A bicycle odometer (which measures w ksbrzv q8u5x-4 s+dqi/+y+u- v)errdistance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whee r4-xrv5u)8sr vbws/zyeu-i+ q++k dqls?

Suppose a disk rotates at constant angular velobpp 7b* oxm. feaw/(e.city. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of eitfb emeawp*x7ob/( p ..her component of linear acceleration change?

Could a nonrigid body be described by a singl.fm ::-iqhu pwe value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger for9 izsc k8oq..ece? 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 2qzi)7rs,y d tj7- 4io mox:asdo a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to ansjsri4t7xi)o 2 dmyq ao-:z,7swer this.

A 21-speed bicycle has seven sprockxmxl8.s (c)i dc(ouicsib36 +ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why?s cmcl.bu (+6)i38iscx(xo id 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 fmv289hdek 7q gast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributionvhg78ek2qm 9d of mass is advantageous.

Why do tightrope walkers+ zoe(zfi; v;73xbc-hqb z9xg (Fig. 8–35) carry a long, narrow beam?

If the net force on an/ p:xi5a+,1b*lc 7 j4xjigxtfe 5 i*sq.wbtn system is zero, is the net torque also zero? If the net torque on n7 picqwb+e1itf5 :5a*x4/g s.ltj* bi,xjx na system is zero, is the net force zero?

Two inclines have the same height but make differe hzr3 0vi9y1chnt angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottomr vy h09zi1c3h be greater? Explain.

Two solid spheres simultaneously start ro 4+/wgo cfwg6x6dvr 3 ;r2avnflling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the o va v+r2xf;n6g fc3ww46rd/gincline 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 sabpn0v9w. hdjpxt o ;.wh(r*-hme 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 h-t9dxbhhwh ;vpw.njo*. r(p0as the greater rotational KE?

We claim that momentum and anbo4cqj (sh.( hgular momentum are conserved. Yet most moving or rotating objects even( jscho4q hb(.tually slow down and stop. Explain.

If there were a great migration of people toward the Eve +m9xu-coxab*f-pmp sr-(;arth’s equator, how would this affect the lengtv mo(x rp9-emaf+c;bu-*-spxh of the day?

Can the diver of Fig. 8–29 do a someh :lx7-+u q1qq zes7uhrsault without having any initial rotation when she leav+7-qhz1eqhu7lsux : qes the board?

The moment of inertia of a rotating solid disk abou9ewd 9v9yyg4v2y n6 nst an axis through its center of msn49vy9y2v ng yw96d eass 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 gm/j.bgk mn 9*on a rotating stool holding a 2-kg mass in each outstretched hand. If yogg*k/ jbmn .m9u suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one iz ly.ypwa-*pyepd:7 6:xy:gfs hollow and the other is s:y yp-gzd *:7.pf6yexy: pwalolid. Describe an experiment to determine which is which.

The angular velocity of a whee(a3gb+yw tvz, l rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangg3yvta(wb z+, ential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on t sv+/ /8szdaovy -gb2nhe edge of a large freely rotating turntable. What happens if you walk toward+yvvs8/2b gs/o-adnz the center?

A shortstop may leap into the air to catch a ball and throw itt1(f 6wp yqx,-42r wamh4rbtv quickly. As he thr ,trhv4rw62qbfm(-t p14 yxwaows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mome3yea(ouu/+n x 9 kypg.7 )ybav-huc;fntum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one yfyg/-c7)bu; ( +y ku hvuo3axe.npa9or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in jq o.mzhaw15 w/8 uhx8radians: (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 coincgrj1v htk/bds/k z90j,0s x *pidence. Calculate, using the information inside the Fr/*sj0rd p9 xzhbsgtj v, /01kkont 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. The beam diveq4 xlfv t(, l5 dfc;i+w7vnif7rges at annwf q+7lv( t4,icl; ffx vdi75 angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rota4qa ram rh.e xbek65e1m4 *tze 8)wl,nte at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladee nl8q6rm.xe4bk*a,rw 5th mez41ae)s slow down?    $rad/s^2$

  A child rolls a ball on a level floorv* v46bb83htticv):a i 5s9vxkx/ipd 3.5 m to another child. If the ball makes 15.0 revolutions, what is ittv5x3*)ad /itvbp89kb4 cv6sh :i v xis diameter?    m

  A bicycle with tires 68 cm in diametu)(3wf;stbjb 6j k libpnrq(:)5m .kmer travels 8.0 km. How many revolutions do the wheels matjfmkrj( b;3.wb i)p5q)un m(:ks6l bke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anguv ,b zfk6,pj*je*0lsnlar vej jvk*6z,e* 0spln ,fblocity 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 4 1f4 yartutsgni74gd4efhr+(up c .-makes one complete revolution in 4.0 s (Fig. 8–38). (a)rfa7gc4(4fni sh u4rtg1 uedp-t4 .y + What is the linear 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 xre/ 4ggx4kx6 n;7hdhof 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 spevccvym/cl x86bq ., )zed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fromjic.zy6 eo2 f)blmb ,6 the axis of rotation is to experience an acceleration of 100,02jeifybl.m )66cb zo,00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly as/+qu b0/ .1cbnae3t0 dh bkgibout its center from 130 rpm to 280 rpm in 4.0 s. Determinend0t/.s c/ k13 q +hbabu0igeb
(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 radiu+ dc4zimh sdxpc jc087 7r/c9ls $R_1$ 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 Aposhl s2aalp:gy-m t /+q8*x n3l*06lmq)wcr r rllo spacecraft put themselves into a slow rotation to distrib m*3all stx:wg*)h pqa+6lcr -nr8r2s my0/lqute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through/sdqf/h ,osxem ;*zy/ how many revz/hd *fexsqo/ms/y;,olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s./ 2i :qjdonv o9lv)li1 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 jetc:7aa j06sqz4a qkv:ys in a whirling “human centrifug aqqcy70jsa4a : 6kz:ve,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diamep6mqapd65xe9 bn,ka/ ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s.d em 6xka5an6 pqb,p9/ How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turn52) vku3 cs5gl9osdv ks 1500 revolutions before it comes to a sgk25 ok3v9l )cus5sd vtop.
(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 ls 1 foaxtd;3ih0p(*n from 95km/h to 45kf(1x pd;o s*lti3h 0nam/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revgf t0 *pzw3jc4h8qi4bolutions as the car reduces its speed uniformly from 95km/h to 4w4p*hg zq3 bjti408fc5km/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 weightqe :3bmqo wi)6 on each pedal when climbing a hill. The pedals rotate in a c):o iwmeqb3q6 ircle 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 5bk/dcjric kqn148-/d 9 x.w qk5 N on the end of a door 74 cm wide. What is the magnitude of the torque if the fodiqrdw/. /jk4xc8knc k-19q brce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39.ao0- 9ip* gtp8 esy;q9ey k k)6rcab)o Assume that a frictiobpgkqy y*a 0p69;roo e )ati)c89ke-s n torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a 0zs(w xws;qx5ttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculat xws(ws0x5 z;qe the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requir;ajhb3a;tpx+ivb +1 pe tightening to a torque of 38p+ ;vapx at bh;1b3+ji $m \cdot N$ 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-kgs8l*b; uzjsn * sphere of radius 0.648 m when the axis of rotation is through isn *ulzj* 8;sbts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.75uw8a9::lq9mp in5kokh ocz : cm in diameter. The rim and tire have a combined mass of 1.255omo:li9c8nk5k hwp: uz9:qa kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, li/rnjg . n.;jrsp+ plzc3fx xsq6:mk: 9ght rod is rotated in a horizontal circle of radius 1.2 m. Calcul n+cmx/sk3 p.p s;xrf6:n9q :rlg zjj.ate
(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 o1d s+t95cqb yfn a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N 9sd 1 cb+y5qtftotal.
(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 poin fq x (aotkif;a2pvv4io. :2q6t objects shown in Fig. 8–43 abou:t xv2io 4qa.(o6i f;f2vqpk at
(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 atol,+7mfrc qm90dyq9na ms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, endz9w5vuld*) pbeh ly )8j1rg6gineers fire four tangential rockets as show58* ug)jdh9lp r v6eyzwd)1bln 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 with a radius of 8.50 cm and a mass of c.* ,k7wdv . lrr):agjl0tdzq0.580d lwj.ag r t,z)cvlkd07r.:q* kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swingstf5rd1:clx:m/ t2q e 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 tangentialn pz8;p76 t-.fqglnsw ly on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of t-np.g86l;q wzpsf 7n760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is e cpp2+ ghre59*ilu,sf ventually brought uniformly to rest bph *f ic2r+9u,5gsel py a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerorfdu17q9c *u5 3aem9qzex h4wk6q, yates 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 action of ths oh4/0zjkvz. / xlh30hkqak2e forearm, which rotates about the elbow joint // 42x3hs .qvkal0zhkh o0j zkunder 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 bcw6ngj99 3 thin Fig. 8–4nt3wbh69 9j cg6.
(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 consistsp)5x ,gx8eyuz 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 acce0yof 9 -aouc5/sxiu;2ij j3bxlerates the hammer from rest within four full turns (revolutions) and releases it at a speec ub-; o3 /i0fa xoy2x9js5jiud 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 moytgzpi5n *ia,o dh9u) ;7h*g bment 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 develops a torqupo /-9y7 p3qe *ruxnihe 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 an 4 sfg-xtnwqmkxai6-63fg((z o1:tfsd radius 9.0 cm rolls without slipping down a la-nigs mx(f6t(fxt-fakwo g 1qz 4: 63sne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with r mpe84 wal0a)(bf -tsgespect to the Sun as the sum of two termagfwla -be ()8p0tm4s s,
(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. How 2**na x)v;qk: lvksja much net work is r *kl*a jvqv )a:sx;n2kequired 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 and mass 1.80 kg starts from rest anudz.bp3inzfu5,x(rg p5 - eh1d rolls without slipping down a 30.01p(55 .3p un b-izedzuhfrxg, $^{\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 *ehm-+ ve2v. +c )tf)gzwxzg) xkj,wqfall and its lower end does not slip. What will be the speed of the upper end of the pole just b)czz egkw) ,jtfhwx* ++x)- egv.vm2qefore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentunz 18cmi+ -c fr;wfj2ym of a 0.210-kg ball rotating on the end of a thin string wfci- 1+m 2;cynzr8jfin a circle of radius 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 uniform cylindrical grindino o i0mh6 df4zuhbnt-jm05-o1g wheel of radius 18 cmnb 5m-o oh o0f64jimdth0-u1 z 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 at his side, on a platform that is rotating at a rae2o:; -jhxtxgp 7wce:te of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation hc ow 2g:ep:exxt -;7jdecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can re fm(-8qqrm6 zbduce her moment of inertia by a f68zq(-b fqrm mactor 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 increase her spin ro pvq,z.leol2k.7a:th t) av(itation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ozlh:p)avvak. iq,. 7 (tte l2of 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 axi9lu4gxl6 i/-udwy/, m haxm g:s6ral 8s through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk /uiga664g8 lw-: hxrm am /sl,9ly duxof 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 m4aghd1 v; b6clomentum 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 :m bv 8c.7.ypa(d4ft :m exeahthe 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 cylindricapdma8l, rwqadf a71umew ;g6(,9x gbw 7k 1lr0l disk of moment of inertia I is dropped onto an identical disk rwq ,padu lf1a9gm b1wr,m6 ;gld k7wea87 0(xrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ,n8x 8v5m,x ,bjkdfjkat 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 salx9g00pg)wf tands at the center of a rotating merry-go-round platform of radius 3.0 m and moment offalw)ggx p900 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 merys)1mhp7e4zd ;3p ziiry-go-round is rotating freely with an angular velocity of 0.spd 1iyize )z4 7p;h3m8 $rad/s$ 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 eventu e5y+gc atpz))ally collapses into a white dwarf, losing about half its 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 nec)+ pga5yzte )w 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 excjj7pn /dh+ 8b 7ddbo5e:lk 8oxess 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 om 9, zd.0(zo4 ojs,cn5cow ji$ 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 plqb:xg)py8qa 8id)2pp atform, initially at rest, that can rotate freely without frictio)ppy 8gqqbdpx:8a )2 in. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands aof 2vi76q*mdpt the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of *m6iv pqfd72o 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 th:vqt:01 g( qaj-tqphne top edge of the spool. A person grabs th1v:t(n gqa0qq:hj pt-e 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 move?

  The Moon orbits the Earth such g f1ne)f925 lwlc;dan l4j)er 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 cel d 2aj)19nf);lnf e4cwl5 rgase, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ppya g0,o.y3;6xq do2v2mhabgzrin 1 d9d2( p 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 uu9z(6hl ug0qf-fq6z y9624ycqsy8a xia v yn,niform cylinder of radius 0.20 m acquires a rotational rate of from rest nu964gx qyqv8zq czsf69 yfi 6al 2ah(-yuy0, over a 6.0-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 0.050 kg anx e d06mi.1tegi0l :ord diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg gt l0ii6.:0e mo1xedrand 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 speed 6ms0f. w0 qxldof 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 timeqd5:vs;9+ u*ft:qinm efo i;ds as great, were rotating at a speed of 1.0 revolution every 12 days. If iu;fv+ iq*nemd o9iq t:;d:fs5 t were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ener l boe a ovkj;-5;-l-yd8sysp0gy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of dalove0okpld8j yb ;- 5y ;s--siameter 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 an8/jr6h e+c:o p*wxkxq $H_2O$ 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 horizsk7md:u j e.x m200 zorpk)l4hontal 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 ignore frictifi4 c--9mqbad on) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizont9f -iqcb m-4adally 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 st6:0uczshy9l c vb3 d2wanding vertically on the floor, and we want to exert a horizontal force F at its axle so that iyc busz9v l:wdh062 3ct will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4h*-ejtg hg13b.2m/s on a flat road is making a turn with a rtg*b e j-hhg31adius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling ofj6/ mybi3ub.yio/ ,iwvr i7(h 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 torhv,jb. /ii yiow y6/ u(73ibm achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as ahj2h 2 p.53)fh ltctrn solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched armstf2p h2h)n .j3rhc lt5, and with arms held tightly against her body.   

  You are designing a clutch assembly which consisx:n d3p*- p .lliwpqx0ts of two cylindrical plates, of i n l.wqpxl0p*pxd 3:-mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and rad 87tiib5j7c*ujs pj-wius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the trapw *78i5j js u-tc7jibck? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but 5co*omy.ubu + 6,*ju zyf. zc7dj oig)do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car 1a:ret8s9j .eeq6x it reduces its speed uniformly from 90km/h to 60km/h Th9ta86jxeq .1sei ret :e 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