A bicycle odometer (which measuretc/lw c 98mpw7e1q k5fs distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 2pe71f/8qlwmc t 5k w9c4-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point f8o;t/ sci9 feon the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude ;t /9ce s8fiofof either component of linear acceleration change?

Could a nonrigid body be described by a single value of the ab7w 0emiepi t+c8 7m4gngular velocity $\omega$ Explain.

Can a small force ever exert a grea,mpbpbii y fx m s1/rl81*0r.zgui+ 7jter 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 your hands beh7ic fot9e 7w+vind your head than when your arms are stretched out in front of you? A diagram mayc+ftive77w9o help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pm)cyytg;890dsvxs:m sp(0n m edal 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 sm9 msvnx0s:d 8 (gcy)mts0 py;procket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender l1 w+ w5v -5mt;txhpbhlower legs with flesh and muscle concentrated high, close to the body (wxtwbh- 5;1h5mlv+t p Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ; wej /szil *h8at4.wpvtyc-)carry a long, narrow beam?

If the net force on a r( qvrzseenm2p0l.0k 1 hyz./system is zero, is the net torque also zero? If the net torque on a 0(.lqhyzpve.r/1 2e mnrs 0kz system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontrl c; /vkt(ewpx:c/ 4pal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explarl;ct/ :c(4pepk xw/vin.

Two solid spheres simultaneously start rolling (from r.:y lkgxr-5ti est) 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? Whichktlxg5-: y.ri has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They sta* oau:.:mhwc8l;suat rt from rest at the top of an incline. Which reaches::oh staw*mu l8.u;ac 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 momentkvq +s(u-d5ts anu)n7um and angular momentum are conserved. Yet most moving or rotating objects 7stv) suund( ank-+q5 eventually slow down and stop. Explain.

If there were a great migration of people toward bawp 7tdl. ,al+ s*9cdthe Earth’s equator, how would this affect the length os.d+*ctwdlb,7 ap9a lf the day?

Can the diver of Fig. 8–29 do a somersault without havinyyyghb+c1 v)/ g any initial rotation when she/yygb1c hyv) + leaves the board?

The moment of inertia of a rke-0mirfhvqny n)5c -. 2 4:l glsl2enotating solid disk about an axis through its center ofi5vrf 4cnlk n)qlm 2hl-y:s -neg 0e2. 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./g f8kw fj 1 qlj6z90tfgg1bgb86rew mass in each outstretched hand. If you suddenly drop the masses, will your0rt.w8f g g6ebf11/kfqzwjl8b9g j6g angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sa9cyzkew orp(6br6(c5dk a;xd*oo) ) cme mass. However, one is hollow and the other is solid. Describe an experiment to determine which is w(yo6ckc ob;kc6)wpd eo9 )*(rdra5xz hich.

The angular velocity of a wheel rotating+p9wxwrg4( kqh na5 m7 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 points east, describe the tangential linear acceleration of+r 5p nwwh kmq4ax79(g 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 rotatingfvv(w5-q+uud turntable. What happens if you walk toward the centerv(+q fdw uu-v5?

A shortstop may leap into the air to catch a ball and throw eccuaw m0 c2ahgwy/8vpf g:w1,j1+.i it quickly. As he throws twfhj m +pa 82, e.ggvcuc0:c1w 1iwa/yhe ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disduflq 3* da+n,cuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the heliaduq*f d3ln,+ copter stable.

  Express the following angles in radians0i;cq 1) dm r.w:+ jwjcqx8geg: (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. Ctffc 8ymk;n/ .d(cx j(alculate, using the information inside thejdmn8f ((cc/x y.; tkf 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, 380,000ykv):jzzh*p)f/o/wo.pc d, h km from Earth. The beam diverges at an.hp dhv/of)c:zz)ky, opj/* w angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motoxr3m .npb8:m f/qb kex) 7r0qnr is turned off xf7p0b xmbrn 3n :q.8qk/)rem during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. gh fbx8ob*c.d: f8by- If the ball makes 15.0 rev*8cf gbbh8 -:.yodx bfolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionkaa0zazsl,,-zvn 6 b1 cmu (7hs do the wheuv6z1-,,ala 7zkncb0 azhs m(els make?    $rev$

  (a) A grinding wheel 0.35 m in dias.chw0z2y0 n, bp0rs. 7criv ameter rotates at 2500 rpm. Calculate its angularz0s, yb 7as2w.rcp hr .iv00cn 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 ka gw.pm o304i(3 wqblin 4.0 s (Fig. 8–38). (a) What iiwl.wagm04 op3 qkb (3s 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 +txrfja , rue21m.i qcu- /lk8(a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po+f vb w.q:,qa ex0nwn,int
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axe 5+eb j 9ygavn*fu.wh:u-vvjz6 g (t)is of rotation is to experience (vw jnye.fuuvgj6 + 5 -zgb):tvae9h *an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abor8sm4dxr6m f 1 n6gs+zcf( iv7ut its center from 130 rpm to 280 rpm in vf+6x1r4rm s ( cgsm86nfdi7z4.0 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 radius 8stv.7q-s7 5 g mkfbmh$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollos/ah( l9 rmwabh7r:nf,b9tz+ spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, the(,sbnbr9 zw7 a fm/+9th:rhal y 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 rp pz0;)qhza9uysgz /: tm in 220 s. Through how many revolutions did it turn in z9g/z : hp0 )st;zuqyathis time?    $rev$

  An automobile engine slows down 6 e iuqioss lo-2*vpfa,*t5(gxy 4je+from 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 stresses of flying highspeed jets in a whirl: ur+1:lb k ghv5myes-ing “humangm lr5hys1k+: b:-evu centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fromc9v ythqsc.8 2j 2lad* 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge * 22 9 vj.shy8cactqldof the wheel have traveled in this time?    m

  A cooling fan is turned off when it i(.5 2ozpov*nnzj/xlr y:ch 0fs running at 850rev/min It turns 1500 revolutions before it n2y:v pofrl/c5 x hn z0.oj*z(comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces itsby. 8t4h1kqvy5wbup 2)l 1 rwm speed uniformly from 95km/h to 45km/h The tires have a diameter o4.ywh2yrm8k5 qv b 1 t1wpbl)uf 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 mapd dizx(gek3 q6h /4hozg; 5v0ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 e3vxdp g/ dqkzh5g46 ;hzio (0m.
(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 weigy ,q5ys)vu-mtht on each pedal when climbing a hill. The pedals rotate inq) m-tyus, 5vy a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the mam(dm8hp0 ( rm(cfpt;ognitude of the torque if the force is ; hcpmmmd(tp(of8 ( 0rexerted
(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 sh173wpr lf:t h9jcx.suown in Fig. 8–39. Assume that a friction torquxw1 9 3.hlcts:7 pfujre of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of z: dcj 0q2hzj4a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate th4dj2z hcj0:qze magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque ofp jj7njeyecxxeltt 6ls08 t3qh-7 d a21s+2j- 38 q+ e t j t-n106xhl3t2jdjy 2spes8xje-l77 ca$m \cdot N$ 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 p6;pml,a,bmamq 0 )0zzt pw2bof a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. l2qtzb6;p,m aapzbm0m,w)0p    $kg \cdot m^2$

  Calculate the moment of is d w xb9kgn2;v08tv8lnertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of vx8kslwd 09nv;b8gt 21.25 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 horizontalfk5 4t(*qyx:hnn d+li1 yfh1n circle of radius 1.24ndkq h f*:l1x yti5n+hn 1(fy 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 constan2nca .yv, wvh-tlm1m50gg ao-k d5ov (t angular speed (Fig. 8–4v- oa-.,g v2 0ycknhm5w5am d 1(glotv2). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. ( b9tufcz3(3e j1cfgc:bf7d /ab8kdy8–43 abubjd:(t ebd97b/cf3cz y 8c1g fa3(kfout
(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 o:f.7ag:69 ac eekqi p ml0v9k1qvaq9 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 c6bidc534 jme06 efok5m7 gbyi wvzj(/ylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to r 6fj ey 0k 56bmec7wzjbi/io(5g4md3veach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.bu :nhe6 o os4e7 a*)hydgyu2650 cm and a mass of 0.580 kguey6:dha2 )nb*shy47go 6 u oe. 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 acoo 7ulxx9:go:va z m 5s.7j/g bat, 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 onosu 5 i:z)kdm8 a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on thed5:)ko 8u szmi 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 brour8rtdnuo rm,wdg2b/6,pa 43)my)di j 4 h7wlhght uniformly to rest by a fr)n43dp y,ihr8t6 m/),hj aw2 7r4dlorbg uwd mictional 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 3hgt8qj8t 8ua0 .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 o:wdgr 7ndz;6pf the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The baldnwzdr;6p :g 7l 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 considg9ns5zw 7,ku- fgcpx5ered a long thin rod, as shown in Fig. 8–4659wn,z-px 5cg7gksfu.
(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,acj )7xx3ke z0 $m_1$ 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 acceleratse,9k,(z1c96g r +x fqdq6fedb x4jipes the hammer from rest within four full turns (revolutions) a zkpe (d g96j94cqixx, +sfr1b,6e qdfnd 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 mos9 (/el:e2h+w1 f3-isvgja ilrv qho.ment 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 torqdo.jul ahs9a1o q4l-h3)woqh o,86rlue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm ro .k7/*,6vi xnsg(mu+e vxnjj slls without slipping down a lane ikmsnnj j guv*e6.x(s,+v/7xat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with d,6alj lhs.)um ,)c5 eatn mq:respect to the Sun as the sum of two terms,,ea tu)6 m c:.mads5ll) hq,nj
(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 nyf 3)h9,cg c/d ans-tlo0.xo ket work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinahf /o)gysct cx,.nk3d l9o 0-der.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from restndm28q* d(fsl-b jv t3 and rolls without slipping down ad( 8 s v*l-b2nqj3dfmt 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 b1lo* k mj.lmc2alanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the m*j.mkl1lc 2oupper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mom-;ykc /n-0yhkt b 3zbwentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.-t0bk b3;h yk/n- zwcy4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.e-qasn,+xh7 i8-kg uniform cylindrical grinding wheel of i-as, h7ne+xq radius 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 sid8kcvp 730ov;)av6a t )a/hfymm k( echsuc. d)e, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48cmcm evy)c8kuf(/3vatpa;7.hko 60va)hsd ) , 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 on +a.l vzpm19 v2zd jmv5 qen,1ugg1f7ce shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.55qg+11zmvpa2m9vc j .vdu,e gl 1nz7f when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase he2 *n*oc,yrn s qfb2y*rr spin rotation rate from an initial rate of 1.0 rev ever2s fno* r,rynqycb2 **y 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 thro 4u0lbzg9vgh,ugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shape, hl0b ggv9u4zd 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 angulac n3crgi. :, qfxok5t*r momentum 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 momentum of thq6 yhb4q5clbe ow: lnsj7:5pmf;i6.me 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 )v*w heuj qx,vg,qz1t0 oua-)onto an identical)-he)u0 aq1*o zv wx,gj,t uqv disk 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 2.4 omqg(94kvf ykzl ,l3* $rev/s$ 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 merryd2aw2ms0-mt.++soknz r (cw +lm zmf--go-round platform of radius 3.0 m and moment of inertia af+ z2n+ mm.2rto (0msmswd -+zckl-w920 $kg \cdot m^2$ 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 merf t4:xv7ah 13+bt z)khq(8srlg 9 jxpnry-go-round is rotating freely with an angular velocity of 0.ahn17sbg+3lx9qj ) kztx tr (8v pf:4h8 $rad/s$ 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 akayjj/sek *- h4 dm,,about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the ne-m*a a4skj,, y/ehkdjarest 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 nhdoog fg 8n- wbndd*31qh5v .0fc0 p9winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass mi /jycq;q;c;teg*qdo *+d: u$ 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, tf7hp adl*7r 8fhat can rotate freely without friction. The moment of inerra7f*7d fp 8hltia 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-w6) k-hbis8d0d:xp htm diameter merry-go-round turntable that is mounted on frictionless bearings ax kdbip8h):t 60ds wh-nd 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 ground with the end of the rieph z 0r.x9y/ope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far dh xe yrzp0i9./oes the spool’s center of mass move?

  The Moon orbits the Earth such that th1epetk 7dln/ -j oq;e6e same side always faces the Earth. Determine the ratio of the Moon’s spiee qn t loek;17djp/-6n 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 acceleratej1kotlucolkv d., g ;10+ai7vpn5 -dis from 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 cylinder of radiutan d5( tqk*nzsbb, 5j*xgj1 /s 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at conb5 dn( j*5jqz1a,x*t ngst /kbstant 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 kgyiddls;jx,,+-*g z u q 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 lid*zu, ix-,dgl;jq+ synear 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 speed of the rearl z1( 0dm;cbun 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 f-r; qcr b2yn)our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were c;r fb2yq-) nrto 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:so6uqfu: 4c,l4a ga7(o l.qo dq2jl(eujy4 z is for it to use energy stored in a heavy rotating flywheel. Ssl ,g (lyuaj64o4:zq7lcu u (oa.e:j 4qfq do2uppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates and4zjw8 rb(m pr qy14)f $H_2O$ 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 wtb8de *gf ry*: u8(lxrc.w 2dspeed 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 f0gb2w: tu )tn7y7gr2cyu c2mand 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 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 Fi0 u n7):cfgt7myb2r w22ytugcg. 8–21g.]

A wheel of mass M has radius R. It is stau q (m;6lrozg.nding 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 homgzl. ;6(q uras height h, where h

  A bicyclist traveling withrbk7t-; n 6eooh (4dcm speed v=4.2m/s on a flat road is making a turn with a radius The forcod4k th;c6( -renob7 mes 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 o c,lc4ctbn .eg8nc::0pc8k dn f length 1.5 m and begins whirling the rock in a nearly horizontal circle abob dnc 4cepc. 88, :ntkcgl0:cnve his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder al ax*a)x,by:i nd her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her):bxx l aai,y* body.   

  You are designing a clutch assembly which consists of2cno, o .-wz-xdjojc,iu*x (q two cylindrical plates, of mass ouo c2* xon-j(ix qjdc,-,wz.$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roq z65 mjz*l qsh ,gq 0n.s(tj/thzhu83gs; /vfugh track of Fig. 8–58. What is the minimum value of the vertical zzt.g/*f0t(sj ns 8hqm 5l ju36z ; vsghq,qh/height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dol ob9o lc9*s5n 2g (ilgotd12h not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca195ikv 0woyjmr reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerati5o 9 i1vmkwj0yon 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