A bicycle odometer (which measures distance traveled) is *q,jnj8w wd c)attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inc c qwjn*8),jwdh wheels?

Suppose a disk rotates at consta3 6r re8ep9 fv9bh5t2)uvsoct 6/6v wuml h1zint angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have rrth8s216ve uvm6uz9f r p tb)ol5/ehic9v6w 3adial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid bodrb fp30l9;vwv y be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than 9na ziv0d82gt a larger force? Explain.

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

Why is it more difficult lcqd2t 0i1i23rn aja6to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A d32j 1nca62ald0 rt qiiiagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the reah+r4w*f x+ bprr 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? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? 4+*h+ fwr rxbpWhy?

Mammals that depend on be*l 2eac02yix wlq0)h/ 1x tjejing able to run fast have slender lower legs with flesh and muscle co2 *etlxqxa e/clj0ih2 jw)1y 0ncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carr bz jbdqjdn j3(p0i3fo* (vj)7y a long, narrow beam?

If the net force on a system is zero, i u+s t);/rih4tt,lzy+t 5 mhyds the net torque also zero? If the net torque on a system is zert)yd+smrthlz+ u,54th i yt /;o, is the net force zero?

Two inclines have the same hez4;+rzh6v ntps b-yo xsk; n3+ight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline willp 3x+znhs;+o 6svtn;kb -4zyr the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouslg qh7d8k1i o e /w4zvk2xo+,q1cw9l nno+n4kj y start rolling (from rest) 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 has the greate/ cnwv qn14io qe8gw4nlk9o ,1okkj2h7+x+dz r total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sa qzf,t3098u7gppg tap me mass. They start from rest at the top of an incline. Which reaches thepp9z, t8qa30t g ug7fp 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 and angular momentum are conserved. Yet most moving or ro /joj4q qx15o1ucpd uq6n1e7p tating objec q6uquqej4 p/o1d5ox1cpn7j1ts eventually slow down and stop. Explain.

If there were a great migratio0zg fidxjx-tq*a7b 2 7n of people toward the Earth’s equator, how would this7 2qxt fia-x7d*bgzj0 affect the length of the day?

Can the diver of Fig. 8–29 do a somersault withoz y8j1f8ni9+phxg .hput having any initial rotation when shgj 8 9pyhfnih+z .8x1pe leaves the board?

The moment of inertia oflmnt-dxsp4 /n5 ;fayr/0 bm7u a rotating solid disk about an axis through its center o5 0mx/lf p7n -rn/ tum;d4yabsf 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(.l5 b7 9acar 2nzqryo on a rotating stool holding a 2-kg mass in each outstret yaqr.rczb9 72no (l5ached hand. If you 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 is holl.eb*l (x/d 8-jrdo;uohak1zg ow and the other is solid. Describe an experiment to determine which is which. zk /o j*.b( xe8uhrod-dla;g1

The angular velocity of a wheel rotating on a horizontal axle points west. a l ;orbj4yc:7h,u9inuhe5e-In what direction is the linear velociu;e, 4ach:ey - irl h9nj7oub5ty of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stand lzsk*3tq/t ;j+zyi)ming on the edge of a large freely rotating turntable. What happens if you walk toward the cent3zzj+; tk is/lm* )tyqer?

A shortstop may leap into the air to catch a ball and throw it quickly. 8 . dxxnwk(h:e2qx rall1n8/kAs he throws the ball, the upper part of his body rotates. If you look quickly you will notice th 8 x.h:8lnk da/x( ekx2qlrw1nat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lae-pg0 wh ie1x 7gs,ix3w of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more way g1x,3es pegii0-x7hws the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30ky6be t oq)knqx2u; 05,4kydb;.t alk $^{\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 bey,jc syeh8:u+ kt)bi :cw;5ji p8 3tnccause of an amazing coincidence. Calculate, using the information inside the Front Cover, the :y5 :yth)8ii,p se+3c8b ;kujtccj nwangular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directen qtk i4zj3-pyb9l5yjyz 9+6d d at the Moon, 380,000 km from Earth. The beam diverges at anijpjz5k ytby9 -n 4 y+6zdl39q 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 rotate at a rate of 6500 rpm.e)sg,4mzlz .*+ :9fduk 7k6aveqywk o When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bl)ym lqk.f:ke*,zw6 va+s 7u4dz9 ke ogades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to ansoz/g et4)uj - *sikhd +k(-jgother child. If the ball makes 15.0 revolutions, what is it+*ksgs - -) u ji(dgj4o/hzkets diameter?    m

  A bicycle with tires 68 cm in dikt:y ;+r zw8di il 8zi0wt9)dxameter travels 8.0 km. How many revolutions do the wheels mi9l8txdi: r+ ;iky8z)zw0 wd take?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotate 7 skfick3c1j8s at 2500 rpm. Calculate its angular velocit j1ikck8c7fs3 y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes oneqsvh0-/g4tr m 3p9mia complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the ceprg0 am4t9qmh/vi 3s-nter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular um)6ikmq )1y 8 heitwfz;rc:.velocity 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 ofrc-o v)1 ,h v)-.)biuljp 9uaxnau2ki 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 from the axis ofzz3oo4 f1ta ;4r;rz4dslvio/1snp( p rotation is to e;3 saio;tl4z/p s1vnrf4zod1( zo4pr xperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheeqgpdn 1z .a 0-h x1fs.8ne1bhpl accelerates uniformly about its center from 130 rpm to 280 rpm1szx10g81hnd. - p n a.bphfeq in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuv69bcevd l0)3 nvxi2ws $R_1$ 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 abk+0h n: 3l61wd( :zxkjvmajqxoard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the hljavmn:63q1wz k+0(d kjx:x 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 )d 01oh+if lww15,000 rpm in 220 s. Through how many r+wh)l0wdf o1ievolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120*p jo6fey-yl*y s1i4f;, zgcs(rcne) 0 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whix wang2*8 dxb,rling “human centrifuge,” which takes 1.0 min to turn through 20aw82,xx gndb * 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 r tob 4zh5sw2frr652mlpm in 6.5 s. How far will a point on the ed2 45r5orf2ztbsmhlw 6ge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runnw7f dnbm(;sbqs(6( ueing at 850rev/min It turns 1500 revolut n(um ;de bwfs(b7sq6(ions 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 rs(feuslj;c )z 2 *ds:t ko7uq,educes its speed uniformly from 95km/h to 45km/h The tires have a diameter ;ssf7 *2 dk)ulu zseq:(t, jcoof 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 speed uniformly f6fp c ( pvc,q-.5gkoplrom 95km/h to 45km/h The tirqp-c5.( 6,kl o fpcpgves 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 weigbf2;olrh vies*z) 870efm* 9qob,qkc ht on each pedal when climbing a hill. The pedals rotate in a circle of fkmef* rbe8ql0 hq7v o*;z,2cbo i)s9radius 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 0 j00+clq) btyup2mula door 74 cm wide. What is the magnitude of the torque if the uylb0j l0q0)tm 2c+pu force 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 Fhg/3r hi/8g;toji ;acf39zx eig. 8–39. Assume that a friction torque of 3;/oehgt3g8 ri9 cjafxh; /zi0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to therpeu cxq1s +1v6w6 r5f ends of a massless rod which pivots as show+1rfe 1s p 6wxrq65uvcn 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 this system.

  The bolts on the cylinder head of an engine require tightene+tjtj+(pi gto9gf(x 5*krj4qx ii .0 ing to a torque of 38 i pj+.k(*t9xoj x0jggf it 4e(5+qt ir$m \cdot N$ 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,edk.4oje )e33tq sjl+qmjz: -kg sphere of radius 0.648 m when the axis of z3k 4jq3) jq:. meosdel+,ejtrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm i8urf m.yjm( k:hf8* qd0a yzn1n diameter. The rim and tire have a combined mass of 1.25 kgj(yz*0kyqm18fnmh d:8ar. u f. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin1*j xvf52 eja *ukzl(smy(qmf 2; 7hal, light rod is rotated in a horizontal circle of rzmls *(*( ye1xqf2 a huf;ak5j27 vmjladius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’sznxf 1+j 43i. l3dovw)yhx q:x wheel rotating at constant angular speed (Fig. 8–42). The friction force between hex.doqnw fxl)x3 vzj3hi 4y+1:r 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 oh/pvv,j nt(oda4 wemeyn 6gz/,r9p/ -f inertia of the array of point objects shown in Fig. 8–43 /vjy(rem9ohga,ep - //nv,pn4zwd6t about
(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 oxygenaz35t -c em/-ypgnr)r 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 spi;7lbh -gjd)w fc)a1r hnning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is t) )l;hbdrw71 fa- gjchhe 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 ofnphblsn,,s neea5 up3 :(35;zhay7xs 8.50 cm and a mass of 0.580 kg. Calculaten5,3nhx:slzpy,e5sa s ;h7 npab( 3ue
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from reste8 6s):n 8qt,nyjyvn z to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is abl /. :zinpynx3an me.o5e 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 hya5nzn.men.x/: p 3 ioas a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating axl5cw h,7ehsh3tb2rg0x,o(t r up/,qt 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque sxgh ,0x,(cuw7p,h/2eo3t lrqh5 trbof 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg z :yt5ak*3pcn svf 3(+0 jtbxcball 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 ack y)+wgs0 :f9:l haibyr- m*aetion of the forearm, which rotates about the elbow joint under the ac ikar * )ys+0f:eamw-l:9hygbtion 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 lons (b)b s.flii*g)6 weg2ddbp0 g thin rod, as shown in Fig. 8–4d0 i .gesbgblwf)* 2s6pdb(i)6.
(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 consists of two masse:s6d) b xle6oa vmq)3es, $m_1$ 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 hamme. umd pmo2*tg4r from rest within four full turns (revolutions) and releases o* mutgpm2.4 dit at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hbzjtm n+/p h9is7f,*zas a moment 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 torque of1:+myahcd y 0w 9fc2vp 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9r;db6h.f h7hs.6 +j c,thvk bl.0 cm rolls without slipping down a lane lbj,s btcvk7d. h6 ;.r6hhf+hat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suakzl0v(r u *( 2swp9psn as the sum of two teraspu02v l r(*(p9ws zkms,
(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 m w;6 hbwdep6;uled 79uxp et4a0n1q(gass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume itpdq0e1 a967 ; h64x ;ebwnuue(gwdlpt is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from r 8ikpzrfymb) r,w xdi;7l+f :txv8g/,est and rolls without slipping down:ir+x bviwf r7,p,d8;gl f t/zmxk8y) a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and ipcby(*x,-1511 aw3 fqnmj;qh ehz.gje dc c-rts lower end does not slip. What will be thf q(h*a511r-hnycj 1.p e-qgwbz3jcm xdec , ;e speed of 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-k9.l s ht3/abvz7 g5l +gblkj/ig ball rotating on the end of a thin string in a circle of rtls/.zihvagbj b 73kl+ g5/9 ladius 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 +zjnu ,i.cgverk6g (0 grinding wheel of radius 18 cm when rotating at 1500 rp.ce uz 06,ikgrj v(gn+m?    $kg \cdot m^2$
(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 rate of7qd9 1ih fz8rv 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotati9h firv1dqz87 on 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 her / n5akx6 l2owyfkjeogc48 8o ;moment of inertia by a factor of about 3.5 when changokljwn4/5 x2feac o86k; o8yging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an 2me (9w1scd+q:win3 ds 5s ixiinitial rate of 1.0 rev every 2.0 s to d(i+ssx3sc1 new 92:5iim qwd 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 verticze9di5m -i c1-l1rqfxal axis through 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 disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clayi mx5-cd1fq-e z r91il sticks to it?    $rev/s$

  (a) What is the angular momentum 0z4 r ljqecy*.(rn;dn 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 momentupru* -pe 7pmfc.5q7kazi(c *ym 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 d.x:3h81 /z :ig+1ko6rle)ppxp jo yg j-owtidropped onto an identical disk rotating at apj1 d.liog )x6z-o y3k :jioh1pxr/ t:w8p ge+ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 k:5vkmj.hip *$rev/s$ 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 rot23mxp j gj59nqating merry-go-round platform of radius 3.0 m m9px2jq j3n5 gand 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 rotatqw5 iu:b(4;ks j qfjiz529w zr tlqe80ing freely with an angular velocity of 0.8k;f4rl 0iqt(98wbe55jqwjsu: i zqz2 $rad/s$ 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 6k l+a+40 frl qtj.h*a/xowvfdwarf, losing about half its mass in the process, and wind+fw0 vokhax */f 4 ajltr.+q6ling 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 irxkdn - (0p+l7: jm -8rlejhoxn 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 g*7cjd- -tpdh6bzh ;o5tnoa, $ 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 pl8l4g4dcyq ,6+ubnzig5 zd 9t catform, initially at rest, that can rotate freely without friction. The moment of inertia of the person p b5dzz64qyc,g tdci9u g8+ n4llus 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 sta(j:w7bg lsv6wm61w, p f4fmjaw wl- k/nds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of 1 kgwlwj ( v:6 l6mbwajpwfm ,7swf4-/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 en wq/, + p9zcds*au g0n4v:gf6hd4lbfad of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person withoa /f+,h04puw z4snb*g:q 9 6ddv lcgafut slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. D9 2 tdlyk7d4l,uej4 mtetermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon atdey749 tkm,ul d 2jl4s a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/gn1uh* .9ln2t .zowbdg eo/f)$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acuxb +f-1-gd uqqfc5thwvg-0l)bt -z6 quires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliverecf- l-hq1-w b vugt5xd+-tb 6fu) qzg0d by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eac:ty j gdv+e52hh of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use consd tj+ 5:ey2ghvervation 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 rear wheel (e ds9-)qk)q l -m-zmzv$\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were 52 a+ypu6 tjx y6rx8gq:7xbn orotating at a speed of 1.0 revolution eveq8j ypx+n6yxu:trxga 5b76o 2ry 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 l sy0ec reu.-5oow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car 5ye.e-oru0sc has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 illustrate tevh d)rzcws+h*/wb(f31o2 os an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v bhq xcbc;po *+q5rp/iy,9dqv0br0c8 e 2tiz 0=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 pi 4p3-( nsr-dqtpvid p9vot 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(-n dp34ip d-9 rvpqts mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M haswf/ian-lj2 26fe hw*e radius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its el ehnf wf2-6 2*ija/waxle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speez8+fb6 - 4hp;adwxxq)npz x 5od v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal x )5zx-whp4 a fz86qb+no;dxp 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 1.5 m and begins whirlidwfo v)39ehi*a g30 j:,)kymcoljj:x ng the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm ajwjm03o:ey )d*fhgv,k):3ox lc ija9fter 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 s (jli.jl.,y3xza v* igolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spin.3ja zii yvxgj(l,*. lning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists oh;a-fcpny -2e f two cylindrical plates, of m-pye-; 2fa hncass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radv6z .5vpjt vgwg3oqwaa 8;5j4ius 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 j;5opqw.w4gv8vjav6 3z5g atleaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumem :l5 jai t(h-6r ra-npb0/res $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces itsztpqks2;/ 9,kg u1 ;l nlhm-sp speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. , skm1s2p;n kt p9uzq-gh/l;l(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