A bicycle odometer (which measures distance trav6r*j pi /r.cwlmgij83f :ef/rq v*nn9n7f-i b eled) is attached near the wheel hub ag89/i nv:*i6 fff rqnebn.-ipwjl3jr *c/r7mnd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constan n ozb34+vab.d-qxeo p /m646pt7 cteuw;e*mnt angular velocity. Does a point on the rim have radial m4e ;b-/ 47tn m*oqt 6cp+w.nu3oeabvdepx z 6and/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 acceleration change?

Could a nonrigid body be descriu zwzx+ ld*q4+ n uytvy:;:u-ybed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.w1s kla dl8s18d//fkn

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your head than dl9jb;,m-k- p 68 jvx3bk.qssfao3ic when your arms are stretched out in front of you? A diagr9,3 vbjo iskdc6p .-3l-axk q;jbf8ms am may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wh x,8 tnlim;u4aeel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprockxul;na,4m8 t iet or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender zi+tr 9qau:8 wgn( (jdlower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thiszjg(tqu:d( 9w nra +i8 distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longnc p.4s np,ox3, narrow beam?

If the net force on a system is zero, is the net torque also zero? t.q, j0 dhwtt6+iymkmozpj)d 19+l y *If the net torque on a system is zero, is the0tjod1kwq6+t,*z .9 +py)mmh j idlyt net force zero?

Two inclines have the same height but make different angles with the hori j- n ozc9qytq4t/z(d7zontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bn9c(j4qtzot 7yq- d/ zottom be greater? Explain.

Two solid spheres simultaneously start rolling (from restgwvtd q9 3ty q7l9+u-tpfuuh*z.m:k 4 .x sy-k) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which hauk-3:qm9fts uy.lp ygtdqt+7* wk4 vz-h9.xus the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius as 6s,sg3 d(npfnd the same mass. They start from rest at the top of an incline. Which rsf3d ,sn (gsp6eaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum ahnbb7rtnj)s/l6 u9 .w,vhv)n nd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.9n,tr 7bhu/sn. j))nbhwvv6 l

If there were a great mivl z5yz+46 55y:vxnqy 7x dircgration of people toward the Earth’s equator, how would this affecnyq:v5v z+xlyrx6zd 75i4 c5yt the length of the day?

Can the diver of Fig. 8–29 v; vc6u ol5j z,wc1y2odo a somersault without having any initial rotation when she leaves; zcv6ol w 51u,vcjyo2 the board?

The moment of inertia of a rotating solid xj9pa; o(0c b*i sjl0cdisk about an axis through its center of *po(;iajj0s x0cblc9 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 rotatj,g96cc xqc0 wing stool holding a 2-kg mass in each outstretched hand. If yojq g60w ,ccc9xu 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,w(1g) 5 oruhh/g)ancf9x m(hp:z - xta one is hollow and the other is 1: axg(mhg-p/h n 59uxr)w az t(fhc)osolid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Iq qqy)juj)-ixtew u;4 d 0m*.jn what direction is the linear velocity odi-qq0 u*; )4xmujetqj yj ).wf 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 o ,9-.9)5;nttdw ol ym jqmcqmlarge freely rotating turntable. What happens if you walk toward the center? t-oqmy99cn,;q.ltmw) dmo j 5

A shortstop may leap into the air to catch a ball and throw it quickly. As h10a0j-iw( 783kffzi:r qvr xxz gpl.de throws the ball, the upper part of his body rotates. If you look quicklyd.0wixzafk0l z:8 v1 rrx qjpi (f3g7- 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, discq.b vzea52r8jhf: n+ buss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the heliconbf h 5zbv2.:8 ejqra+pter stable.

  Express the following angles in radians: (a) 30vo jq 6ucml+11 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculat5/1kej)vaj ylr*rib; yglf- 2jy 7o k(e, using the information insi2gj- yy*krf ke5)ai;l(yov l 7j/1rjbde 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 ag/ 0,qz 34rpyty xyv0bt the Moon, 380,000 km from Earth. The beam diverges at an anrb,vg0y4q/py 0yz t x3gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Whenwc ,w,t8+uuef the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration a,we,+u cuwt8f s the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball ma 5ii 68 9tki;iqtg3vigwg,6ckv .tv q/kes 15.0 revolutions, whagi.t; it 3kt/k ,g986icqiv5qgwiv6 vt is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do tns:h8kabd *- bhb ab8dsn:hk*-e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates aticgo 3 nx:u v/vm(2ta5 2500 rpm. Calculate its angular velocity invgot5cxin : m/ u32v(a $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolutiozkr: x4umb8g+i *qh i:x2dz v(wa/c)on in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m 4xgi)o*abq:zv:/xz 8 mdw u2h(+ki rcfrom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earte3 jqo;l8fk9a u-h 3bkmyui63 h (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe( otnqqzq0c+z: /u .ediv (ta5ed 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: kw0b gxwpco4i0 bfh2o,k p9) cm from the axis of rotation is to exph:bwpk04c0 wx2b ,pf) oi ok9gerience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abouthl w.oie2- e:3nwn/em its center from 130 rpm to 280 rpm inwn.m3 oe-n h ee:i2lw/ 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 radiug*wn5 jqf8c9i5:z*tgih q5 fys $R_1$ 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 them+cbxcm(tv a )q)fbez sk*: s2p3x0oc1selves into a slow rotation tosb1(x q mz:0bck2 cv)e)o+3*cf xtaps distribute 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 auod(p7u+jt* q* 7m0: wojzi cto 15,000 rpm in 220 s. Through how many revolutions didaij7t +**u0 qum zoj(p:7cowd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ce( te(*f bpmaz3 4f*ukalculate
(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 tested3).bexqjw .i yz4bxl 7 for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throughi)bxzyql. 4.xw jbe 73 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 inky. 7 2pkyj8xn8)ke jl 6.5 s. How far will a poin. njeklk8 87jp)2kyxyt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runng + 7oykfu9sm/7jwll ,ing at 850rev/min It turns 1500 revolus7f 7jl,w+ omgkl/ 9uytions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reducesk33pgd8z oh wn5c d dv;i:w/ j459bznf its speed uniformly from 95km/h to 45km/h The3h d/pd3:on 4dbwjfg 8vc9nzz5i5; wk tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed19xje ls pxbxd(17ngfga( e5 8 uniformly from 95km/h to 45km/h The tires have a diameter of 0.8 b(pgng19dsje (x517f8xel ax0 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 +dxw36rrgc)g puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius w6rxd)g gcr3+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 th,s 6i, bo2aezwe magnitude of the torque if the foab 2esz wo,,6irce 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 thbz) t+k3y 9d ,0flshoye axle of the wheel shown in Fig. 8–39. Assume that0y)zd39,tybkfls h o+ a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod w uphr:9zt2/rahich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on 2tz rrhau:/p9 this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38y 3d02p ;e(fpw beonul,q he83e;u 82dpp qnywh e3felb(,o0 3 $m \cdot N$ 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 inertibf,c5gv7abur g9 :7lbfxd7w 5a of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through 5 avu:9wl 7cdxbbfr, g7f5b7g its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheeugpp /:un*qu 2l 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ig uuqn: up2/pg*nored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a :rvpls;jah ;2ap 3 -ubhorizontal circle of radius 1.2 m. Cal ju:br2lv; aap3s; -hpculate
(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 rotating2kr/ b/h3d ngcomvbk3rma+8, at constant angular speed (Fig. 8–42). kv8d+ra bn2r/gbm/c,3 hm 3ko The 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 poydn2m 55akerc:qo 3+ cint objects shown in Fig. 8–43 y5m3q5d k:c 2 recoa+nabout
(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 oxygoo-yo 1m*dfhlh-0j, ten 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 spinning at0x dkwixso zr 2b7kv;z3tdg ul**7;c ( the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a x g;oux*sltwzi277krzv*3; dkbd 0 c( 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 m)wn q4l,udx8k83r kg v1dfg(lass of 0.580 kg. Cdq4w 83k1g,)rnlfkl g 8v(du xalculate
(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 swifg3rrlcbk()z j: d, .0z2:vf /bvkaiyngs 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-driveax31 gtzw*m04t*/ ulne8bvrln 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 760 kg,gax1t8/*3 rwv0*lz tneum4bl 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 at5gq z-qpb i0u*ioch -2 10,300 rpm is shut off and is eventually brought uniformly to pg5iz b0o c-q -u2q*hirest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelm5aa bhn- +nq4+z gdx *teya((4e gd6nerates 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 i0kwab x:ju zyva a-zo- ht-+k6f l8:k;s thrown solely by the action of the forearm, which rota+tzao l-h kk w x0-fz:vk-8ua;ay6:jbtes about the elbow joint under 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 vt k c1 s0m4*)u qi+miuwgbd.9shown in Fig. 8–46tqg s.41 u kd+9*u)0mw ivbmic.
(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 .lmkt uzd(-u.sists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from re/s /irp+4,m36;h altsr rmou bst within four full turns (revolutions) and releases it at a bsrt +3r6s ur; hl,m4/moai p/speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inerz ixtt0 1yaq0m6(06 dfe b ,1m-flofowtia 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 torque ofm9/1 to kleq7k 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mazzi 6ec.5d60nf 2p e ,p/qsvimss 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3 dn.,f 0e6czsp/5m6evz ii2 pq.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resistbb+a3za5c-jg + )spect to the Sun as the sum of two+sa+gtbai j zc35-s b) 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 7.50 iwnry ) ;ja1my. (u*vxm. How much net work *)nuayx ;(yi v.wj1rmis required 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 2s k8ool- seutp;0m)4i0.0 cm and mass 1.80 kg starts from rest and rolls without slippskm- 4ost)l8i0o e u;ping 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 vertif -wv(2occqo;cally on its tip. It starts to fall and its lower end does not slip. What will be the speed ofcw-qv 2;fcoo( the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en dd nk*+q0vl0rd of a thin string in a circle ofq0 nl *v+rkdd0 radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of v *da/ 4zw:xo+xu7cpla 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1xxc7 :d 4zupao*+w l/v500 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 side1dj p -evgo+: ox3njg2, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizo2 vd1:jg e3go-xjnp+ ontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce 2 numu3 x3 (:b7p;u fdr:kmwflb:6hojher moment of inertia by f7fmhju3;m3( pd x6: bwlo2:rnu :ukba factor 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 hev:3jywswkj8 hh7 iq + v,p7it6r spin rotation rate from an initial rate of 1.0 rev every 2.0 7ij3p+hsv7w :qwt hi8 kv6j,ys to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center aj7 z ;17zjtxh*m */stwrmab:z t a frequency of 1.5rev/s The wheel can be considered a uniform disk of mas* tsb/*x h:mtaj ;7rjzz1z7mws 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum x ;o1c3wb df1pof 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 momenj-xl6nl yy*( ltum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an /an(6,zsms r/tk bit9identical disk rni,t/akz6s9 /rst(bmotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at nz * vuids* i5kb2pd-*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: safqw v)l 20if;vl)e of a rotating merry-go-round platform of radius 3.0 m and moment of in ff l 2v:)vi;q0as)wleertia 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 velocity of 6m3:ocpq ovgwye6 f *+0 vyw6c 3 g+:fqmo6o*pe.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 collapse /v n h;bfuw*tv;0h7w.ecki *2vb pd*hs into a white dwarf, losing about half its mass in the process, ;w.vw fe*0b7k* npc;u b2 thv*dvih/hand 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 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 excx*9w0;h vt/v- h o,mo6ccdrj dess 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 xwc)5l h(gi,w $ 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 freely withr0iozfyw s, kh /we+0og(btv0sh8q-) out friction. The moment of inertia y()ft0gris8kz+q bo0 ehwws o ,h/0v-of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at ;(dhjh-smsduz:pu3s 9es4 , othe edge of a 6.5-m diameter merry-go-round turntable that is mounted on fricti,4 u;zdo3 mhu-j:e (sds s9shponless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on the yo//h sf: zbrzkc46 m36b3ihatop 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 do hs4o/hb3f:a3 k z 6ry6icm/bzes the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Determj1nhy41vv:gp )0plc4+v ognl e9kr0rlr pt50ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital rvpk19ne nl yt1jr:5o0rp pvlv0hg0gc)4 +l4angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ofju,*xx;iueiwo*n v ; , 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 -vc*nfj (h jr;fux9qh+u;3 yu uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliver;j-;* f(nfyhx39q u urh+j ucved by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cxc0r3yq w8j 7zj7 v)- ided w-nd*q)ybylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (conqv ijw7 y0je-7zdq d3r)bcydw8* )x-ncentric) 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 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 of the rearxive3s) ydle3 e468cv 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,n5g,i*rml ( nj but with mass 8.0 times as great, were rotating* nj (irnl5m,g 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 times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autfam4 8- wa 4k6voi82fv x 1xpmr(-fjkbomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and 18afp4vkb-f 6 mk xa(j-4f82oix wvm rmass 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 /gcvudv 0g1 6han $H_2O$ 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 horiv7z )vqis( )buzontal 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 0gtutmh (sv2 ge;v.8hand length L can pivot freely (i.e., 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 the force of gravity acts at the center of mass of the rod, av(t ggvh 2.e8uhms0; ts shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticz:aj8faevs b8vp3*j 6ally on the floor, and we want to exert a horizontal force F at its axle so that it wiv 8v8z6a3 :*jsjpf eball climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn witynqka*d 8x8r 3h a radius The forces acting o8ay 8xq*rd3kn n 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 slind5(1p avqy6zw g of length 1.5 m and begins whirling the rock in a nearly horizontal circle abqv1 yw z6ap5d(ove his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin ro(dms-w kxwavn bn.q a*;*2t/g ds, making reasonable estimates for the dimensions. Then calculate the ratio of the angular spgaw*k m;dn*bwtnx/as( q.2v- eeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutchwx:u )x3y spri,tar l,y 2qp/.j*s4plbc6z m 8 assembly which consists of two cylindrical plates, of masstm/ rr.c pyq,wy4s,z62pp aj:3lxu )sl x8i*b $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t) bylj* q):6mvl dqyj8lqm562zz tju;rack of Fig. 8–58. What is the minivj jm :d65z;2j)* qblq)m8ylqz yu6 tlmum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n y:pjp3wynbe4zkx: 84 ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly qmoim73*coe. 9v 5 odofrom 90km/h to 60km/h The tires have a diamet5 vdc. 3iemqo*9m oo7oer 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