A bicycle odometer (which measures distance travelehjq xo1 /thd+d my4y8xb17i t0d) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inchj7m8dy+h 1o10xit/qt4 x ybhd wheels?

Suppose a disk rotatesb lzdle39bj/9.e5jd 8/w uvf+r3w rdr at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? Forrwbdwd/zrr8 d/u93e5ej +3 l.vj9lbf which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descrltg d;+3 gf7rao 4ri8dibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever-7kwl28j0t2;wib ssaj;k etn exert a greater 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 behin a7 zvm6lr, (wy-i kmzywlgymuf(9)32 d your head than when your arms are stretched out in front of you? A diagram may help you to answeryl 2za37 ((, wiwgruf) zmmyy 9mvlk-6 this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalw89stsazaf6(a)j, gss4 gi , p 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 large front sprocket? W f8aa,)s p6sjw (4i,gg9tsazshy?

Mammals that depend on being able .v3,gep7y4-vbktp w h to run fast have slender lower legs with flesh and mutv4 pp37g.wevhbk ,y-scle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, 8xvq82gf lndosbu)-.xa(ij .narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net toky (ukc 6- qy-sndw4lm0k(dm 2rque on a systelk- 4w0cdm2( kuy6sqymk (dn- m is zero, is the net force zero?

Two inclines have the same height but make diff6tg4a y4tt 9hzerent angles with the horizontal. The same steel ball is rolled down each incline. On which 4h6zg49t tya tincline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dowbma/ 5gqz 3xr ig:g3u.n an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greate3g :muq/r3bi xg.5azg r total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. ew953tom bjpo vc5f*) 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 total kinetic energy at the bottom? Which has the greater rotatioo*9)wt53 bc5j p oemfvnal KE?

We claim that momentum and angular momentum are conserved. Yet most moving or ro 8k,v:*:aigcqrcwexo q9 cx., t oric8g,xax. vc w*:, qeqk:c9ating objects eventually slow down and stop. Explain.

If there were a great migrat bb.cuh9jvf(a(ysdpo8. (t90kvy ua jt, c:/iion of people toward the Earth’s equator, how would this fbvv( 8bu d(,ctj sat/yo j9phk(u90 cia..:yaffect the length of the day?

Can the diver of Fig. 8–29 do a somallk s8bu868s5l d7zbersault without having any initial rotation when she leaves t87al 8k6ld z lsbsu5b8he board?

The moment of inertia of a rotat4lurgj0g;be0 ing solid disk about an axis through its center of mass ie0;l40 r jggubs $\frac{1}{2}WR^2$ (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 sittintnao z4fa v8w,)4bdm7g on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity aobv8d nfam7z 44)t, wincrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,: 3gi aq:+v g8rjms 3gq:0xovd one is hollow and the otheox aj im3d:s0vqgq+gr:3 : 8vgr is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotati;xdr :,b4nhs1i8gf vyng 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 this point at the top of the wheel. Is the angular speed increasing or decre:dbvf1;g y xhris ,4n8asing?

Suppose you are standing on the edge of a large freelx rf a :i)1octkucv9+e ;kk+ b-de6mu:y rotating turntable. What happens if you walk toward the o6 c:c9:vku1e it ef- b++)kdk mur;xacenter?

A shortstop may leap into the air to catch a ball and throw it qql;s.)uabye cb gs,*1si;lr pyy5u2- uickly. As he throws the ball, the sy.cge)bqr*alb; pl,5uy2 ss y-1iu; 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, discuss why a hely( (gunt25t adicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the h5tgda(yu2n t (elicopter stable.

  Express the following angleq;z n:nv.9xawqt 78so s 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 because of an amazing coincidence. Calculate, usihqa h g (29w*l,dl-jpk g9.citng the information inside the Front Cover, thak.j *q 2hi(g9chdpg -lltw9,e 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, 3xfgyqpn : t-z:f3148kmslx a/80,000 km from Earth. The beam diverges at an anglfm4 p-3/8xznx y s1qlfg:tka :e $\theta$ (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. W v;x00i)aa0imchtv z*hen the motor is turned off during operation, the blades 0*itz;) x00vai a hcmvslow 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. If thqmc ddk:aso(0d gm2 );e ball makes 15.0 revolutions kga( ddq:s0odm2c;m), what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How maafr9 c 6v7izq.;+4 ayg9laqkkny revolutions do the w9kza4 gr +.c l9q7;akaqviyf6 heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpmhyw no(nhg . oo))1j5m. Calculate its angular v(hon og1 5jowy)h.n )melocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fiqszmf hs6n5 )p n u2qsx:5mr-(g. 8–38). (a) What is the linear speed of uq2-nh6 s q )(xmssmz55:npfr a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbi / 8k 9wkqeu54tzj6k+vzoqvw6t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s *ug0g/(lx26mfzq o fol +f+axpeed of 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 part0ugh xsa9aoj- z+r0r+icle 7.0 cm from the axis of rotation is to experience aas- 9 hjrg0x+0+roazun acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to i ruq4,7 2-m2+vljbhikmoje+ 280 rpm in h,4+- q k2+uie72lvimjmojrb4.0 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 radiuf(k*f:flpu(4cvq aj *s $R_1$ 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 Apollo spacecraft o tlwb kqig(9t7nkv41 j 43uiatg*,w/put themselves into a slow rotation to distributeu/wvn1t,bwktg4kg q la*o4 ji 73i(t9 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. 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 15y.id43a qpwy u6qz/n*ie+ hq5mng)z*,000 rpm in 220 s. Through how many revolutions did it turn in this tez.ihzyp*d*gn +quany im w4)35qq6/ime?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2gh8bdly0 :w b /(fcnq 2kld3v8.5 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 wh9pr8qy(r gca8 aq zk98irling “human cey8a9rkqza8 pr(89gq cntrifuge,” 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 from 240 rpm to 360 rpm in 6.5 ,8akpc64 sb9dz,qin t s. How far will a point onp98abi q, z6c kt,4dsn the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off whe ir.vi10yjd wjk(d );hn it is running at 850rev/min It turns 1500 revolutions before itrdiy) ih.0wd j;v j1k( 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 +9mah:,d vzx pzh8dv8 b)4extcar reduces its speed uniformly from 95km/e4pav)v m:8zx z,h8 dh+dx9tbh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly c,rvw0/uq ;tcrk6o:zfrom 95kzqwr0kr u/:,v;ot6c c m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts al-lxyi( 5jf+4exu,3q :g jbnq il her weight on each pedal when climbing a hill. The pedals rotate in : + qj -eq,y3iilxgbuf45nj(xa 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 uv/aits5, 8a8cnb rpy6l3q ,jcm wide. What is the magnitude of thl 8ntic/qu3r, b86jv ,sa 5paye torque 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 axle of the wheel shown in Fig. 8–39. A icutjgy/v.avog,*8ty. 4k :ossume that .vj otgg:*k ,8u vc 4/iyatoy.a friction torque 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 endsbj6qqbkt+ o/ju(4c 4obm 7j8v0kx8dh of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torquedu/b qq h(v 47m8+jbj kj4ck8oxb6to0 on this system.

  The bolts on the cylinder head w.caw+ vxkm3d l k2j3n 18sz-eg/yv+t of an engine require tightening to a torque of 38 gv18clw2w m3j t -3e.a vkx/nks+z+yd$m \cdot N$ 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 sphere of radbr3xcp k 6u(h9ius 0.648 m when the axis of rotation is through ih3(kbc6 x9p ruts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter.h. b60fl0fpf v The rim a.0 l6bfhp0ffvnd tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball g8 aa-i jbu91 oquqk90on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Caa09qquo ak1gi-9 ub j8lculate
(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 pbk.4:yc3 i*qn f4n86ng gwosp otter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force be3nbo:y niwfnkc s4g q*864pg.tween 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 shown6o( b7 rusvk/: xcjd a-,hq3an in Fig. 8–43 abouv( , 7x o-bd/rncaqk3 :jh6usat
(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 total g uvo6 ..cq;kh-ffk8)z qzz q0mass 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 satell(lo: yzgum9h9ite 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 u y:9gz(lh9mosteady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm anw0 5gzqp1napt e )s,z;d a mass of 0.580;t gzw0z pn,sp5)a1eq kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest thr*wbknpq51 (th2x; nz4 bplky c8,v9o 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 hann;mb6nmg(,wm)c-k3c nm c2(fnm 9ja wd-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 f 62nnc)cga3m n(,w-b n( wkm9mmcj; mof 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 i0o :bmu7p;oyqou 2vk p4 q(te6s shut off and is eventually brought ub 0e( oo2oq7 m:u uv4;6ptkqpyniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-,/jm-ey:*g j78abd uvu 3vsoy 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 .c;zlsp5io;n 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 res ;spc. l;oz5nit 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, yz jw9p0 p3rgd+* 9ogcas shown in Fig. 8–46.c*99jwdpy r gpg0 o+z3
(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 l snqn;kull+/azn(of *. 1+jrpvr. 1wconsists 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 hammer from rest within four full turns (r4m+9 ak qob,ojevolutions) and release b4,akjo9qm o+s 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 of inx4i .hj5c5ggg.n: klb ertia 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 touz9 ks ;ynno:(d-5v mcrque 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/ci 6 rrf+gjl/ :c8 +yt:ufe(vsh0mge radius 9.0 cm rolls without slipping down a lane a+rc0 hvirg/:8:e(ff s6c gulme /+tjy t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as tfsfpl)sb9 v mmvu ;0;j:o ,eotu.4/awhe sum of two tee:fsof ums b;m l,4.vjw 9)/u0oav p;trms,
(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 radiwdf;+r pg/dw 2us of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is/ frp+;w ddgw2 a solid cylinder.    J

  A sphere of radius 20.0 cm and maoy/ ),yi 5ena9tut.sse+x * alnx r1-jss 1.80 kg starts from rest and rolls without slipping down a 30.0ryl se- n,9n 5x t/a )y1xs*.jiuo+ate $^{\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 tagdr8z8,l52u q 3vh,ytax4z io fall and its lower end does not slip. What will be the speed of the uppdyi88,,zzuvatr4 aq 5 3ghlx2er end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on thado3aulor/ ;ndq3k31 e end of a thin string in a circle of radius 1.10 m at an angular speed of 10ua1d 3dlnoq33;r /kao .4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a (kjptzr 0iy-.ww,dv 2 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rprkyv.-i wjdpw (z0t2 ,m?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, onimm85nm*gl )pz8sp0b06s hn kos )/xn a platform that is rotating at a rate of 1.3rev/s 88*mss0)z p 5 homs6lnki)0g np /xnmbIf he 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 1 iab sq6bsfi a4m5t6a4d l+e,d-4uwv 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 l6da,twv5aud4is44b6+e1b fa -miqsin 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 9x8hz g jy-q/oincrease her spin rotation rate from an initial rate of 1.0 rev every 29 g-jyz8 ohx/q.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 througyitcbq 4jz- u)zm:w .8h 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 the frequency of the wheel after the cyit 8m qj.zcz-) :uw4blay sticks to it?    $rev/s$

  (a) What is the angular mo5ws,xp5.kag 8/m zr kamentum 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 o19kruki o9n 2y3s(sc uf 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 Ir 3tptfndw+1 pp,bl.m vx r7+-da(4jx is dropped onto an identical disk rotating at ang -f l3 nadmwvx 1tdp+ 7r.4,txj+pp(brular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.osy gq7 c,u*b24 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-gsyjo1 qw;u5y l wi zv7r6+ qy60*ut6dvo-round platform of radius 3.0 m70yt ;uws qw+ 6udyjvo lq*6ryi1 vz56 and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocf ,5lk;rvjgp (ity of 0.8j lp(5fgvrk; , $rad/s$ 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, l4-h) ho sc 3fsjegls7*osing about half its mass in the process, and winding up with a radius 1.0% of its existing radf3hg) -sc7e4o* ljh ssius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winkl.3i uvs)lv) ds 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 (5xay osp jgay i)h*g;zql*q1+(-thh$ 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 p79pfk +pcehdpiw5 oy +jd 0djw 7m6y(2latform, initially at rest, that can rotate freely without+0hp5 7djpd w6o ( +2fdw7eyki m9jcyp friction. 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 t9xqt hv 5bk*/5cf /w8ybu jl0ehe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionle/ vlh q te*wj5 by9bfx5/c80ukss 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 rope lying x:f3f*qwv 6 3rfv7joqon 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 spor3 xf *3q: wjo7vvff6qol rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thycv 0:tci79it e same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to itsi yic 7tctv90: orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rl(rq(*tut j4llq ib,i ,u( u+iate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 1 .. spvjci()sdm2 8qxhbkoy;w -dan60.20 m acquires a rotational rate of from rest over a 6.q 2w;(hjd sk.-p)b6 inayd1x8m cs.ov0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg,h 54k t/vy0)/:onnordeqx c h 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 qh5/v40d )eo, k:rcyn/hnto xlinear 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 7fj*vsi/o vzx3/s3ke5( ps 4 zjybd-rangular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mz 7ou: b-z ouy)s9f*iqass 8.0 times as great, were rotating a i 9ofu)b*q uzyoz7:-st a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a l-dx*nvp g6)zheewa a8e(t()fow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uwevent-hfz ag8x a6 p(e*)(d) ses 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 u/ppe0e4pzrvr .(es1;7eh d t an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface arom *o+y gt j/.cw3c75y n0 zh2j*dilmt 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 7rwmg+tjz q3.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 showqgr.w7j +z3t mn. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius r5q :j/,y5uoo 7(elc 8jamjqc R. It is standing vertically on the floor, and we want t:5( jccl8,qom qou /aejy 57jro 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, where h

  A bicyclist traveling with s-is 4 jt 9p4pmq*d -f mia+;yttk1u0uipeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyc4-sqj+t0 i u;ty*ap4fi- md im9tp 1uklist 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*tc9sb gzz0ch 5mu5+d-kg rock into a sling of length 1.5 m and begins whirling c05 tc*zug 95b+sm hdzthe rock in a nearly horizontal circle above 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 1dp7 y v/w5zs)2nq;t) jknyo iand 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 ar7k d25j;siqt1y o)) n /wyvpnzms held tightly against her body.   

  You are designing a clutch assyx5 q(,jbfsafaq75u) embly which consists of two cylindrical plates, of mass (5 bq ua)yq5sfj 7fa,x$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 *oc p4jwped: 7qftj7 9rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Adt 7 opj7qwfp 9jc*4:essume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ashbq dh 8ltu) q)q s-x89g,wp;bw.wmh5sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions (8dbk/ pwm;p( tf,x orapo8z3as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter p( fxb8odp w/z 3p; rtk,a8mo(of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s