A bicycle odometer (which e 403*+ s+ sik5sffvslj. qvrsmeasures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on ssj issr4l k+5v*sf.+0qv3 ef a bicycle with 24-inch wheels?

Suppose a disk rotates a tjql)( s95 xzpv2 md3qgjli++t constant 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 radial and/or tangential acceleration? For which psqdm5) j i3( +gzqjll+v2tx9cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descritjg5 -ha7j u ptw0ld7l:n3jw7 bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lg+qjdc*kd 53:1q2xts -xdzu-a ,rowf arger 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 to do a sit-up with your hands behind your 2t9)vp ch) sn pl6o0pn7+l4yqlbzq d 9head than when your arms are stretched out in front of you? A diagram may help you top +pq6s)td 0l7 lz2co9lvn)9y p4hn bq answer this.

A 21-speed bicycle has seven sprocke slm2 *fei(zyn:xtxs-,ge7*y ts 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? In which gear is it harder to pedal, a small front sprocket or a large front spro2e-y*ge7,sm s tlyxni *zfx(:cket? Why?

Mammals that depend on being abw3haw;/d bw-x le to run fast have slender lower legs with flesh and muscle concend a /bw3x-h;wwtrated 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) carry a long, na) c.yo wr1pee,rrow beam?

If the net force on a system is ze/ aq38kz ,,zd3ihwadr ro, is the net torque also zero? If the net torque on a system is zero, is the adh i kzdwa3zq8,3/r,net force zero?

Two inclines have the same height but make different angles whm9e m85b)asuith the horizontal. The same steel ba)huem 8m5sa 9bll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultanedj ee (5+8ymdnously start rolling (from rest) down an incline. One sphere has twice the radius and twice the 8 n5medjd(ey+mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They st(pgc6vm;1z sw pb 0pj,art from rest at the top of an incline. Which reaches the bottom first? Which has the ,cbs6z jg mv0wp;p(p1greater 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 movbl c4y t(ckpt1op7y -q9s6fq; jyw;-ning or rotating objects eventual(y-kqyl o9t p 4;w- c;j fbtcpny6q17sly slow down and stop. Explain.

If there were a great mwl.o6 4s acg;oigration of people toward the Earth’s equator, how would this affect the length of the 4 .wo6 o;aslgcday?

Can the diver of Fig. 8–29 do a somersault wivii8s9ksh5.u t+6dw1r.zm g wthout having any initial rotation wheud wsw.rm16t+g5 h9zsi8k vi.n she leaves the board?

The moment of inertiap 7ccak:eq u0zr;h30y0vg +ac9 dg4j r of a rotating solid disk about an axis through its center of mass is de0j3zc7;gy k+0rcr:q a49v0 g pucha$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in eacs ybm795:k1mjd o p3bbh outstretched hand. If you suddenly drop the masses, will your angular velocity increas31dkp j5mbsy: 9mbbo7 e, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeve-zhb0/w lrscnf + f 891zzdp:/ug)bkar, one is hollow and the other is solid. Describe an experiment to determine whi )klazwp/z cz9f8 ug0b+:bnsf/1 d-hrch is which.

The angular velocity of a wheel rotating on a horizon /z4/:wzt rf+4yhyqu ytal axle points west. In what direction is the lineqztf/uw +r/zhy44:yy ar velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largri*o.4zxhf )b5l yn i;e freely rotating turntable. What happens if you walk toward thx5z.i*n or ylh i;4)bfe center?

A shortstop may leap into the air to catch a ball an91lfc6 4tq1tkm lwn.bd throw it quickly. As he throws the ball, theqnt 9ctb.lf6 k4mw1l 1 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 angula+jmijm6+;jj hvhbmy4- m9l (w:.wrmp r momentum, discuss why a helicopwi:h v .;-jlr+m 6y( jwjbmjmm9hm+ 4pter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radia o/+yy2s3lsysbl p (2dns: (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 amaziau/c 7z;dt ( ,kt6qydzng coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as/atck7uyd d zt zq,(;6 seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direct 2oa3xqg j8 yayj;a(b-ed at the Moon, 380,000 km from Earth. The beam diverges at an 3qx(y8 ;oyjg2ba -jaaangle $\theta$ (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. When the motor is tu4v3cemvg8edw 93ubbolpb8 0 3rned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow du8v b3gb8lbc3ev o w49dmp3 e0own?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Ifp(zcdu s6qn-peh8,h 87:dwmg,iv1t s the ball makes 15.-:dhmv 1u,pss6n7zcp(e d8h ,w8 gtqi0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho2c fm5gp0fx*cw many revolutions do t *0xf2gfm5c pche wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atk6 crjv;u.hv 5 2500 rpm. Calculate its angular velocity in 5.u6vh;rc vjk $rad/s$ $\omega$ =    $rad/sec$
(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 revolution in 4.0 s (1 .6 feep.fyakFig. 8–38). (a) What is th.a pk1 6efefy.e 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 of the Earth (a) in its orbit around rmr4t nd2pgu-49a 9a rthe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o8zlt.hyi9iyda a5)f 0ssc9e1f 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 centri* 1 -pn,yza43pj j)pwn z)xxvufuge rotate if a particle 7.0 cm from the axis of rotation is to experience an azxuanw))pjjxpv pn *,z4 y13-cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a4p(* 5ost 4t/gse ( ykumzwl3nbout its center from 130 rpm to 280 rpm in (mgtps3os nwl 5/u*4tz yek(44.0 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 radiu*fj4i(5ob3 emmgyf4 ds $R_1$ 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 aboam,zqkuqvq88ht,jwg9 ,ppcq is 6d3 04rd the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 pjdh,p 0w3 qukv q9 cm 88,6qszgit,q4m. 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 hdi+:iz ,ya8;ykxdi g( /heps,ow many revolutioigs;y8ad d zh,:,(p/yx+ke i ins did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal ct dh60sahonwli o e0(9e+5d0culate
(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 e mi renu- 4dy3qm27 y7.6dljga whirling “humu3m iy2e-gq4. 7dyerm76nld jan centrifuge,” 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 diameter accelerates uniformly from 240 rpm to 360 rpfi ;kvpv j89p(p f)yt2m in 6.5 s. How far will a point on the edge of the wheel have traveled in this k(yv ti2pp9)j8v pff; time?    m

  A cooling fan is turned off whexu4l5 ym;ezgt twg;pyji(0*-wz-b)r n it is running at 850rev/min It turns 1500 revolutions before it0 ;lr -)j uebx;y( zmz5ytpgw -wgti*4 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 reduces its speed unif0f ,*wz0 oeecsh9,po uormly from 95km/h to 45k u9*swoe p,co,he 00fzm/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 revolutions as the car reduces itfd.id8j2tvn0* ltq9; ;y qyy b5dve.a s speed uniformly from 95km/h to 45km/h The tires have a diamet98 .d02 ty*qfdi n qbye.;a t;dvy5jvler 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 allsljq,ea* c ts47f7u wk :b,h79quk5sg her weight on each pedal when climbing a hill. Talu, ujg ,*bt4qsw77:qkk s7sef9ch5he pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is thk m 0ufp)tr l0ws:,y7xe magnitude of ther t)f7k0w u0ym,pl:sx torque if the 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 Fig. 8-ivl of82mc7f –39. Assume that a friction torqufm2-v 87lcofie 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 enn v j)m )wysu.cxmq79+ao 3in,ds of a massless rod which 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 o.mow+u )j cqsvn mnyx9i,3a) 7n this system.

  The bolts on the cylinder head of an engine require tightening to a torqueal ir x vo6n1a(*jy(/o of 38ar(v/jxyl(i6a1no o * $m \cdot N$ 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-ky 2+y 1a;40zof a wmi)sb+vobrg sphere of radius 0.648 m when the axis of rotation is through a1 bo+r0 ya;ys o4 +zimb)fv2wits center.    $kg \cdot m^2$

  Calculate the moment of inertia of a 3e)6h 006dbgx bur 2ol xoop5obicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why? bp0gdeb63u hx02looox5 6o)r).    $kg \cdot m^2$

  A small 650-gram ball on the end of a th+n3 x ah2jys; /hw;mdbin, light rod is rotated in a horizontal circle of radiu +/a ;yj3hs ndhm;2xwbs 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at consta8;85v4lp1 n5mll p p/ zedynpgnt angular speed (Fig. 8–42). The friction force between her hands and the clay i5mvp1en ypdnlp; 58z8l l/p 4gs 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 72qfg 7o*afv0w+/vqbamw e ) dthe array of point objects shown in Fig. 8–43 /fm0o*7) wqgwvbqade + vf7 a2about
(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 oxyge2b/3p9pxcrpu k8fre 0xfz;c 5n 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 gedj8crbe ln.,r4bh)gd z .b(z6*) qn spinning 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 b6ghc * (.) qde,ljr .4bn8gzrnzd )eba 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 kaot 7qvwae ;(boz0 x 5hs()e,a mass of 0.5ebsov ( )xk;ao wq0t5za7(e,h 80 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 swings a bat, accelerating it from rest tpxojf: uq+e:+o 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 tangentialll-h,mb5v hd 7xy 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 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 mb-dl vhh,57x frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shuteli(wof5x w7tt( br djk*w2u:k lo5.0 off and is eventually brought uniformly to rest by a frictional w ldbowe7k(t(rf0 5o:x52t li kuwj.*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 accelerarx:o 9i4mchhnb1g*3 ztes 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 bae8a )zqjcac 6q:0qhd* ll is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8)6qdc q:a0jqe azhc* 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 in Fibcxcn lwh8te6 +g- :i;g. 8–46th- 8lcnb:x6+ie;cwg .
(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 coz(ibzbrb hoky+7m7ejk6 ig+e5q, td 329 rww8nsists 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 rest within four blt3(/2/afkopy 6lgn* +,j yohjg n,-;z hamcfull turns (revolutions) and releases it at a speed of 2*c;mo,a ng,(+ n z/- 36khbhyl/lptyj2afgoj8 $m/s$ 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 inertia of rwsnld(80zx m,ki +zj2i ds25$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ofcpi )doy)( mefc33n)l 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without6yh-luu ns:m 0q.p q :b/kr-my slipping down a lane at :hqpn/6-mm0- b k.ul:syr uy q3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with firsda: j rh*/u+.es 6pfcj+a5kq(1:cfj)bf respect to the Sun as the sum of tk bpafe5*+fi fqcs: .jfc/+r 1jr6 j h(s:d)auwo 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 (umizgu47 o 7em. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a soli ez( umogu77i4d cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rol0zmt2tb(6 tre ls without slipping down a 3b t60 tzrmt2(e0.0 $^{\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 ix oa6wus ;t/2,v,g dvgts tip. It starts to fall and its lower end does not slip. What will be the speed of the upper e vuvsa 6g;wt,,d2/x ognd of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0./om0 a1-wdxy7 :7pmqmgrxlm2 210-kg ball rotating on the end of a thin string in a rlx /17gywa 02:mmpmmo-xqd 7circle 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 tnma1 ja)elh ys0 /p49a 2.8-kg uniform cylindrical grinding wheel of radius l90 4y/maj1p he)sa tn18 cm 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 tha f7 :5h:5phhfb u4qfcnt is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rnu5h5 7 c4:qb:hhff pfotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one show:b4srt 7sbk1xn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she bs1:kx t74rsbmakes 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 initzkgs(fol*r: zp4 4;mi ial rate of 1.0 rev ever(4rs izg4;oz* k:p flmy 2.0 s 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 verticn;a i*eo,v 5nyal 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, appea,i nv5 on*;yroximately 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 j:96 n zicbh,jmomentum 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 angulaqqpqy/(x-o 0u3ofk 5 dr momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an idenqijm;n 3i-ncsw 3vxm u9yf *) v(k*u*ltical disk rotating at auvwyn; vknm fuq9j3 *c3x)mi(i** l-sngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns15mas2wchji *abp. 8rb (/epn at 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 o5ho*x )krgggbc*9t( jf a rotating merry-go-round platform of radius 3.0 m an9jt oh5r(x*c *gg )gbkd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular yb )l150dprw8o qs5z5fzaxj 7 e.y.gl velocity of 0.87)f5 o8 lzz. ds.a15bxj y5pw ger0yql $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing abkt 7 80vvye4lpout 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 momentup y 408lvkev7tm, 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 involvf(3v ,h1nftt8n n ivi,e winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass ictj u ,-a 5f.9yn;pqo$ 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 roq)ac7o 4.1xg9bmsbszz ) yjr6tate freely without fricmxzs.)c az7 yro1 s 64bqgb9)jtion. 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 at the edge of a 6.5-mk8e lr v4yj,iaz0t*(q diameter merry-go-round turntable that is*ir4e,k8 j0t(qal v zy mounted on frictionless 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)ob-w xyhca55 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.- co)5xwyb a5h 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the kf9i7ke;i:br ,kk e w/Earth. Determine the;ke,rkk/k 7 9 e:bwifi ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerateotee* n8 v4*tq ai/7 lu8hby.hs from rest at 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 ij - bomhnlmi,yc 7l)7xj/wt- -n the shape of a 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 d t y-cn-j),mbxlow7/7j-h m ilelivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of maq / :zrqxzz,v4ss 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 conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its rx /4z, zqvzq: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 srv :fcwd1 0ba; 6)wcjhpeed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, ka,59p (tt c:wbu)p0xv + iudqwere rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphe pbtuc+0k9txd (p5a, wu:)qiv re 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 energy stored bdv937 vhx8pvgyhn 42in a heavy rotating flywheel. Suv7vy4xn 89d 3h bv2gphppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 illustratesf4k: h 51a9app rat j:;*ztgdc 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 7 fjg-6co/ fzq9fqcg2v=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 massa9:pkbqb ip 6( M and 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 p 6(b ka9pqbi:at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. Ittlbd(2q0 q*zr is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it wi r0*tq 2l(dqbzll 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 turhq kh;f8n35 4nc* u3m(jb ayjdn with a radius The forces acting on the cyclihfj( 4;aqdykcb3m35j u nn8*hst 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.kro .5:ke z w)g5w6gzs06lcyn50-kg rock into a sling of 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 required05gk5)k:z slc n.ewgz ow6yr6 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 rods, makt :byzx t8m:y0zy/ 1ywing reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tigy01/8zt:bm y:wyzyxthtly against her body.   

  You are designing a clutch assembly which consists of.jthf-kp:9u zs8l1w m two cylindrical plates, of mas1jmwt s:zu98fpkhl.- s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls a6 2-+qidj seyplong 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 looqjs+p i6yd2-ep without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d8 9fi,f x*;j o:zfjaboo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 rp o3fjgw;3m65 e ) uatuv)i-yievolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accpgt yaj; ii 6meo53)fw -3uu)veleration 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