A bicycle odometer (which measures distance travele8 )sock *voiq3d) is attached near the wheel hub and is designed fc3*i8 sok qo)vor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulark *nf9j7ak4z mbx*x(n/i7uwk 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 tangentnu* 47jxnxzk* /f(ikamw7b9 kial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value6 0x ofjg9v/lr:o4tld of the angular velocity $\omega$ Explain.

Can a small force ever exert a 3 78t,3ex9fphugm pybgreater 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 sit-up with yh6 uj(uxa*vi -h (eig7our hands behind your head than when your arms are streha 7e *vxhgu (6ji-i(utched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at tlvfz h;+ys 9w)he pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a lal)szh f9wv+y ;rge front sprocket? Why?

Mammals that depend on being abjy fyvpr ;zkd4 + 8,0gif8svu5le to run fast have slender lower legs with flesh and muscle concentrated high, close to the y0i v zjf5; ,v4ry kdg8f8+upsbody (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrrzk :gkfl 5k1j-/1zxtw ;x9z 3jvgm/vy a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If apdftg, ;8zokph19/n8m.wf v the net torque on a system is zero, is the net force zerw,89 h./agfn ovtdfpk81pm;z o?

Two inclines have the same height but maek7snx ,y r 9r2m4zpps13ns7nke different angles with the horizontal. The same steel ball is rolled down each incline. On which3pzrnse4px rmyk9 7n,n1s7 2s incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down as 1j,h)5: lyilqohvr ;n incline. One sphere has twice the radius and twice the mass of jqs l)rv1,i;ho:5hl ythe 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 th,7gaemns3 i oy7 *ls7ue 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? Whigo 3esy7l7u na7,sim *ch 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 moviw(6zuma)veoukse1y/ rsgy : 0 2oh74ing or rotat s7 wu:e1koh)goa v(6iu2mzyr/ey s4 0ing objects eventually slow down and stop. Explain.

If there were a great migration of people towzfcjrigg,i76 g8 ex4 )ard the Earth’s equator, how would 4e 6zfgx7 , ircj)gg8ithis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without n utnnrq5 ega.+4i3l+6.llu rhaving any initial rotation when she leaves the i.4 lrl6t5nue.g+3 nlnr auq +board?

The moment of inertia of a rota 71ujn4 cdakp4st8 (eating solid disk about an axis through its center o k4j7ctun148e(p asd af 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 rotating stool holding a 2-kg mass if 8ad+1sd(yjm+c km7c n each outstretched hand. If you suddek (dm8 1amscjycd++f7nly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havettzk ,.l0mq gkh9xxq)d.zw+od5 +m4 m the same mass. However, one is hollow and the other is solid. Describe an experidm+,qqm + 5hzo9zkm0 .4kxdt)tlx.w gment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. If cz1ddkcffuff2 dz/80x-(- h n what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangentialz dfufx(-h1k fdf/fc-cz0 2d8 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 fqt q:ddwnp bp0+ 2s*b g wr q97+mo9;bcr,ba*rotating turntable. What happens if you wa dtar9pr gmp9b2;:c*0 no*fb bqqb +qww7s+d, lk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asyy3 r)mz6qwev k9 ww (om,)az* he throws the ball, the upper part of his body rotates. If you look quickly you will notice qewam3 *y r)zwoz,6y( kvw9)mthat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momk0j+ao t8u,-imt+nz2h egls . entum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelle mult+t8nzi. a+sej ghk, o2-0r can operate to keep the helicopter stable.

  Express the following angles in radians.mplffti/(v n,abo a;- ,x cb5: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using cb7x4dra)au5;e1 n, z2)t*szdjola x the information inside the Front Cover, the angular diameters (in radians) of the Sun a5bd)n4a*s,2 uroceld;x) tx 1 z7z jaand the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directe.c4qt3ro-lb ox8 eh cyo(1w/ar7r *fyd at the Moon, 380,000 km from Earth. The beam diverges at an angle yo/4r h3xro8c* -lce7owbytaqr.(f1$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm( xu;1dy1lb p:)5 bps8vepbap. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow dbl():y appepvp1;1u b5x 8 bdsown?    $rad/s^2$

  A child rolls a ball on a level flooreu8 a); nol)d7p)mht s 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamp;d)87) e nauhot) slmeter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dol-p;z dza q,zi . q.vdgj2f0f; the wheg;qa lz.dz-d0fp jv.,f;q2zi els make?    $rev$

  (a) A grinding wheel 0.35 m in diameoivylgh/aouc9xo+ 0k 5-( 4gv ter rotates at 2500 rpm. Calculate its angular velo0-a(k54hc x goyilv+go9ou v /city 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 (F (3ar7tm;vglr ig. 8–38). (a) What isagltr3r( 7mv ; the linear 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 theqbv6ova7/ wn:.) b 0w grhwj(kbox6 1c Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointgbffdz dk: 6c) ib 5;1*apm+/mok,rvq
(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 cen1)o8,po b-rtiw z1koftrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an)ofoi r-8,p1ktzb1o w acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleret+ *rzmf :qx3c-ex7d1z g7gl* zh ,pcates uniformly about its center from 130 rpm to 280 rpm xtglg *fzdc1,77cz3hp eq m:z- +re*x in 4.0 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 radius6pwvpzum7* 2(mboo *bx5fy t* $R_1$ 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 ad wjryr mf8cww-8 o:g 6z718pnboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. D8pg1:w nycj- 7w86w8 dzrfrm oetermine
(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 indhc5hostl ca0.g;* 0s 220 s. Through how many revolutions did it turn in this time? st;0 csd*a5l.hho0c g   $rev$

  An automobile engine slows down from 45fm s-bptk;3,t00 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 whirling “hua9cf-4 jbqun c9;; xxqman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions bef qx49bqu;-c9njcf;a xore 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)8g+*ucwmq3o l(r dpbys2y( j in 6.5 s. How far will a point on the edge of the wheel have traveled in this t( dg 2rp(*8 wbmsj )qcly3yuo+ime?    m

  A cooling fan is turned o sq .kg,25ab4 tz5adioff when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop.agta5 45dbkoq,i2 .sz
(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 ee8+li l/)2.mz:b)n ufafvurwr* qw ( as the car reduces its speed uniformly from 95k wae8u.i+r zmbe)qf /l*nrfl:w2u)v (m/h 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

  The tires of a car make 65 revolutions as thesv 0l5lzs1 *am car reduces its speed uniformly from 9z 0 vsm1al5sl*5km/h 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 pedalm;c,h-p)bbr d3lvh0q 8jfs wrz*;t7 x when climbing a hill. The pedals rotate in a circle of m;l)z,;rh0 pstq8vd7-cr *jb xwbfh3radius 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(4wr yj9u4nfc of a door 74 cm wide. What is the magnitude of the torque if the forcuy4cw9f(4rj ne is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown inpv y0spm6x8 xjpvj-xw) **z*dw sfs8 . Fig. 8–39. Assume that a friction.vp x wpps wsdf-xz6vm *j*8j0y)x8s* torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a:8df e* 0ijso+ :meovbttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this systfm*0e+8o sb:vji edo: em.

  The bolts on the cylinder head of an engine require ti6mw bb-el8/i hrv9 *0 o:)fahu6 lucygghtening to a torque of 38 *uha6vl beu 8ich/yw )6 og9:frl0bm- $m \cdot N$ 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 i1zclz z--no+b j9xh+ 10.8-kg sphere of radius 0.648 m when the axis of rotation is thr1b+ij z-znzh- ol9x +cough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle vsgk o9qtk ljz -8lu mh27ki9s*o 5+b7wheel 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?)ku+ gl2vj-s sto5 i8kq mo79*h zk9l7b.    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in j:t3mvu xuo8p 78-wqm a horizontal circle of radius 1.2 m. Capx 8:tquwv -u37m8 mojlculate
(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 skj y8ytfa;9h(-kjoow.t87bpotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her has-(t k8 f9j8 bo yy7oakjwth.;nds 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 po/ yy j(f-r0 fmt3)cqloint objects shown in Fig. 8–43 a-tff )0m / jq3(olcryybout
(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 coeuo fp, r6slr.wz3i 6/nsists 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 satel-j9gsq*;wc p vlite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a 9gpw vj ;cq-*smass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a r*oml i(6eakkr4,p ). w/itggsk n. n9jadius of 8.50 cm and a mass of 0.58a,. gj*irk)t4(m/ k6k sp9ogn l.en iw0 kg. Calculate
(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 batxr;f3c wluxggu3 3, /s.ilni6, accelerating 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 small hand-driven merry-go-round z/l,wd 9ctj,r ro o:cw/q/q.q and is able to accelerate it from rest to a freque,w.tczqo:wl/ / ro9jc/dq, rqncy 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 rotating at 10,300 rpm is shut off and is eventually bri6h,ru mb7/-+t6vtdwbn8e jbd.3z 6m zhmo; nought un d-w+u;m/rvmb o 7bi th6e nj,n.b863mdhzt6 ziformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a hc(kfnw.)xj *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 sole8qlb7 6rtt,h9s s)ibkly by the action of the forearm, which rotates about the elbow joint under the action of the triceps musclet9l87bssh it,)k q6 br, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blambc bqj0a4h*6i(xc1 loqs-b-de can be considered a long thin rod, as shown in Fig. 8–46cbjbl16--c i hq*(qb amo xs04.
(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 t/rr/hyoe pq/fp45/tn o 1)nu cwo 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 w h+eyu.p8sv 5hithin four full turns (revolutions) and h s8.+ue yvp5hreleases 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 hif5:4 .dx wbugy, je.j6o)hh fas a moment 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 engine develops a tov mgnwc 4)wa;4rque 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 ao .s-rc 1nxz5;s 2 1dnz5gzz/ kgszd6dnd radius 9.0 cm rolls without slipping down a lane62;sszzzz z5.krx d sg/odc 51gdn1 -n at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energhjju 54f28a-olrc (zb y of the Earth with respect to the Sun as the sum of two tr5-a hcb2l4(u zfjo 8jerms,
(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 ra45vpe) pxo2b cdius of 7.50 m. How much net work is required to accelerate it eopv45c xp) b2from 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 rwb*b pjge-; 4; )putrsest and rolls without slipping downb; p4 e-; *)jtusgpbrw 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 balaz0r7bnq.p 2 npnced vertically on its tip. It starts to fall and its lower end does not slip. W.0n 2bpqn pzr7hat 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 momentum5 ef dpibfags-.x/6 ef*;2an d of a 0.210-kg ball rotating on the end of a thin strix-5fnipsged/f;abf* e6 d. 2 ang 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)gga* yq/(8sx tnp0lo of a 2.8-kg uniform cylindrical grinding wheel of radgp( 8x*q/ts0lyga) noius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform -e9b;uxy,k u mthat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, theemu;ku -,b9x y 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 reduce her moment of iner:twhiub4e , .+0jbnf p(ed3sgtia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2t0hse p3.uib + jdge, :fw4n(b.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 incre, j,ovvvz./*e cmjo pd08f ff.ase her spin rotation rate from an initial rate of 1.0 rev evef*cej.o jmfzdvp 0o8,fv, v./ry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through xa op7n2n+)jqits center at a frequenc2+apqn7x oj n)y of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a0a27r 3fxavuit 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eartx ;h:gltjg-s2 h
(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 droppux wmbv 8i/kauo/0 5.hed onto an identical disk rotatinoui.5h mu abvw0 8/kx/g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.43d:n,kmxgre 55jep7(n c ut3v $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round pl)p on*.z cpn3catform of radius 3.0 mz3no pn). *ccp and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocz hxsv k/ulwxx7i/3 006gejy2ity of 0.8v2wlx3 x0ygx/6 szh/ki j07ue $rad/s$ 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 irh dwpmymsk 9d(s1 zr+.n. 7p2nto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existd1ksy mn.7 .r swp2(r pz9+dhming 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 ix2hb/e dni:) srsq,.e n 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 .29cztto*9nkgm s4;0bf y;- sxdzrzo $ 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 platfgc7 7 l iodm3)3wun3ymorm, initially at rest, that can rotate freely without fri7mylm3ogun3c 7d 3w)iction. 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 a 6.5-m diameter merry-9me 9*e(t( ia bkkr2qr. mt5ilgo-round turntable that is mounted on friction ae*5kl k9e(it(2q9br.t imm rless 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 tyfal w8)a5e9whe 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. 8–50. The spool rolls behind the person without slipplyea98 fw)w a5ing. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side a, :qlv)c : zzcori+jb-lways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle qj)zicrc-lv,o b+ :z:orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate rkpuiu.ql8 62np1e 7fof 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 radffvn(p5/hqj :t ekmk y4/dg81,am, gcius 0.20 m acquires a rotational ,4fmqa8t he vd/p/ k(f 5mgn,jk1gc:yrate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and difoe.nx)lw ..c-v;m atameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050ctwn;- oal.fv) m.x.e kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speea 3a;x *mubjs3d 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 qix/fnyphf z,) (m uq+qm4)2 btimes as great, were rotating at a speed of 1.0 revolution evey+iuq (z,4 fh mp f)xq/qnb)m2ry 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 automobile is for it ty-djlnq:77)foo wvr 5o 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 and mass 240 kg, and should be able to trnwofrlyo:-qj d) 77v5 avel 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 illustratg5/1chv:d8:s7l/l7 or hqk s poqae 8mes 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 /yuy/ k vo404za*;bjtqogac h wyb 52;5co8zkon a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M 9 u 92yrg b13c2cdsjq/ywb9 knand length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally yjkn23 u 9/9cbsy9w 2r1qcd bgand then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, asma+ w ya6 o.6s9ss/mhnd we want to exert a horizontal force F at its axle so that it will climb a step against which it ress.ssy+ w m ha6s69m/aots (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with ab8tk p5v0vp5 s radius The fotpvksb05 v5p 8rces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length amqj )f-f9zrz+qh6cx74 +ejza/ 9d dtjsom* (1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it sqo47zd/jf+ m9xjaj)m d*+ eh z6 fqcaz-rt9( from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and h ;e4-*)ozel k d-tjhd4bqg7ki er arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a ;gd-4*zeiq7kbetl4 - jh o)kdspinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembqjg5a46cfaj qa+h5v2k ;c ,qd ly which consists of two cylindrical plates, of ma6c5cq;aqjgad4v, 5h q a2fj+ kss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t0b) kxd33qj0hj jxf)w he looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach t 3)xd3b0xqjj kj)fhw 0he highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do -2b2:y42m f h bb xahbkwhw9e4not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolution ok 8j*w /0k.hj2meozas as the car reduces its speed uniformly from jke/* . 2ja0hwzom k8o90km/h to 60km/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