A bicycle odometer (which measures distance travsyo d60 xgt8zoh3m,p fxw94p)eled) is attached near the wheel hub and is designed for 27-inch wheels y8sofz90,xwg3h) od tmp4p6x. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pom;: x-vzi gh-e90 oaxvint on the rim have radial and/or tangential acceleration? If the disk’s angular velocxi;vmo : -z v9haxe0g-ity 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 swi7l w/,a( gk,ygwja(bfr- lz1 w- b;pingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larifip p 8*vmz:b7./mqq ger 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 yourswp+ b:01gm un hands behind your head than when your arms are stretched out in front of you? A diagram:gb0+musn1pw may help you to answer this.

A 21-speed bicycle has seven sprockets at the re -m* -v9 wacfpafd; ggiz,0p9qar wheel and three at the 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 lar-cpf0qd- wi ggmaf9;9a ,pv*zge front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flesojuri;q h:)5fl e:om-z*mh4 lju 7gy;h and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, mlru;hz;uf*h o:lyo :5m ei jg7)jq-4explain why this distribution of mass is advantageous.

Why do tightrope walkers (--wtjj3*m pq1in ar+ fFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zerta,6 vd.p se2:oxj *kso? If the net torque on a ovd2,s:sa .*ep t6x jksystem is zero, is the net force zero?

Two inclines have the samenqaas4m(i arx+2f e14 height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incli+4a4 (amrqeisfx na21ne will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from uaa-lo7j4 +m7n0u z t4+vy lgarest) down an incline. One sphere has twice the radius and twice the mass of the other. +zuvt4+7o jam a l u7g4lnya0-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 and9gdvo.1o0 btv the same mass. They start from ovg 0ovtb.9 1drest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentu j4x+k **wgi(e8wrs vwm are conserved. Yet most moving or rotating ob4v*(ji ew8r+wkx wg*sjects eventually slow down and stop. Explain.

If there were a great migration of people toward thehj 3 *:3ps8 0ad z1ncybst8pgm Earth’s equator, how would this affect the length of tsn8c 30m s:bh3t p*8d1pgyjza he day?

Can the diver of Fig. 8–29 do a somersault without having any inil 0v0jfqf6u6 rtial rotation when she leaves the boarff0qr v6j0ul6 d?

The moment of inertia of a rotating solid disk about av1a c0 tt2fqbtxb:afd 7q-ox:4q kl pu5/(d5g n axis through its center of mass isgxdx- :/c5td(b2tf:0lta a7kv fqq54poqbu1 $\frac{1}{2}WR^2$ (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 va,+wzy1 7j( ofsg5cg71 zn e-9dyyhain each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or sta15zn7d cgy1yw sgf yz9j, o (+-heav7ay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollo ,xq1c-t/8tgks 1mbmbw and the other is solid. Describe an experiment to determine whis 1b- 8/m,k1cttmb xqgch is which.

The angular velocity of a wheel rotating on a horifrj5nr. sbd-qu* o0wje7e( .uzontal 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 this point at the top of the wheel. Is the angular speed increasing or decrsqwu 5jue07(onrf-jdb . . *ereasing?

Suppose you are standing on the edge of a lajb1;g2s4 yxi c9kuj4g3mhya; rge freely rotating turntable. What happens ify4;mgj3i cgsj9hyaub4k 12 ;x you walk toward the center?

A shortstop may leap into t emc1dwi fo6)8ik5v )vhe air to catch a ball and throw it quickly. As he throws the ball, the uppek1d i)ofiv )5wm86cevr 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, discuss w1x -)h98w -oc d(th/jyv.,zrf eivr w wab7e.shy a helicopter must have more than one rotor (or propeller). Discuss one afw.9i dro vxw.j/s,e(c1b w 7y t-)zv8re-hhor more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ()ju e( ysb.:icgoud3in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Ch np-h4 yfr03d eyn*ytbwg( 04wuz)/galculate, using the infowb gyhgh z npt-y)n/r*dyf00ue 3w4(4rmation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M/2 lka 8h *q blj,dztft9jg9u.oon, 380,000 km from Earth. The beam diverges at an angle zdlqtjtb ,ja/f9 982u.l*khg$\theta$ (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 rpmi ;+fa;;bdb pzg 6.zmib6zaoyx21/uv . When the motor is turned off during operation, theaxzo;b;vz;pz6b +fy2 i./iub gm ad61 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 lev 0ecwlu*8a)zz/h p: g iu+h fkp:pn/,gel floor 3.5 m to another child. If the ball makes 15.0 rgi)0,hce pl puagz8/ hkpu*:n wz/ +f:evolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diametjhuwj0s66h1x er travels 8.0 km. How many revolutions do the wheels make? xs6 601 jjwhhu   $rev$

  (a) A grinding wheel 0.35 m in diameter roh wc m(ycg2l-8u4uu9 x u)yxmm1sb4z*tates at 2500 rpm. Calculate its angular velocity 4umu2zhc-g)uc mb94 xxyw8u (*sm ly1in $rad/s$ $\omega$ =    $rad/sec$
(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 rei+i7 zqvb:f9 3mdftt(volution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 17bi( t qf+zd: 3v9tfim.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Suon+l y)t ;fa/ln    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed 6 pxj-cdlv4 6qd ;9xs.elx1l9- wgy0cguuz2 zof a point
(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 th t1) (3h/o8q j igys)n1hddthpe axis of rotation is to expern hti qs)3)t 1d1/hydjhp8(ogience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13vs/ofl s;k1*pqye .; y0 rpm to 280 r* y; vs1poq.kes/yf;l pm 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 radius*9t hp,d7ke lvv/m9dr $R_1$ 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, ast(4.mpohjk1 9wqajs2 zronauts 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 d.19sqamw zojh ( p42kjuring 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 unif ixpyzf7y */:zc0gq+ oormly from rest to 15,000 rpm in 220 s. Through how many revoluo0y p:xz/c * ygz7iq+ftions did it turn in this time?    $rev$

  An automobile engine slowed et qnm70k7(u(4b m )oi4ifp9r7 gays down 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 stressesmct xg653oi9f of flying highspeed jets in a whirling “human centr3m56 9oif tgcxifuge,” 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 acce4 v1uvx . rpb1k 5oo.:qvpjj8qkpw1*elerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a x:qp58r bqjp.vv.kj1wk*1ve 4o pou1 point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8502u62q e /)3*0 gjpytrxqmzhhppl+ th-rev/min It turns 1500 revolutions before it comes to hq36 xqp2)prh*up-ge2 tzy/lmh t0+ ja 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 ps(+4zl:lm8ph q phu8revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tireq( uh4mp:lp8psh+8zl s 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 the car k 9 +(3z swzdr8o*ehs:u2cx ajreduces its speed uniformly from 95xedczokz8 3 2(9hwrajs+u: *skm/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 tt uxj(0i6o t8her weight on each pedal when climbing a hill. The pedals rota t0(uj8oitt6 xte 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 it t,vn5*w m9/sal7xq a door 74 cm wide. What is the magnitude of the torquwxq *7m5/9tiva ,n lste if the force 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 yu8d*cxp9j8g /lcjn e7(7 ehu axle of the wheel shown in Fig. 8–39. Assume that a fr7ecjdluc*gx(8 8 /7 h9e ynjpuiction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,z)o:s v( 0ur ;8-jubvd68 wpqf ltct+u are attached to the ends of a massless rod which pivots as show cr:6-0dl+uv8uq(o 8 jtu;bpvst)fz wn 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 system.

  The bolts on the cylinder head of an engine require tightening tkus8h,d bvjy+wn /3kt8iig )(o a torque of 38 jy)nt,iudk(kh8+/ sv i 8w3gb$m \cdot N$ 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 10.8-kg f:r *6bh7s5rodgc)1pt qvn f;/ks lb8 sphere of radius 0.648 m when the axis of rotation is through its centefkn;1b *7 ps6d t5oc:hlq/r) 8gfvrbsr.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyceb2/at hy30kt le wheel 66.7 cm in diameter. The rim and tire have a combined mash3ka0yte t2 b/s of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ballvg4ki/(yqw3vprap 1x 3y fr). on the end of a thin, light rod is rotated in a horizontal circle of raaf3.qv 3ivr4pkw 1() gpr/yxydius 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 boidqqg e65ahdb)mw8 eg +h 0m1ruq3+5vwl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her he vdbgm 0r i835h 6aq+u qqwe+5ghd1)mands 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 4qw+-ky5 gxkiobjects shown in Fig. 8–43 axq -yk5k4+ igwbout
(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 consi +;n-i ,ss jfzcs)evg6tpm 7o4sts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rat cvwq,jw721 + rlx.lkbe, engineers fire four tangential rockets as shown in Fig. 8–4cjl1wb.+ klqxv2 7,wr4. 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 uniform cyf6r9- u i8 hpqje5i0wclinder with a radius of 8.50 cm and a mass of 0.580 kg. C5 e- icpfu68q0jw9rihalculate
(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 f lbumn.j -zfdtj -rnq58;j(p4 rom 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 and is abl3ok d0 omt 84fnv:c;bfe to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictioo 8b3dm ;t:kcfofv40 nnal 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 br kob:e m4xe8 :t31lhtjought uniformly to rest by a frictional torque 8e:tmkj1oexl:34ht b 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 accele hp2s)ggvp+u ;rates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball00wtzeq7c09vxjnzzry:xq 2i +38 f bs is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest tox0z93: j zqnfeb0yq7+2xi w0tzs8cvr 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, as7dm7+z0lk 7 wt9tlz1k a +mxet shown in Fig. 8–46 zl+tkxk 0 t7l1w7+a97m zetdm.
(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 consist q:y- hhnohm+ 91lob/ktwg2d+s 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 the +c/q.f ky+eco hammer from rest within four full turns (revoluykf.q o/ +ecc+tions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment oft(.)o:1l1fp klbz zcq 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 torg /l j xy9*efgum+*w0rque 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 down9fr +ei:zssx4yo -8t d a lane at 3.3s s4e x-8zt f9+ioryd: $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eng)i.wlxkll, 35t1mu gergy of the Earth with respect to the Sun as the sum of two ter)ll ui,l5t1w xk.3gmgms,
(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 )bfh sxeq4f b(-i480un 2gbjy of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it fh (fxbbynfeu 8q )i-0g44 sb2jrom 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 c)nl:x9ufrok) w48 4ocw.k xxj m and mass 1.80 kg starts from rest and rolls without sj)4 cxr.owkfnk 4ou:w8lx) x9lipping down 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 balanced vertically on its tip. It starts to fall anx 9x8bzi3abiq81( kz.cvy,f wd its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of ewq(xiyzi1bzfb.,9a8 38 c xkvnergy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin stsgw(zhz*6 6f 7-b ngzpring in a circle of radius 1b fwz(np zgh7 -s66zg*.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8y8 ttzv.pf-wp/g 5z(ya(bl0d-kg uniform cylindrical grinding wheel of radius 18 cm w/ gtad f l5ppy(wz8 (-b0zt.yvhen 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 .);ogav q,qxikx1arv// egxj+xj9px7v3d 3 q side, on a platform that is rotating at a rate of 1.3rev/s If hei gv)pg9q av.qex+oxx3q1,jxdkjr;x va/ 7/3 raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her momen;h: c.s;segwew pz)x1t 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 posi;zws he;gx. cwpe:s1) tion, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate tebg qtm7c.gsa l (r9r /x)c:()uh* dkt qx42iof 1.0 rev every 2.0 s to a final rate or.mtu/bsd)xcrx a 9 2kl:gc4t( tqge7 hi) *q(f 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 aroundc,v hl1*q.n+7w iy6 dc(wccz n 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. What is tvd 7i 6.wnc,nq yc+whcc *1zl(he frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3. 8wlvmt 5u2qx65 $rev/s$ 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 mome jw.5m 0ec,orakjr bi;8l2h *entum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped oo2msj(dxkr p9 l-;3t+xb 7h wsjmee-rxz - ;p1nto an identical disk ro(p dwsel xrbjxk--37 m jhorx9 ;1e-2ztps+m;tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns1(.gq8-j*uctu m uzft at 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 rotatv.6fq :y1akodix 5:e hing merry-go-round platform of radius 3.0 m and moment of inertia 9o : xq6:yvh kdieaf.1520 $kg \cdot m^2$ 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-rog.fqf fmd v dpa06t8,zii2/tr37l:mm und is rotating freely with an angular velocity of 0. mgi8qmf7it f:. lm0zpda ,t 6vrd32/f8 $rad/s$ 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 about half itse /jql7wj(ad) 02j- rqnqy )bg1vysz 6 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 nearest iqzl0jevjy72 yb(q/ )jndsq r)g-6w1 anteger)$\approx$    $rad/s$ (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 120 2usm5k 5 8ffl.ljjma s3z(r)w $km/h$ 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 vmv pnhyg,73 p(-m fu6$ 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, initial 5(vc;:pyi4 uszq2f nlly at rest, that can rotate freely without fr5(fcvizn2p y ; u4qls: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 edvwn uz41 pw((yme7/h3 djq u6zge of a 6.5-m diameter merry-go-round turntablqpm vuh /1den4(3zu(y6zww j7e that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the ,mgjn;zr,vdyw e,84 q0s0a xqrope 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 tvz,eqq;0 48sx ,mjwa,g r0dynhe spool’s center of mass move?

  The Moon orbits the Earth such that the same side always :3tn nxy-l:80c nefcz eu;k r fh.3ba6faces the Earth. Determine thencnxh 6len0.a8r:u:zkt; fb eyf3 c3- 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from g5q qxrvu8j ;7rest 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 cyli a:;/obz;syei2qkah*nder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delive:szk aoq;e*iyb;ah2 / red by the motor.    $m \cdot N$

  (a) A yo-yo is made rva:6weapi*l3 4j .nz ,u5oykof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy t*y n.r,avoa i 3le5kw4up6zj:o 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 speed of the rear wheel c0y5 5fxh wl)n($\omega_R$) 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 Sun8v/)yy krz 8-so/rfnj , but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, lr)v r8j/nky/ -oz8fsyosing 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 automob-b.b mp/p1*zgfoac 1oe g8.av ile 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 o8o1. pca z*evb.fb-gpga 1/mbe 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 illustratenjjq /v, uua*-s 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 w6) (eyvg. ;rf0jzslron 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 and length L can pivot freelywlo;9l4/i w *hmt2p v-bsdi3p (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 ofd9*wp 2 blv3lt-4h si/poimw ; 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 therran* 7 ;xph9koi0-a o floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, wh ; ak-oh or*nr7ix9p0aere h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tu lgdgu/1ce+v) rn with a radius g1d)ugl+cev / 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. rzwtv-ijq- *:jtr )u1u7(jga;ut *hj50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleru *rvtg*rq u : h-j;t(-i)wtjjaju17zating 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 cylin+*(wdv nenq i/w7kowqm+22nn der and her arms as thin rods, making reas2i+/2qwwnnw+q n o*en (vk7 mdonable 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 body.   

  You are designing a clutch assembly which con00c bw5atmn y2+wb 3owsists of two cylindrical plates, ofn3 05acw0 tm2wbw yb+o mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rollsv . vzzyzi34+y along the looped rough track of Fig. 8–58. What is the minimum val+yi43 z .yzvvzue 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 d9- z(q9ug.2bq ye4 qpecr g7gho not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as t sa-s 7b++mf25rlx aqfhe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the an-fs+ m75+lbrf sx2qaa gular 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