A bicycle odometer (which measures distance traveled) is attached near the wheeyt(e8o y.y vj6l hub and is designed for j.yv 6eto(8yy 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the9 r6g5q yx1f5a) hva4vc-rlyh rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the mag 65rchx5-q)ya 4lvr9fa vh1ygnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sin h c62najs05hwgle value of the angular velocity $\omega$ Explain.

Can a small force ever exer 8/kyfcl: 5bworp+6 iot 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 sid-14lr 6qt, gydi-gki t-up with your hands behind your head than when your arms are stretched out in front of you? A ,gk-rtdg l 6 q-y1d4iidiagram may help you to answer this.

A 21-speed bicycle has seven sprockeu 9 eucn/6zs:lhmb ( rq4inq r a;+w)nc15ck;ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal + 4r1s65krw;nm(uiehcq9nz qncub)c;a/:e l, 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? Why?

Mammals that depend on being able to run fast have slender lower legni+3qurq z bld* :(1zps with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain whyznl zqrud*ib3: p(1+q this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n:0 lj 5 ,wkq0zxdi*ndrrk+vm,arrow beam?

If the net force on a system is zero, f)xym u9;ie1k3 )yak qis the net torque also zero? If the net torque on a syske;m3 )kx91iq f)yyua tem is zero, is the net force zero?

Two inclines have the sam9dfjfkj 7/ tkz2 pih58e height but make different angles with the horizontal.ip 7/k 5hj8dt9jkfzf2 The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from reykdfo im8. 1s2st) down an incline. One sphere has twice the rad8.kdsiofmy 12ius and twice the mass 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 radius and the same mass. g5;(mr +i ryj/odnu.fd3 zn 9nThey start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at theg9+ .y zd i5o(fdn/;r n3mnjur bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angulagncf73av5wirul : )z,r momentum are conserved. Yet most moving or rotati7a:z) cvri,3n ulw fg5ng objects eventually slow down and stop. Explain.

If there were a great migration oto2;fn nus)*l f people toward the Earth’s equator, how would this aff2*fonu;n l)tsect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any ini1 .csrbf kf.e:tial rotation when :r1skc . ff.ebshe leaves the board?

The moment of inertia of a rot en;nclz b6 28wbni7;eating solid disk about an axis through its center of mass iec 2zw87nlib n6 ;b;ens $\frac{1}{2}WR^2$ (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: g. r0exiogm/ on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocrx i0e go:.gm/ity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and thcl)s 1ga*n e,bcor 4icy606e,lew/ aferc61lb)fe 4a wance60i y,lgc,e *o/s other is solid. Describe an experiment to determine which is which.

The angular velocity op6zi8u /p*qq6g 8rx kdf a wheel rotating on a horizontal axle points west. In what direction is the linear velocity odpgzk8 8pui* x66/qrqf 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. Whp x w98tb s:;c,g.byi2rr jch9 fke.1oat happek2o b9x9prhfjbw ;gysire 1:.ct,c8 .ns if you walk toward the center?

A shortstop may leap into the air m we7p.fpkh40fn ua98to catch a ball and throw it quickly. As he throws the ball, the upper part of his b f h mk0pp8n7uw9ea4.fody 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 conservatp.. axs5tkp341 p dkbfion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more wayf 4s5 pp1 kak.tbxpd.3s the second propeller can operate to keep the helicopter stable.

  Express the following angles fn vkc-7.r;il 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 coi3*yofpq- h qwp, t,l1vncidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, asvoqtpp3yh lq,*1- ,wf seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges g) ha8q+ae /tiat an angle / +qga)ae 8ith$\theta$ (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 rpgi 5k6)bt5 zme rw-.ohm. When the motor is turned off during operation, the blades smwb 6kr)5.e-5 ozti ghlow 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 ano p7up6l (oe,kqther child. If the ball makes 15.kp,6 plu (qe7o0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameh(0dk1 xzbrw 0ter travels 8.0 km. How many revolutions do the wheels make?bx z1hkdw 0(0r    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. mf) 0c 5lmx;igt/dfx8Calculate its angular velocity in xdm)gl0 5m8;fit/fx c $rad/s$ $\omega$ =    $rad/sec$
(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 revolution2 )uq q6 geiz;b0,gpub in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child ge0 z,bupq)ug; bi2q6seated 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 ot6og,j5uz ( w k0ceuj)f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spene /6j(bofw x6ed 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 cent*mh84m1 yty;sps*zzk rifuge rotate if a particle 7.0 cm from the axis of rotation is to ex4hps mz1msy*;8y zt*k perience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheelcbov5 bc2,qtx 7lz-9gfv 9d0i: 0mhpb accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determi: 7f9 gvodl,ibc0 q2mvxbb 59tzh0 c-pne
(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 radius40 a8.uiell me $R_1$ 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 they-h5 2 i1en98n9afmrwvnp4zp Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy nf84imen9-n5py2 1p vz rwha9evenly. 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 15,000 rpm in 220 s. Through ho hj qc**,jilr,wchj il,,rj* q* many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calc 8 oewtnddd3zuhq0j1:2i6 zl8ulate
(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 fl3ps18w-yu3j0b.ngstu9 kfz x ying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speedx 8zy301.u9p-s gw3u jsbnftk.
(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 ikxufe 3u/ws .2n 6.5 s. How far will a point on the edge of the wheel have traveled in this time/esx2. 3fkuuw ?    m

  A cooling fan is turned off when it is running;mjau3by85kj n5 ns+m at 850rev/min It turns 1500 revolutions before it buanjy s+ m3;5nmj58 kcomes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly f nxjhj8- 7dg ykuc c7hb1w2/b,rom 95km/h to 45km/h The tires have a diameter of 0.u7 78jh w/db chyc1b,jgkxn-280 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 6x )h(r4s eiw hk*j6(ks5 revolutions as the car reduces its speed uniformly from 95km/h to 45km)rhs4( sei6kw(k* jhx/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on e1yb 2w kw x 3wouh:(aq;b/xzjs3j9+qi aq-v+cach pedal when climbing a hill. The pedals r(q:au1+w z2x 3obx bisw/qhy3w- ;qck+ j jav9otate 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 wide. What z/w tbmdc70ji5,swi / is the magnitude of the torque if the force z0/ wj7bt mic5 sw/d,iis 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 pbw3nkffej*q;0qo o,kgc . -0. 8–39. Assume that a friction torque 3 ,0-nojqf g.0k pco*e;w qbfkof 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massles7a;faq dpd,5-xgn, s vs 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 net5dfa,-vsgp,x adn;7q torque on this system.

  The bolts on the cylinder head of an engine require tightening to odffxaot 0t .yw/tla:cs)73 (a torque of 38 osy 0aa/t) 7d3fxwf(tlto:.c $m \cdot N$ 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 oe7 74ekde+sy ;cctne;n-b w8of a 10.8-kg sphere of radius 0.648 m when the axis of ro y;wkn7; bencte sdo- +ce478etation is through its center.    $kg \cdot m^2$

  Calculate the moment of ine-*8i j3lyqblp rtia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be igno lq jpi3y8l-*bred (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizon olchg/ l3.3.ut7sl wbtal/h wlso.c3 3tl 7g.bul circle of radius 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 bow3lh776b p qavol on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the cla 3opq7abh6 vl7y is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown ib(bp . j:fupl :-vy0vun Fig. 8–43 a uvp0bb ljp .-::fvu(ybout
(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 ofq wd0 6x 62xm+n7hcvvl 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 c7sv 8 pd3lbb3eyosy7*ylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as sho*7s s87vdyb le3by po3wn 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 uniform j(.x+n,p4titahrn:*(kcgmq cylinder with a radius of 8.50 cm and a mass of 0.580 kg.+nhkgi*jc.:( an4( tqmpr,tx 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,bz1fk 2jxt -m9 accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-reu3 wsam5ud( gs-v )y,ound and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. 5()adgy,w3s mve- usu$\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off elfr db9vf5zo )e4.(7exb zuo;a6 v0zand is eventually brought uniformly to restodo7z6a 0 4v v) zb;5r(ez xu9.ffbeel 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. 8–47- fkkvz hu.ill(dsf4)/ y )vs5 accelerates 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 ball is thrown solely by the acx/6x ebtnu 6 g,p5o+wetion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniforml6 tg+epnue x,6wbx/ o5y from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be cv.ajj i/9 7:5w qo1fo2peoivronsidered a long thin rod, as shown in Fig. 8–46.o2r7/v9 po.ia v1jfj5eiow:q
(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 consisusu389 pc ig8pm(oxl* ts 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 .8m wlh6 rrgb9within four full turns (revolutions) and rerh mg.9w6bl8rleases 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 p1o +4d apvu(binertia 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 de: psqa 4+bq.d1ancs9cvelops 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 gqk . y9hu/t:cor:c4w 7z db)mrolls without slipping down a lane a .hkygtc49cw:zq7 bu:m o/ r)dt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t2wm ztry,;fi 4 djj.z4hcj la)6u .a+nhe sum of two termsj zw.f6tjzl2+ancj yr, id 4.hm; u)4a,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. H7xo+ltnekq0tq et:y e 9u5n0( ow much net work is requle 7ntk50e:+n0t xute9oq q(yired 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 anvc*w uj* 0qtev:dx6j6d mass 1.80 kg starts from rest and rolls without sqct *6x6wu :j e0v*vjdlipping 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 atgjez 2w+ c67ob3d+cits tip. It starts to fall and 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 enero7 t3z2+ g+dw cjba6cegy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a 1 vwe63w) /axgd)kq9gy.s ankthin string in a circle of radius 1.10 m at an anywwa n3s 1exg)9 gdka/ q).vk6gular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel qqza:uecl /-lt:2x5 bof radius 18 cm when rotating ae :qcall/-z 52uxbq:t t 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate4cep f0,nqh*t of 1.3rev/s If0e *c,hfpq4nt 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 Fi+f2bx*4k0c et(k pkv rg. 8–29) can reduce her moment of inertia by a factor of about 3.5 when chan*krp0kckb vx 2+ te(f4ging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an in7*ecm5g6ti ud+po hxh 4 l:mq:itial rate of 1.0 rev every 2.0 s to a final rate ofh: c:mg 4o7mhld 6+*u5txipeq 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 at ejvd)iz303n xis through its center at a frequency of 1.5rev/s The wheel can be cons)jz03detin3 videred a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater k)*llh:y0k9a uugcnj1 n, yk+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 mo2yigcl)ztjiw/w( a/y9 z:ob 0mentum 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 inertal k 9gffv3pi4v6hng83vg3s4ia I is dropped onto an identica pvsh v3fkg4av649i f3n8lgg3l disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.ta2y*d s 6b,i iav)hdr2)1z:aqsin t54 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-ro)xra58h ca4.dg r* ldiund platform8d4x)larg d5h ca .*ir of radius 3.0 m 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 vepl0 dx4u)ry*fh- v4zilocity of 0.8i4dzu*xyl r)h vf-p4 0 $rad/s$ 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 its mz4)wf d( d+bqhuxx3 g9ass 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 th9bdxq3 xdg)+u(4whzf e Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 12.ht1dk39oj(ft kvfces 5 (i)x0 $km/h$ 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 fgs e, fmw(yr1ygsirkm;) b +q(:94 nw$ 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, initfnp -b+t xx(lp1o)e 1rially at rest, that can rotate freely without friction. The moment of il - (1benr+o1x ptxpf)nertia 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 men ivexgm;0iy-9,g x,d fg np:2rry-go-round turntable that is mounted on frictionlesix,ngg2-ef09,iy nxp gvm d;:s 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 lyi9*p yee/370zq 3 qemnkg p sptnu/dn,5ng 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 rq/ 3udp mg,3s5k/tezn *ny9pe7nq0pe ope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Eav)9:t p e 1c;onmg.wmzrth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular mom zg:v)9w cn;. 1eomptmentum (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 8c gbaw5v5 *zi t+ssebrz/clel ;l 473rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius8;zo+* m64yzipl tq nhe/s s7: csvr9 rg.y)hs 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torq4nmg7hq:y9s6epzlyst;zrs v / 8cr so+i* .)hue delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical dif1 o2 zwl+tl9 s8yxri+sks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical h+l lrx8osy ft2+w zi91ub 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 sthm/ vyn7,7* plxp9sl peed 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, werjjs;t/9zax ia 6ne+r h .8(lsge 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, losing three-quarters of its massea.9(/gx;tarhs8j+l i 6jszn 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-poi0j-r pn3m ;ikllution 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 35i3n0-ij;m kr p0 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 tll+jp(d y25kaq x 6d9an $H_2O$ 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 surfa7j5 q)r7ukkax ce 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 freely (i.e., we ignore frictim:v):gsis usv7szd-/0z f0i a(xc o *ron) 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.v girf0) *vi/zz (0sdcs- xa7us:mso : 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 m*)ag-ex.a jdstanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which je.-md)*a gxa it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/smueo qdij .7bx*gdo9) jux4*3 on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal for u)b i47.gum9 oe3ddxjqoxj** ce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 kplq*jjfuba ,.im 6ihluopuy8w)dg90o, ,7; into a sling of length 1.5 m and begins whirling the 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 t0 *, ;p b)puhla , u.ok9 ofg6yjii8j7,muqdlwo achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms2h)ud0xtlclf/wm 0,ad/ sm5r as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spi0s)lx5mm duclra/f 0wh2t /d,nning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cy5+enw ,py zm0/q3ucxa.;u kmclindrical plates, of m.;0mpaxecwknq u+u yc 35,zm/ ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 og;v/s +ywpju9e 0 w/5hisqa k5f Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is sv/iwah kguewys5 q9j+;0p /5to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, b/+sazsgg: f9 p lo.tc.w/4 ir.k/tc wout do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as fof4 ehw3/14oh gp3rb8gsnar w 5v:l1the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a wof haoep1/4w3s1bl:3g hrf4 5gr8nvcceleration 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