A bicycle odometer (which measures distance traveled) is attached near the wbv jl,2htbj .:heel hub and is designed for 27-inch wheels. What happens if you use it on a bicyj : th.2vblbj,cle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a .- l0jzy (pzvypoint 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? For which cases would the magnitude of either component of linear accelerat0zy .ly vjzp-(ion change?

Could a nonrigid body be degrpmz(frgh6j4t(59n/ el -hb scribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than x3 c.ov*li/ t1n kuxi8a 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 ts9ddxdxls 7r0kx *l1i-r9 4poo do a sit-up with your hands behind your head than when your arms are stretched out in front of you? dl *o-slr9i 9sddk7 1x0pxr4xA diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheeltf/ -6wuz3klt , cx) 6qvpf/rk )nba:r and three at the pedal cranks. In which gear is it harder to :bzk/nk6/ aw 6t)tuqv)prxf,c -fr3lpedal, 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 vd/ b*)1dbzv.ozjb n.slender lower legs with flesh and muscle concent.o z /dvb1b.n)v z*jdbrated 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. 8–35voxs )) sqn*md;z p2/+yspfk3 ) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero?wx+q9 ;h*y1k2v uvknc If the net torque on a system is zero, is the net force zer*qx9; uhk+vv 2c 1nkwyo?

Two inclines have the same height butl3udqr7:m ttj ng 6wr:(e: )fq make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wil763:rwnltq:qtfu(d)jr g me:l the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) d8 t:z x n7)ueslmqczvs*.j.2 eown an incline. One sphere has twice the radius and twilv.7 m.c)et s s:xuz*n 2z8jqece 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 sam h g mx )*nwifi/m3v8.prbp l0:oyg0t+e radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the gli+:owpg.*3i f y m0b)/tgxp0h8r vnmreater rotational KE?

We claim that momentum and angular momentum are cob9 ,q(npu-.g8)+ vuy* iemnemy-emlf nserved. Yet most moving or rotating objects eventually slow down and stop. Explaem-yb +, u8 qv9y- le*(img.)efnnpmuin.

If there were a great migration of peoqc0 *s 5tzezv/ yg,h)rple toward the Earth’s equator, how would this affect the length),ez/z vc*g0qyhr5t s of the day?

Can the diver of Fig. 8–29 do a somersault without having yo6a;szh-0ypw76fnh any initial rotation when she leay7hhsy p6f; -ownz 0a6ves the board?

The moment of inertia of a rotating solid disk about an axis through its cent(7 7o(nfni ks jrvr:q7er of mass isrrovn7n7 :q(k 7jsf(i $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstqm 2iwmy+j/y 0(adiui,: sfa1v:c-r gretched hand. If you suddenly drop theam +-,2/y:qswd (mf0 : ij1giauv iycr masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. How2)xo8m .bcox;d t,xt bu9o7fm ever, one is hollow and the other is solid. Describe an experiment to determine wo2.ftm t xm9x7u; oo,)cdbxb8hich is which.

The angular velocity of a wheel rotating on a horizontal,hm;8mu jcv,)i cyxm *r,: vpt axle points west. In what direction is :m pymrv,8m c)xh*u;i tvcj,,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 decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whatw 7nzw/+kr h9t happens irw zt7kw/+hn9 f you walk toward the center?

A shortstop may leap into the air to catch a ball and tc bua-3gj xj p4m(+qj3hrow it quickly. As he throws the ball, the upper part of his bo xj ac4g3(-b3 m+qjpjudy 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 q.sbc) cuk2h1 helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable. b.s)c2c 1uhqk

  Express the following angles in radians: (a) 30 kk.+rqo4r0e . ohg :su$^{\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 E*8s;fe1k z8h 0n uuuinarth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameterk hui nu8*s08en1uf;zs (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam n1c,mun3 wvj 9diverges at an a 1j,9nvcunwm3ngle $\theta$ (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 2-alkqgv2u x*w y ,p(arpm. When the motor is turned off during operation, the , l 2vgpkaxy2au-wq( *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 ln1 ugnrj tbub;9+ v4p;evel floor 3.5 m to another child. If the ball makes 15.0 revolutions, wt1bu+ p;jbngv 4n9r;uhat is its diameter?    m

  A bicycle with tires 68 cm in zmip* /q 5/ hpfe (k)pzj;6fufdiameter travels 8.0 km. How many revolutions do the fkj*; qui/ pe/zfz 5fpmp(6h) wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angi9n tqw v8a49z8/g klcular ve/acn 9zkvgi 48 w8l9tqlocity 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 completcerplf)sg-( ntya bn u/5 ,30de revolution in 4.0 s (Fig. 8–38). (a) What is the lin fpdl3y 0),au-/ t(srnb5ncg eear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of h*xlvka*. 8vb *moq w(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 speeqy yn/ieq5n 9dwz2*1 gd 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) mustyn vler r /a7pzlxi6u;p3g21) a centrifuge rotate if a particle 7.0 cm from the axis of rotation iyulz2n16/e;p l7irvr3 g )p xas to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm (e)a/llptlw-wjh:tn ++g 2 qk)gp 0ztto 28:g)tl+h )wntq 2kl0pwa l / tpe+(j-zg0 rpm 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 mc1vs-y78 m t(dtt8ii$R_1$ 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 Modz4f1.-ic *cv;p kog ron, astronauts 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 revolutip4-*cco.d k vrgz 1i;fon 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 15,00u 9/py4 aurh m9bt2uzb1/-hny0 rpm in 220 s. Through how many revolutions did it turn in thi4 nt uhy 9uy2buz9/mb pa/-r1hs time?    $rev$

  An automobile engine slows down from 4500 rpm to-wwrso(wh4,c 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a)6/jksan zxq6bj i-633wisjd whirling “human centrifuge,” which takes 1.0 6s /3jks)qj6x-ibnji6z wa 3dmin 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+k0hcn8 b1 djs s. How far will a point on the edge of the wheel have traveled in this j0 bk s8+dcn1htime?    m

  A cooling fan is turned off +hs44lpr9izpbwd ) i 77lhsv6 when it is running at 850rev/min It turns 1500 revolutions before it comes to a hvih49lsr6z lp+7sw47dp) b i 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 uniformg0lo*:kv 5+zyqja)dxpn7b 9i c,,azbly from 95km/h to 45km/h The tires have a diameterzla, kx+p9d ai:bjc,gqy 5 ob nv)*0z7 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 36 o osmslcjwga, hc1n9zt 2--as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ofcz,j--ot 6m acl9go2h3nw1s s 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal w8tr xf f*p4 g6yrfxtjy1+tj/ 7hen climbing a hill. The pedals rotate in a c1yf gy+p 7rxfj/4j8t t tx*r6fircle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the magl r5 omb t5lpzfd6- 86qi:5lkznitude of the torque if the force is exez5fmo 65: -pl8llbt 6zqikr5 drted
(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 b)-xt7tffz q li:o em*2knthg),b*.rin Fig. 8–39. Assume that a friction tor.tmt b t*eohf)bq:ix lf*r)7zn2- k,gque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached qoxlc e:srglr8t2i.* am;8 o 1to the ends of a massless rod which pivots as shown in Fig. 8–4l.oamsxq* o12 citl8:g;rer8 0. 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 he)xez ; zya96wb.yan rld28 x/rad of an engine require tightening to a torque of 38ey byx/l.2azn9wr ; x6)az8dr $m \cdot N$ 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 oftcut r2k67,opd20xc wx 2q69 f.laeeb radius 0.648 m when the axis of rotation is through its center.feq2 x2. x k 6dcprw0c2l 6tbaote,9u7    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm inubasq*w t9*2xcrnb; ) diameter. The rim and tire have a combined mass of 1.25 kg.wa2xncb 9q*)btur s;* The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th3)i.h9kq /jxrf 6m4n1vwtgdee end of a thin, light rod is rotated in a horizontal circle of rre)g hwx3t 9mj 16/.qdk4nvifadius 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 bowl on a potter’s wheel rotatin xjg-ce* n5qlwt4oy -9g at constant angular speed (Fig. 8–42). The friction force between her hands and the cljgcx -t9 lne54y-ow q*ay 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 inero( 3dpodufw5: e)qhs*tia of the array of point objects shown in Fig. 8–43 abouth:dd)(o 5* fp3wue qso
(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 consistm1zqwt1e 8mv ;s 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 spin-36 2y s8)li5 loqqxqnnmkf4f ning 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 th5 yim)o nsql8 fqkx2q4nl36 -fe 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 cylinder with a radius of 8.50 cm and a mass oxq7qzjh :m0d3nrgd 2i( w( esg,c/j6 sf 0.580 kg. Calculate,gd/sj qidez0ws73 6rxmn(j:g(2 cqh
(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 rzw2o*s0 mbvd2qxr6 ;*.k5crhv n7ao yest 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 ablae+h,h( 8 5 ibf8oszxne 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 theh a 5ez+8(xnsi,hbf8 o acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is event es z n5aloc+)5) ridmyp*vu v0uj3k,;ually brought uniformly to rest by a frictional tov+j l03u)*v ci5p mu o5rsa;ze)k, yndrque 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-kg pr8wg33e2my z e0x nqa+bgr(3 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 skfdane: z :r 7b/u:c.ball is thrown solely by the action of the forearm, which rotates about the elbow joint undecd/sr :: ebuf7n:.kzar the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown i(lvogz3m krc):i s4 iz/zr*:y n Fig. 8–4iszmvi (r :3cl*r zkg4z):o/y6.
(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 consisxakk243k: zw7 zzio 9fts 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 accele2 rge:ial0 ps9cd*pu 4rates the hammer from rest within four full turns (revolutions) and releases it at a speed of 2adr0lgceus p*p4 i9: 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 momen(k 1rlgo* 2l(z9 xluout of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine dz3:wt b r.c c+9zqq9goevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping dow7 ;k /px;8yp5w;sflcqt4tue hn a la htp/4;;e8y5ptslk;x7ucq wf ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun +r c50rk/.mljn5galn as the sum of tworn 5lkj. +0crm/ lgn5a terms,
(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 awrg wi+86+ 2fm lkjqc(c,n4 28oscigf radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate og+8w ikrjq2,l wm (nf2+ cci8gc64fsof 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest a9/1hr jnqs( iind rolls without sliq/j9sn ri(1 ihpping 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 and i g5m0.*u2 bf.jgx8k:tunr pvx1q+hvf ts lower end does not slip. What will be the speed of the upper end 2pnt0vr*gu xk. gq jx5:f1.mvf+hu b8of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a )o / e.lp2lvanlb (2ek0.210-kg ball rotating on the end of a thin string in a circle of radius 12 /epe)2l.nva(lk bol.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uni5lc)a-.6921 yhj upynv wikq iform cylindrical grinding wheel of radius 18 cj6. 2 q)ncpv- w5y9ikayiluh1m when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands uc-xtd-z x/vli 93*2k 2i1hbnzrp w2 tat his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positionzi vx9* 2ti2bt krdph/c3nl -x1-zuw 2, 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 Fij71 /:,a nazlc 01,kjbtl brqpg. 8–29) can reduce her moment of inertia by a fa pkqljrb a71,:/1n bz jtc0l,actor 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 position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increb *b)1awg3 jw2z*nweft okm39ase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final ra*agwb f93jn wm)ztb1 w3 o*k2ete 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 dc7;v)/zo 1i czdks,yvertical axis through its center at a frequency of 1.;zkc,1zv s7i ) /dydoc5rev/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 clay sticks to it?    $rev/s$

  (a) What is the angular momentueybi64 3yzd q5f58yf ym of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the )ui74xec. cg iuim5dsf1 u1w 6Earth
(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 droppe(m4 8pzup1i emd onto an identical disk rotam p1(i emu48zpting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 wro,gtf1f-a l 4*m na;$rev/s$ 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-roup fq1:tz6se)f nd platform of radius 3.0 m and moment of inertia 1zsq:f) tep6f 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 vhtbiq s86 qr58elocity of 0s5t h8qr8b i6q.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses intoags* k(2ddg( o6 oa/m vh8f(1hwaqq,e a white dwarf, losing about half its mass in the process, and winding up with a radius 1.o2 md( e fhqqh6ko,s1a8(w*/aadg(g v0% 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 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 17 d.,zhzy gk6* xb.kpqsu)ag720 $km/h$ 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 /fy )vakt f8:4;kg ocy$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freelkd2kkc.()lh4t mty +by without friction. The moment of inertia of the person plus the platform is k)dk.b24l h( mkctt+ y$I_P$ 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 thef645.wrykhn6e dr s2wh fh.rjx 4)3q t edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of ine)frf.w6 wq y3 6se n4rhjht.r5hx4d2k rtia 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 59wpgg)j-4 o) p6nim y lfe2e.tntnm/end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig.n/ t6eyo9t2pnig ))m5fmg-w ln. 4jep 8–50. The spool 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 the same side always faces the Earth. Determz,i y0/etez. ov kn8+hine the 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 E,zz0/h ovtni.+ke8y e arth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratn1x ih tgj8gd //.e6 oxqx6yi.e 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 0.2l51vr 6 od:jvclc c9+i:o z0di9.zdz e0 m acquires a rotational rate of from rest over a 6.0-s interval at constantid.0r9v:1o +6l:ie5vzo zz9 cj cdcld 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 3 gkiig.zvz i+ff)iy4 e*,9cr0.050 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 to calculate the linear speed of the yo-yo when it re fzv gkiyizi9girc4*,e3. +f) aches 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 t+ujovgm lg q5 ,4c(w/phe 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,p h/(gs4t/() vizzhrm 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, 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 t4h(vzg msh p/(r)z /tiimes, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ep8 7 tznzuebu1 -*i+(h)g xztvnergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without(8zuhu igp +bzn7)t*tx1 -evz 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 illustrat3y zta;.t* )hii xaxf/es 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 8xi y/vv23e9 ufi;hy:t)p8ai:.mlp 6u mkvjqsurface 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.,v29qfkay51j vws0o n(; w rf0s we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that(0 5 fw vqk a19wvf2rs;0ynojs 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/b76c,rakl;d k g5fut R. It is standing vertically on the floor, and we want to exert a horlcg;f /,5u6d kkabt r7izontal 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 witgt /5ejni q0c, vu8yr,-4rvoch speed v=4.2m/s on a flat road is making a turn with a radius The forces aui g4joy8er5c-q0rn, ,vc t/vcting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock intoy7)/f41hln6 xoee:mwh8hgw o+ a+n pcx k8yn. a sling of length 1.5 m and begins whirling the r+:4o+. wn)f18l x e ox6 yypk7wgn mh8hchea/nock 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 cylinderap m ibd3:(hha 7me(rt/a46ot and her arms as thin rods, making reaa(t6ha i7rmamo(/e :t3bdh4p sonable 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 consists of two cylindr 9h1,ong6pw3l 9 cek(vyjey/ dical plates, of masyednlovc9hp(3 ew /6 gjky91,s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alik /azi;r/ :t3sc8qweong 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 hig/e8qzis3 ctw ar/k;i:hest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do no,qzs/sk v/b v/t assume $r\ll R$ h =    (R-r)

  The tires of a car make 8,vyiq h pel,55a5bu/a5 revolutions as the 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 acc5 u,p/, eqh aai5vybl5eleration 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