dmnioi :eudtr*z7 l4 vo( e-*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 angulvto r*-d uez7mi7l4 ei(n*od:ar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to ane 4bsilp 077sx3tzd ;rother child. If the ball makes 15.0 revolutions, what i;x 737zlrtpssd4ei0b s its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.02wryzvvftr,wz4 zm, dk;d1(7 km. How many revolutions do the wheerf7 zw km4 zrdvyz12(,v; w,dtls make?    $rev$

  (a) A grinding wheel 0.35 m in hw5f)/ 1)swem*ivisadiameter rotates at 2500 rpm. Calculate its angular velocity i mfsv)iwhea5w )s/1i*n $rad/s$ $\omega$ =    $rad/sec$
(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 :e* pu mnu 1d:51vxv-b va2bbhrevolution in 4.0 s (Fig. 8–38). (a) What is the liunhm2* badpbe1x-v : b5: vuv1near speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun8b sgy+lrz)7/ sbh /lf    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointx0dj5e-yq+c+rux8 nef 6p3; zns)z sf
(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 ro;mea pmguf,0i xmo o,p41( n9vtate if a particle 7.0 cm from the axis of rotation is tmp,ao; i9oxvf 0gum, (nem14po experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelqs*gizor - /j+erates uniformly about its center from 130 rpm to 280 rpm in 4qisrjg/ +*oz -.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu:ak-m s35p9gtzsr1xcd e;g 5js $R_1$ 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, astr.nbu11j+b2kl rsqmz3onauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their tr+ jq n3mus.kl b1br12zip, 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 rest to 15,000 rpm in 220 s. Through howw ed/ w(-ge*zfa1 u9 ev 1:lvzt8ffd:z many revolutiot18(:zd w ag:veeful1 v z9-df/ fwez*ns did it turn in this time?    $rev$

  An automobile engine slows down j:sab.- vqu /za98 t xtsp*rw)+d1ig zfrom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stressea u xw(*1qbq zvm)) j1ns+1jprs of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revorxvp+ *z1 j)nj11 a(w u)qsbqmlutions 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 in7/)za 6ycis f-i9e0bfx b;bql 6.5 s. How far will a point on t9-af; cbfs7b/qzbi 6ylexi 0)he edge of the wheel have traveled in this time?    m

  A cooling fan is turned ofh e:ena/7 prs6f when it is running at 850rev/min It turns 1500 revolutions before it comes to 76a:n/eres p ha 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 uniformj)i jl 2sym44h y0.pylbp;*8sf6e rs wly from 95km/h to 45km/h The tires have a diameter of 0.; 46)fj2lr *sp py.hmj4iysswly be 0880 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 revoluvawn,2i)b s3;. ,urnjgye 6wz tions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 mbvw2s ).yin,6 , wga3;eurjn z.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a h; gpy4awu mw)hvuo1/ 4/pt)xdillw)1owt/yxmv 4uag4;p up)d/h. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm ndlyh xd4l: w1e6 8(dfwide. What is the magnitude of the torque if the nedhdll(84fd 61 y w:xforce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wha/o(loqg72 r +f gg-d b8ver1oeel shown in Fig. 8–39. Assume that a friction t-bqgo1e7o(+ov/r r8ggdlf2 a orque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of ma:panssai0*pweh , qk/(up3. sss m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially ew0k :ppsq /.(unhai p3*s, asthe 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 g;kra-m8 r +vu /;hnx6yukbw5 head of an engine require tightening to a torque of 388mvr5 r;/w 6kxu;-k +bnuaygh $m \cdot N$ 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 1h*/ndn 1ih xsapr 344s0.8-kg sphere of radius 0.648 m when the axis of rotation isn p /4h3rh1 ssai4d*xn through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle en6y) fzv - ynuhy; bq3;7arn4wheel 66.7 cm in diameter. The rim and tire have a combiyrnynauhfn3 4)-ez7 6v ;;ybqned mass 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, light rod is rotated in a horizma f ey35oqic;qe1e2zlg 4l/vw *11hmontal circl c1mv1 l3qf51mwoq/ ;la2e4g*ie ehzye 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 py04y 0x0vuenlkpbe (1 otter’s wheel rotating at constant angular speed (Fig. 8–42). The fr 4y(e010ykl npxeu b0viction 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 poinyvo4 /.m zf o-8opv;abdz )/gat objects shown in Fig. 8–43 abb)/afo a gp z;8-yovzmo.4/dvout
(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 tot . f zzv1yq4i2vh7.bdval 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 satelln,+ o+om auuugy+et ;+ite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3 + on+ouage;+tu+muy ,600 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 cylinder ba+7(*uzo 4:u y etnzwk z4n2wwith a radius of 8.50 cm and a mass of 0.580 kg. Cazn* wbuytezk a4z wo7:2 u4+n(lculate
(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, accele k;jpxd2 mlf2 ssmc619rating 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 tangentiaup dkutpk2e(;f8 0pa(lly on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequap ek0u pf(ktu8;2( pdency 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatxv42jh dohja/3.)o zfing at 10,300 rpm is shut off and is eventually brought uniformly t jood/jx a3.h2z4 )vfho 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 acce*wr2v, u2 b v rowf4fit27lhhj.tht21lerates 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 ac/jyib74cwdo3h9r h-42 . crkmjv ,eiction 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 rej9rch7i44w chodjy 2-bcimv ,/3ek. rst 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, aakrtwi9 6(p;y s shown in Fig. 8–4tykw 6apir(9 ;6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,hva+7/ziatrex h6kd +otx.vz, *p9u- $m_1$ 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 k)l jxu i8dc 0*84r azghn -o)6tgr9ymfull turns (revolutions) and releases it rl-o)6x0ya 8zi4gu rkh9 mt jnc)8d*gat 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 momt;x0q- zrvu s.ent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile enginef/tmuurw a2-1 7ethfyq o6zij9 3+(vk develops 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 t gxrpn5+n +ddr i5t5 fg.h8y8rolls without slipping down a lane at 5n8 tdhd5+tg+yig f5.nxrp8r 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eart67:jghtbs :g. l x/,owlpj.d bh with respect to the Sun as the sum of two t 6 xgpdjoljw bt./g:,.:h7bslerms,
(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 mas/ig ,akw 8s.dum4h2kc ss+itovq048qs of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest t og2cawi +h im/,ksk4d8t0v s.8q4us qo 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 kz:-- iw*vgw rzg starts from rest and rolls without slipping wi-w :-zvr *gzdown 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 bal fc*4ln;1jfk p.3zntij; as/1wxo m/lanced 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 hits the ground? [Hint: Use consernjo;smwkxz1f/3jctl1 p .*n 4; fi /alvation of energy.]    $m/s$

  What is the angular momentum of a 0.210 .1mn*;7 qgt, )ulnqxku)y mpk-kg ball rotating on the end of a thin stt7mm,qyk)kqu; )l *x .pu1gnn ring in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindriznxf5* xh.dhi8ok2 j6 cal grinding wheel of radius hjh 86n.k f2zdi*5xxo18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that io)kc)hxjjdx /app q-fa5 5:n;c2wsb 9 s rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to9 fnbqawosjxch)pj d;2/xp5c -:)ak5 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one,/v:)dy8b qhtg qsvuli ,f 0z5 shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her uvi, zbsyhqf5tg: 8/ )d,ql0v angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial lruprrxd f9p)9o9-)t rate of 1.0 rev every 2.0 s f9r) od p-ru9)trxp9lto 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 verticaly* gu ;t4ep+hm axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg ;yt4pg+*u he mand 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 clay sticks to it?    $rev/s$

  (a) What is the angular mombanyhc2t 6, .) zbnat9entum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentugl riy 9 84z;la4l/odkm 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 dropped onto an id gp. md:s4iys,entical di 4.dpymig:ss, sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4b 4kzl74,+za od7nsb t.(gsrh $rev/s$ 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 1vl 8,conbwaen uip ab9wp++h5 )4i( srotating merry-go-round platform of radiuwnc9ah,b+wle+nuips bv8 1a (i o )p54s 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-r*0/-(oknov6m wuv owaound is rotating freely with an angular velocity of 0ao vwm wo kn-0u/v6o*(.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 dwap)k:e bb4tee( rf, losing about half its mass in the process, and winding up with a radius ) kb4ee(e :pbt1.0% 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 12ws d* h)1qpbd/wtm6,am* n)8w zwhu(x0 $km/h$ 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 mgah(8cgv b 4-asr 6v*$ 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,u+tb xb.y.z .uqbayc ,4d5co , that can rotate freely without fricty, c+ d,uy.a. zubxqo.4bt 5cbion. 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 zx*5f9g 9s9( qztbjqbedge of a 6.5-m diameter merry-go-round turntable tbfg(q 5j bs99z qz9xt*hat is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the qw(/zyt8c,n)h t2z uk ground with the end of the rope lying on the top edge of the spool. A person grabs the8 c)h/2ztq z(y,tun kw 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 from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same siin3njm )/m24pai7i0 /xbyhvx de always faces the Earth. Determine the ratio of the Moon’s spinxh /3x)miy0/ n a7bjnmp 24ivi 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 from 05 qfz-c t0(liiy+ afhrest 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 cylinder of radius 0.20 m acquif0b0ah*-y17 hcm dppares a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by t1h7py0 0 m-dhpacbfa *he motor.    $m \cdot N$

  (a) A yo-yo is made of two 32(m,w5 )y9y*dg vmc2t+ck qn xautbz solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentk)y2cdq cvg*9,b3zaymw (+u 2 nm x5ttric) 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(t;r0 v2 +6pt c*4fbdal fmzqn the angular 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 size of our Sun, but with mass 8.0 times as great/rlwxfv0 1+6f sgiz *e, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carri 1 li6*xzff0erg+ s/vwes off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored qv1 u;h(c4ni zin a heavy rotating flywheel. Suppose such a car has a total mass of 14;( v4nqcu1hi z00 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 illustraf: /lcb :km:fg1ljzk*tes 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 horizontal surface at wut2ko;ukw -) 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 ju3hit+* m9e) knal 3iM and length L can pivot freely (i.e., we ignore friction) e3h3mn +ku iaj9*l )itabout a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiu/ rs mea*+ds:ds R. It is 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 *rsd/m +sdae :step has height h, where h

  A bicyclist traveling with spo f ztbw/x;,x1eed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cycli1/w ozxxf,; tbst 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 begins d,bw(h5 cml 0lud:x-h whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a radclw ,l(u h:mh0d5-xbte of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a soli6 kz/sg mig 7yd,twh0-d cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinningdgh6m k -, zis07/gywt skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two s 55 d.,e/r rbagdm3 r7knr6zxcylindrical plates, of masrs5zgk d5,ax7m/.nb3 rrd e6rs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58() q 3xq5.7lih72j8gok:m8n fvlqz f mu1xtdi . What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the qlv:zx)f 77g oh12 5qi88m(3 djqx.m ftuinlktrack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but2ra)8n q4ft3 ) c.r+w anbz2ooclp/at do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed x 9f*x62o hrrum+ bk(luniformly from 90km/h to 60km/r rlmk*uhfx(x6o2b +9 h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s