A bicycle odometer (which measures dista78ci,9)*hrqnq.tz kr tfp3 ae nce traveled) is attached near the wheel hub and np 7q,fr czt*hiateqk.r3 )9 8is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doessv0mlg gi85cyq8 8o5r a point on the rim have i5g8 8mq 0c5g8y volrsradial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valuu-d2g7j hxo3n e of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater6ok: wo hd2kc) 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-u0k/7u9u tycj2lr 5 okqp with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to 2 k7u/j5 ocr lkqu0y9tanswer this.

A 21-speed bicycle has seven sprockets at the rear whe xs)usq/z 3r rjbtxf/jk3w2 -,el and three at the pedal cranks. In which gear is it harder to pedal, a small reaf3qrwx-s3ts)j x, /jk2 zbur/ r 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 legs with fl5 diap bbt1485hysmtjsf 0m.2 esh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this disbyh5bm. ttd 1p8si520mjs4 fatribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l( kcg xhljjeo 8pen l( o.04:+dg9p9ktong, narrow beam?

If the net force on a system is zero, iuq :r. c /pmf4 ;d7o;bpo3motjs the net torque also zero? If the net torque on a ;dp u:j po r. f7ocm;b3t4qom/system is zero, is the net force zero?

Two inclines have the same heigh f3+b 4y6 j,jhpe0s)t g6sjnpglkw7n-t but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thng whk6sy j3,7 lbp0+j e tgp-4j6)nsfe bottom be greater? Explain.

Two solid spheres simultanenlot :y o,gx gi2c3:k( 7dyx4uously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mayyxix 74dtc3,g2og(k::l nu oss of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same mdo( k7l/ej8yva gr 5,z+c:lx radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater + z(ajvl/ oyel5mc :kr,dx7g8total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angultv kij7uc9eq* z o*.;re *+k:zoiu1 ewar momentum are conserved. Yet most moving or rotating obi;o*rvzo9+ wuice q uj:e 1k*7t*zk.ejects eventually slow down and stop. Explain.

If there were a great migration o aegbixffab5 tt/ s9a.(tq)b*z*r6 v-f people toward the Earth’s equator, how would this affect the lenqt/za a)f*xi9b rg65-fts.bat v b(*e gth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial 4p y3./ k yx:vnuz 7pr6msx76acr dl7mrotation w4ckrlns yu6px7y /:x677 armvm.z3 d phen she leaves the board?

The moment of inertia of a rotatin (5,0y8 vmjnf1r yon.ew 8sbbtg solid disk about an axis through its center of mass ownm0b81 n. ysetfby8,v r5 (jis $\frac{1}{2}WR^2$ (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 in each outstrjb) dl0bn nf7v59 .njketched hand. If you suddenly drop the masses, will your angular velocity i b.f7) nd5kvnbl90 njjncrease, decrease, or stay the same? Explain.

Two spheres look identical and have t7;;ez g0-o bnqje dr.sdh;xk:he same mass. However, one is hollow and the other is solid. Describe an experiment to deten-b qez;g;reoj. hx;k d :sd07rmine which is which.

The angular velocity of a wheel rotating on a horizon5.3q v k8*k,lfs hfojbtal axle points west. In what direction is the linear velocfj, hsolk5v*f q3k 8.bity of a point on the top of the wheel? If the angular acceleration points 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 standing on the edge of a large freely rotating turntable. xt mn 2c)d8rj37, rl mrvi8zb/What happens if you walk toward the c tr3) nm,b/vlzcixdm28j78rrenter?

A shortstop may leap into the air to catch a ball and throw it quickly. As hs(tdr , pugvmb.:iy w2v55j9 we throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite dir,bm2wvu59pdigy5 wjvt(s: .r ection (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a htjcyl1q v a23,u73 achelicopter must havaj 213h 73q,luyvtccae more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followin ig75p .ggjqx,g angles in 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 becauw* xw9lqwyu4 )se of an amazing coincidence. Calculate, using the information inside the Fro wwwy49qxul)* nt Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergeuer d0pqo,6q6 ukzt2,7esn0 es atp,rzsdtu60e eqqo7k,e6 n02u an 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 65c( *qa, ve..fqwxsh ip*xnk-xr (;f( p00 rpm. When the motor is turned off during opns rfhxv (q.w( a*ie(qxc-.f*x,k ;pperation, 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. vn+k ,c2+ mh2(wcutpd If the ball makes 15.0 revolutions, what is its p huv t+n,ckcm(d2 +w2diameter?    m

  A bicycle with tires 68 cm in g) lf0)q2lih- d.-ce /bir ymgxpk8l*diameter travels 8.0 km. How many revolutions do the whee x.g8-fyrg*h qk2b p) il-/0le ldic)mls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter ro6n h/)uw1c;)p uabm gdtates at 2500 rpm. Calculate its angular velocity inp1uabhd6w n mu ;c/)g) $rad/s$ $\omega$ =    $rad/sec$
(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-roundy b-;lxp ts2.e makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seatx pel-by.2s;ted 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 m :xk44wk ova,Earth (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 p1lya n5 e 9f9vc., nl;f-6 x4vjlamvyv1uncx9 oint
(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 unm pjw9 zgj((8;mp.0cn jgw-centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration ofn8 (n ;-mwpwj juz gpmgc.09j( 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce xyl77lr e(:ta7djghc *uiy-+lerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine77xcg:7yhrde* ll j -+ayi(ut
(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 radiu0 i7mt1xxfk5tu alv8/4c0mawicj 57as $R_1$ 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, aave+k,d7muk ,c45e2e avs6j7;e (p am 7spjkestronauts aboard 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. Deter+pp kdvc s5eev,e2 ,7 e;jej467am7smka au (kmine
(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 unifo)u9zj- :clsw grmly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn g )cwj-su:l z9in this time?    $rev$

  An automobile engine slows down from 4500 rpmz)zn;ascx6* crfn xy8fq)6t6 /o7 tab 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 us780 s (y: eeappt:yda whirling “human centrifuge,” which takes 1.0 min to turn through 7:sps0aey:td8 ( ype u20 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 acceleratesa;jcmt6 f) ow) uniformly from 240 rpm to 360 rpm in 6.5 s. How far wfacm ;)w6tj)oill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running q*x:pq :v8z92usnfb pat 850rev/min It turns 1500 revolu 2 n8p:u:z9qvxpfq* sbtions before it 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 cas0zdlx0/r 6p5g ah+ adr reduces its speed uniformly from 95km/h to 45km/h The tires havea0sdga 6+/xh05pzl d r 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 themli 6h*o(pgs, car reduces its speed uniformly from 95km/h to 45km/h The tires havsh ol6*m(pgi,e a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbys juz6y+gdb, 0ww427ji-4hbnzn:auing a hill. :wu4d+2n47,w6 -a0 bzjhibyygjs un zThe 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 wtw skue+yg5-h0vw0 4side. What is the magnitude of the torque if the forceh-gsy u0v+ew 0t4wk5s 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 a/r9*kqvc( uswizm+y9o0: cxwi2x0 tuytm8y 5 bout the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4 0/ux5qr 0yiwk8:wcvt2t 9c s9iy(uxmyz+ o*m $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends7p9;b btwe 0qkfdj7 k1 of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position a0jtw 7k f9epbd17 qbk;nd then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cyl*vdy pzkg ey,(2 a :s.llns5 w.9w9auqinder head of an engine require tightening to a torque of 38d 9 y:ww*9uzl2q(a k.5n.yse l pgav,s $m \cdot N$ 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 inerti)* wzsylmvgvr3,8b y3a of a 10.8-kg sphere of radius 0.648 m when the axisy3yv3,bml z 8wrsvg *) of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelbt90 fh7 nd/v;a cvnc0 66.7 cm in diameter. The rim and tire have a combine;7 nvah9 fncct0d/0bv d 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,ds 7+fzi9)b tz light rod is rotated in a horizontal circle of radiu+is9)zzt d7bf s 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 bo11 osuuzez:, e69 mhl0kv -x3y tfv9tdwl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwxsk1,m1ezhul9 t :0 t6vv-odyz ufe9 3een 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 ariul7r)78 xelac/sg iam 2r y8h syq:2:ray of point objects shown in Fig. 8–43 aboutgh/ 2muc eyr8:sl2 ) i7l:qi8rys 7xaa
(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 w *y0em(ta0tv ohose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinnin.s qxu1/nvjw5ats:p c, kqs t60qq 5x*g at the correct rate, engine,*tqst /xswn pq.1 5kq0xvua5 scjq :6ers 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 reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a ;n cu/i9n 8e3ukly z5uuniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cal/uu89 5n; cln3u zykieculate
(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, accelerating it from redd m6 (j;py.4qctovw+y86c gi st 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-rotqkw8tx /q6tboq*qf5+ 9v lt*und and is able to accelerate it from rest to a frequency of 15 rpm i5t8q vwq/ kqf txlb+ 6ot*q9t*n 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 brought;4d uqfzmbn;zm b 0x4*vosd9,q0 i jp 9qc-0xx uniformly to rest qx0bc 0zsmfx;-0q bd94, ujiov;4zx *n9pqm d 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 3.b+l-c),j;v,ikz gxrj 6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of tjvao sh zo+cv0q/8rr;+1bg7whe forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from7o +v 8ch1qbzsog0r/arw;v +j rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in F)ek -seko4+pg tu.wc 2ig. 8–44-cwosep+guk 2t ek).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 consian lcu7a808gnxfnn31m av c6/sts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates t ocs5gqyt /)+pmb;s2x he hammer from rest within four full turns (revolutions) and releb2s5;p co+y) gm/qt xsases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment 3(/y*xy a rga(t p1/4)lus(vo envkjcof 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 develo5)i6sto h6g,zxmtgyudfe fy o 9: :9a3ps 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 rolls without slipping down a lajcrsfho9nw 6b dw96o5 0n2( nbne at 3.3 w b9(29ownfrochd 0nb6snj65 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earthgl)/)5clw/ke dd,xyu;5 el; lmpc kn ( with respect to the Sun as the sum of two termspln)uc;k /ce5 /we) l xkdg5;ly,dlm(,
(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 7.5 ;1h):y xh6vugm dmrjh b0a+yyjv)b5+0 m. How much net work ishr;5: 6hd)0j1 v+b uvjmy ba yx)myhg+ required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without f1q(l;8:+oet*ulvu s lr sj7w slipping down a 30.;*f7s: re1tjvll(uoq +8slw u0 $^{\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 balanced vertically on its tip. It starts to fall and itqenxy c*h*/n ea* wkp,1 ua57whj8 6ivs lowu1j*/wv85wa* 7hxp6yanhi *ce,eqnk er end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momet95 lv -,q:dd1agzpd9 wvi q.mntum of a 0.210-kg ball rotating on the end of a thin string in a circle of9.m a tdiq z:dp15lv-q,gdv9w 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 : c6kve; wfis) dkt3o:uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? sv)w:3f;te: i6kc d ok    $kg \cdot m^2$
(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 thja-x cwbs06y j1yk)yt* p1 ;anat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fi 10 y*takxpw61y-sn b)jja;ycg. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce herr+i ddpr- a3*e;acey 7 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 positi-icd+p;redy3*ae ar7on, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increasp9mk15)pfk: j ; bnrkqi:am *fe her spin rotation rate from an initial rate of 1.0 rev ep;9:b :5pmqfmrkknj *i fk1 a)very 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 ro;oynfn4e 45so 6ijy)ltating around a vertical axis through 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 shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. W5n6yono jilfe)44s y ;hat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skfxp1 t+ 2ch32nn.k rqkater 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 momentuhmsw 9uz (,vn(m 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 mku:* dfgs.oq) lg68aof moment of inertia I is dropped onto an identical disk rotatinglmof8 a6u .g dsk*qg:) at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns avq m6mjkiwv/ul +-gwr5l8xmmh. + 22t t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotatingvc3:riqhnl ti /17h+x merry-go-round platform of radius 3.0 m ahnx rc1ii/7 +tvq:3l hnd 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 azu5ll s(i:z lp ;u8qe,n angular velocity of 0 e izl z:,uu5p8ql;s(l.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 wh8 p 31 oovfkbnyof33n,ite dwarf, losing 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 8f ,kooy3fon31n pbv3rate 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 12mn6nn5om co., vqg6 n,w34 sge0 $km/h$ 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 h0c;k2fvu 9fa8830lw oczdxu$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotatyuxmzuu o-r9z2+ :yza2q5 3gbe freely without friauz5b 3xugmz2o +ruyq2: y-z9ction. 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-g.8l9j lq4fbffo-round turntable that is mounted on frictionless bear9ffq .llbf8j4 ings 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 ground with the ens*)z di2,tirzl mmc;+mop0-h3 f(p ))qpsq jl d 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 slipping. What length of rope unwinds from the spool? How far does the spool2mo *p 0mihsrp-( ;l)m+,lzq3 zq )fjit)scpd’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detdpk) 0 r,scs(l:vy/re5z gq 4qermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case): 5ypszde4g(k s 0lr,/ qvqcr, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from reset7vkna v6 afdw:5.4rt 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 shapzbh 0g5w4.r5lq3nv d ae of a uniform cylinder of radius 0.20 m acquires a rotatio5w5 vzhd.4 g3nq0lrba nal rate 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*c4s fo1kne 6tar4*db disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentrdo14r snb *fte4kca *6ic) 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 the angular spo* k9v kch)u 5:+p (tsgmi9zjeeed 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, were rota wbyw;upbwqo/l +42l8tz i2j2ting at a speed of 1.0 revolution every 12 days. If ity4lil8u2w/ ;q2wb 2 bp tjz+ow 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-pollutm .hri65p() hmg 7vwp2 u2tdadion automobile is for it to 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 travel 350 km without needing a d.matdp2rhg 7pmiv(25)h 6uw 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 (eam*8bud ) wpan $H_2O$ 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 a(xxd /fhb vtgdz59* 8ut speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fricray ;p hp1gj4d3yv6 /rtion) 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 F;63 1vjag pdp4hyr/ryig. 8–21g.]

A wheel of mass M has radius R. It is s63 gm(3/ hb*gmm n,e fkza,arwtanding vertically on the floor, and we want to exert a r, hefgz 6/3 *w(knm3b amam,ghorizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roavmnvv s wxei)( ;1:x3ud is making a turn with a radiu3i(v1;sx)e:xuvwm n vs The forces 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 in1t2s1 mh9 qwk+tki46vd1 grrk to a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle abq 1tw4hkkts91d ir +kv1rm2g6ove 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 s0o(5mft3 elupt u*+o tolid cylinder and her arms as thin rods, making reasonable estimates for the dimens oto(tmu0fe3p+lu 5*tions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricalic r-*6gf/lv-or- ujx plates, of massc/*j i foul-v6r-r- gx $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. 8nu1zd3 e5o qpsp *;gn;oml)/ 1i qebi2–58. What is the minimum i* nne3mo1p1dl)bpug 5;; /s2qioqzevalue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu3u n-/6jvgto zme $r\ll R$ h =    (R-r)

  The tires of a car m+zarib1e ,tvj ;2y,f xake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m.,vr;jitb21afy+ x ze, (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