A bicycle odometer (which measures distance traveled) is attached nea 7p5cr5bzxmae(b8p z -r the wheel hub and is desi( ep5r a5-8cz bzxmbp7gned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a )m. +1fukhl*m 4kvcd gpoint on the rim have radial and/or tangential accelerm )umk l.h g1vdcf*+k4ation? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sibfp tr503 icaa*8bw)4sv8t n bngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a large 9.sb6bxerb pyix- e2:r 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 t,cwc 7 bgwk6r+o do a sit-up with your hands behind your head than when your arms are stretched out in front of you? Ac 7k6w,rbwg+ c diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the f ,cbk*i1f21nprq d2 srear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocf p1r*122si,fkbcn qd ket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on z/ 2 dmmu tirbs918e ic;9a.wzbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (rcb udize2szmw 991./m8t ;a iFig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ca sai5-luec.b8 rry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the 8agq jew lh3l30 q87lhnet torque on a system is zero, is the net force zere 8galj3qwl78hl3 q0ho?

Two inclines have the same hei(ws:fc,8bnja,bh6a iylul 6-- ee7z/l r b5mfght but make different angles with the horizontal. The same steel ball is rolled down each incline. On whichb z785-jb m,-cb 6sa, yfr l 6hlwnuli:a(/eef incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an iv)x8yrsugy3;h( (qip uoz.ud4; wpq, ncline. One sphere has twice the radius and twice the mass of thequ; gx)s,3p(pq r;yz o4wi8 d. uvy(uh 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. Theyol)rc m(;h(9 uyv6muqm3b/ v,kmcjuu a3i+d1 start from rest at the top of an incline. Which reaches the bottom first? Which has tym9 v;/(iqm mojcu u3 6 la3kh v,m(ucrdb+1)uhe greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mome:eeufx8 u6r)p1 qu- hs5.p/ebqaqa:cntum are conserved. Yet most moving or rotating objects eve hq-/qpe : x.86:qau1e) bfspucr u5aentually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wo7 9 lhlm5() ta,p+k0awrhdgvt uld this affect the lengglpd 0v )+h7a9wkh lm,rt at5(th of the day?

Can the diver of Fig. 8–29 do a somerw1jq p; bh fv5 qp+ej-slt3.s(sault without having any initial rotation when she leaves the boap q pb+ ejh;sf vw.t1lq-s3(5jrd?

The moment of inertia of pco iruzx 0r-jj01 /6 a5)acpya rotating solid disk about an axis through its center 10i rcaxc)ra06/jj-pyo z 5upof mass is $\frac{1}{2}WR^2$ (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 mas yln7.e0eudvhih82df,n l 8 3us in each outstretched hand. If you s8,h un.nflidu382ee 7d0lyhvuddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicxop,uw,7r w)a k )y4qdal and have the same mass. However, one is hollow and the other is solid. Describe aqox74 ,urwkw),d y p)an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointsqblt+/)l:rbofk m)3o west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe +qlr kf:mb lb )/3)toothe tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on t csx7,;qzn ir.he edge of a large freely rotating turntable. What happens i r n,i7z;c.xsqf you walk toward the center?

A shortstop may leap into the air to catch a ball yr;5u yh6k9drand throw it quickly. As he throws the ball, the upper part of his body rotates. k9rh;dy r6 5uyIf 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, i.nk-1 -uhhko discuss why a helicopter nu 1.hho-ki- kmust have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglv cweu1r-.sn (rwn txi4;z-5m es 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 ,3sdok w* yiaz ebxa8;*ox 9gv;5)pqk because of an amazing coincidence. Calculate, using the information k8yapke ,sxgi*;a9bqvxd ;)o z3w*5o inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth.bcfdy.v5i b.djw3 (kg (;iyb4 The beam diverges at an awgb.c(yk4bijyiv.5 db f (;d3ngle $\theta$ (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 u-+c 1avv3y jdoxz w odwaof)1+im 029of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleratiwz)owcvd0 xj+dva1m2i+ y3o1 o- fua9on as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floorue 22ns +;5radvf - k0xty qur0rm+o8g 3.5 m to another child. If the ball makes 15.0 revolutions,2+ver rnu+k; 0y5 af-qxd0org2u8stm what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revop2cf9g:0fx n xq9x1:p ua4m 0d:somsg a:ln1ulutions do thexg fo:921num4s qf 1cnpa0:us9g l0p:adm : xx wheels make?    $rev$

  (a) A grinding wheel 0.35 m c n8rm2 cp0a(qin diameter rotates at 2500 rpm. Calculate its angular velocity inqr(2cmnap0 c 8 $rad/s$ $\omega$ =    $rad/sec$
(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 cmgb /oh6/v.a ed99 dz) 6bp:lbm tfer1omplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sea: g9 rm1mehl etdp /b6b6od9vz)bf/a .ted 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocitxnhjl+ (0is:sy of 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 speed ofdprcgc4c5 s 3endx +mc1*23 hz 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 rotj0 /8bs(aaxyb4h/t lr ne et.-)t 0gqeate if a particle 7.0 cm from the axis of rotation is 0) elh.8gj stb x0neq4t r-y/aet/b(ato experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its lbsw -mv6-pz+.j. r evcenter from 130 rpm to 280 rpm in 4.0 s.b6.-welr-j v vzpsm+ . 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 radiusg00r p6d48ylt ijsk7gr* jvq m j/l*g; $R_1$ 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 put the.t* r;8p svrjjmselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute duri;p 8rst*rj v.jng 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 2w d26keo taf p2aspe69* v3a/tr4isdqlqg0 7420 s. Through how many revolutions did it turn in9s44s0 pow 6q rl/eiataad7gv2*q2dftp k 36e this time?    $rev$

  An automobile engine slows down from 450)q4ag. 7.v5l shsr ;+pwppaws0 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying e1ynr bq+fq6: bu 1b)ahighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befb61qny be+rqf:ua) 1 bore 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 ix3sq3 tk,v 9v1gd4hy360 rpm in 6.5 s. How far will a x3s4i3k,dvvg q9 yt 1hpoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned xdh: +ms9zm7q off when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop. +mdh s7mzx q9:
(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 makeut0o xk -w+mxo.((txw 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have au( x( w0o om.wtx-tkx+ 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 spxs:6 h 3d wqbk1.)trkteed uniformly from 95km/h to 45km/h The tiresqt stkh6 .w1)xbk:3 rd 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 bikdonzx p/.xo2 . pw.c:be puts all her weight on each pedal when climbing a hill. The pedals rotate in a c:o pp..o xwx./zc2 dbnircle 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 nimltv 9p -hdf,ey+ u;b/u8/wthe end of a door 74 cm wide. What is the magnitude of the torque if the forcetev/+luy; h8,n 9 w-dfubmip/ 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 the2* ;ply n1xrl99j16iu cfxk ww axle of the wheel shown in Fig. 8–39. Assume that a frictifx;lc6nuiyj *9rww9 p12 k1xlon torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachf l)u 4vu3r(a3uf ,zhred to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the h azuv(l 4h)ru,f ur3f3orizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque /ij6j gj+yx0x of 386yj/ix+gjj x 0 $m \cdot N$ 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/vm( v s8ofkippk:2-o sphere of radius 0.648 m when the axis of rotation is throug8(o-2 : pvmpk /iofksvh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cabv+0i74qf b9lckr e9 xwh1 7im in diameter. The rim and tir+e4b 9 ra1l k7qcif7xvw09bi he 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 on the end of a thoa1dvt 3q* zi3in, light rod is rotated in a horizontal circle of radius 1.2 m. C*3izovdaq t31alculate
(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 +s0klz(j w4db on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The+sl(zwd 4 kb0j friction force between 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 obj bqcvt7fc- *0tects shown in Fig. 8–43 aboutq0t b7c*c vft-
(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 twh0 /ha-6 o q*y34gag5w7bixx p+cdj notal 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 cylindrxdib3kt -o der4;5f/8)ag jnu ical satellite spinning at the correct rate, engineers fire four tang-/8bn 54 tj)gxefd dk ;uar3ioential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder xx0lt+yp7g3hfb 4mu - with a radius of 8.50 cm and a mass of 0.580 kg. Calculaxtf+hlbu yp734 xg m0-te
(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 swingccy7 b):ikj g*s a bat, 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-round aesj+z z0/sb/u0 k8fx-h (llfynd 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, neglecti0-ue yz(ff/xl l/sbk0j +zsh8ng frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingjrk yi m03e7+q at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fric0r+7yjqi ekm3tional 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 t ;uic2ox;a5u j,d+ts+p r;dlaccelerates 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 yj k5bv+v *2xg51j8yf,zm u a9yv5wgzball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelev x 85gj,yj g m*yf9zu5bv v5y12akzw+rated 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 considerqx2, 1j vwev2;-tunip *x:9w ck ..vyro 0ejrxed a long thin rod, as shown in Fig. 8–46.u:r02vrt -*jn1;wp eve.kij ,x 29vxyoxqw. c
(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 con/y80gtcpg +n2bedh,kh 5rfk2sists 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 ha9l5 ysin(5sq) ja8:fx* tj sdy(l7efkp1g /cp mmer from rest within four full turns (revolutions) and releases it at a speed jgl9 al5 pqpsjtk)xnfc7yyf518s sdi (e:/(* 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 inertiap+) 2qhddgw x( cm3i/b 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 devb2a+ca -eae*cqby9 mxo9 w,6helops 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 radius9y0ux ;;n lv /o58zwq.icu odm 9.0 cm rolls without slipping down a lane at 3.30zumnyvliu9q x; do/5;.8 wco $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect tol(/m- -4ii5tcisc cvd the Sun as the sum of two vitc / c-(c-i5dm4lis 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 a radius of k6geuea3t)/zfrx6sc8)6e ee 7.50 m. How much net work is required to acceler c/)u68zta rx k6 e)ef6eese3gate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and retl4n0pitbp1mi*t8b u .7i+golls without slipping dowtt7nit4.108ip+ mgl e bbp*iun 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 stw..ru r.e 6h mj7-tiytarts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just befo w.uj i.tr.6tm7 -hreyre it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba +ks2dy .xb1v- )w ozjl12zbtzll rotating on the end of a thin string in a circle of radiu-xw+)z1sy o212 dltbvzk.jz bs 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniqyg yuigcr ,ts 2 13c7.gu6by*form cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm,gsc2g qi*gyu7r. tb 13cyu6 y?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on,90rk 1bj+vw p8kdqwr7j rh4 c a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decrek4p vq wdrbr hr+0,1 j7cwkj89ases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the ta 8l2a bgccssz;zq 0t01xur6 , a1xg4one shown in Fig. 8–29) can reduce her moment of inertia by a factor of abo8l,g s1zb z a 2rt0txsqca4u06 x1ca;gut 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 increase her spin rotatisvjh/ ukp17r8on rate from an initial rate of 1.0 rev every 2.0 s tvj/hs87up r1k o 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 through:n-8zq7)u aqar 1s ja;unxi+t 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 disa; a x8-1nsur7zatq+jn) i :quk 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 spinning at 3.5ef nv(j8hi;p: $rev/s$ 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 angularirlpyp1 l 61+jh+z0 w +h5f9d iwhwcs6p il2c) momentum 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 mom; 5mokxyvh7w,6qbx x/awyv*1ent of inertia I is dropped onto an identical disy76/ v; wxy5,am*w1bkxxo hqv k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ad 0 zu azu612w9kj7:iih8gexat 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-roun6 d8 kddkahz./d platform of radius 3.0 m and.kh 8dadkd/6z 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 freelym/ynva4ali17 with an angular velocity of 0.8m/4 y7a1an liv $rad/s$ 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(e3c;3zmhmr ,4s 2fajmz; h mk mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rk j;c 34zhfsmea r2hmm(z,;3m ate 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 wind)t0/ + 0azcv:gp qfheus 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 75hhvp*xh/rp7tig(-k ll5 w y$ 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, thatcm7 k(1s)bseb:k (ul q can rotate freely without friction. Tskec1lum7 (k)(bq:bs he 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 standsbzsz1k)4a mv cd vb;41 at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings a44azb vv)mcb1s dz1; knd 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 on th6kz jx.kuh fbpci+ 7, olw),j3e top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass movb)jpch3o xz7 ,j.kwlu+ 6,fike?

  The Moon orbits the Earth s k3ivd6g/4 knsuch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, td/ g3nk64iksv reat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a i5vwl),1w0g6kqz qb3 mt d;hyrate 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 theu:pxs 8of v1r. shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 61f:r8o vpxus..0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0 +ouf(mnj jx6--1zdpn6o xpt850 kg and diameter 0.075 m, joined by a (concentric) thz6d xnu1- 6 mfppjn-toxj( o8+in solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is 6e0; l :e9+ p8jat sgu6rkdkbhthe angular 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 mas jt y3mb*14cyvs 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravita 3 cy1b*j4vmyttional collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to u,ww/v jje v/w-as* s*fse energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diametevww/ssj/w *vae*,-j fr 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 a;aw zb n*5js0s5t,s/vyzxbx; n $H_2O$ 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 roll- p6n-dvmkp d.x8goxcu 4,n 8hivl (0iing on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ig:bik8ziv )(kz nore friction) about a hinge attached to a wall, as in Fii)kb:ki8zv( zg. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on t;l+b q*fpitlj9m4asi u5 81 p7h0b htehe floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step4p5a*7bi; ftqplue h +108tjil9 bshm has height h, where h

  A bicyclist traveling with speed v=4.2m/si; y, ijmk))tm on a flat road is making a turn with a radius The forci)m j;iky,)mt es acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.k0tj 3g3g gphsk kq7t( mz64/t5 m and begins whirling the rock in a nearly horizontal circle above his headjtq 0sgpkkg34t kzg 6t(h/37m, 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 solidm -1,koskjt 3dhy(5p k cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then cas,k d k(5m1hjo3pkyt -lculate 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 0*x /fm1i5gxtha9r z8 esqb 8oassembly which consists of two cylindrical plates, of masseb/ oh88iq91r0fma gs*xzt5x $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Ffcof0eedla 2 f(5s; v6ig. 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 2of c0ve ;(5salff6edwithout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but ,ltcgevcu45v9 r* 9hm do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th*hcuj5atwsv7 tsj k7laox:;7) 4pt3ze car reduces its speed uniformly from 90km/h to 60km x w ouvc7;p *lz:)asj4ts3htj775tak/h The tires have a diameter 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