A bicycle odometer (which measures distance trve0 vsdl6e k5aie ,.32+y;wavllm ( svaveled) is attached near the wheel hub and is designed for 27-inch wheevea5v l .2 lal d;36swevyvk(0 ,m+isels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vsi1 mdsyw 072welocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity i7ywsswm 021di ncreases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by tq wlamm:8+6ca single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explabotc r14:tii1 ) dq0vyin.

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 sv4oon4xuz gn/t 09 3oyour head than when your arms xoo v4o9/ugz t4nn0 s3are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pzxf8b0p86 xm ukw -mw;edal 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 sprom;x fm 8-xpz0bkw6wu8cket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower leg8yjw gagd: o83s with flesh and muscle concentrated high, close to the body g:8 j3awdogy8(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 q m9.w- .biazylong, narrow beam?

If the net force on a system is zero, is the net torque al2) qe;nu:xuhyyh 3e9h so zero? If the net torque on nquhe xy:92);ehu y3ha system is zero, is the net force zero?

Two inclines have the same height but make different angles with aim 3s1b+tb8qxp a p21the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain qsx 1bp82+aapm 1bt3i.

Two solid spheres simultaneously start rolling (from rest) down an incl9.mppvmt; 3y9 c.vzy.hp9qsmine. One stq9 vp3yv;m9s. cmp9p.zym .hphere 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 greater total kinetic energy at the bottom?

A sphere and a cylinder have the same ra5wlklf2y;) k:/g r zdjdius and the same mass. They start from rest at the top of an incline. Which reaches wgr):l/ 2y 5jl;kzdkfthe 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 movij xihr7e;jp/z57 g7 i:fcpm5 bng or rotating objects ev z5rx7iip5f/: m egjpcb7;jh7entually slow down and stop. Explain.

If there were a great migration of people toward thiggm2qm +q. 6xq/*yfae Earth’s equator, how would this affect the length of thef*2qq yx+q/mi6 m.gag day?

Can the diver of Fig. 8–)cibc3aa ( fe529 do a somersault without having any initial rotation wabf5c)3c e ai(hen she leaves the board?

The moment of inertia of a rotating solid disk about an axis th7; u)ajt7s bawrough its center ;aus b7a tw7j)of mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mamht)n0/:7p/vdwv e h3 1dqry lss in each outstretched hand. If you suddenly drop the masses, will your angul nm/dh: /v1wt d)q 7rpyhv30elar velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one )3da j , ebkr8fx6/sb ogs4k0j m42skais hollow and the other is solidmb4k0s) g3 bjds ek6s8xa2ak/fj4 ,r o. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontalv(bspj7 :l d- o9c.vex axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear accelerat :x-p(.vj s obd7ev9clion of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edges3 mj tjx)06j6 6zgjjb of a large freely rotating turntable. What happens if you walk toward the centerj6bm)t3 g6sj 60 jzjxj?

A shortstop may leap int() p(*lo amqmlo the air to catch a ball and throw it quickly. As he throws the ball, the upper part of him)l(lo p*a (mqs 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 l vvo1s*2ne6cha tuj,9cp*-.5mgo su l aw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). mcs*u,* 5 .louvce j1tva2og sph69-nDiscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angzg 2lq5o)yzmtk21 /qsz.3a 0j pc2whyles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence.fxxfe3m4 dp7 0 Calculate, using the information insxpm0fdx e43 f7ide the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divrsrf4i*5q8wu j6 oru+erges at an an5io6swur+8qr*4f ru jgle $\theta$ (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. Whebtdc10dcip y )rh.en7 vf.51bn the motor is turned off during efy 1.b)0tc .vpb 7icnh dd15roperation, the blades slow to rest in 3.0 s. What is the angular acceleration as 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 makes gt8xqp s ,.vr(j (-lq)njstv415.0 revox4r.gq8(, (ljq j svv)np -tstlutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many)v y+jd0-eyvj revolutions do the whe j+ v0j-ev)yydels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rps:x -rdpwp 11cm. Calculate its angular velocity inr1xdp-csp1: w $rad/s$ $\omega$ =    $rad/sec$
(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 (Fig. 8–38)n29c22 nknnav . (a) What iv2n9ak2cnn n2 s the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arl zh ;bo ty+b.blw,7,oound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a m2(bq 6yamu m)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 7aj1 of rq rq.o3kdb1*7.0 cm from the axis of rotation is to experience an acceleration of 1 qoqo *7r.3 fb1kd1rja00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboueh+y dg6r7c1 lt its center from 130 rpm to 280 rpm in 4.0 s. +rd lcyge 71h6Determine
(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 radiux xsa8( jt 939lezby5:j,heces $R_1$ 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 abokw; lldz j n/4)3-dtjxgnb)t +ard the Apollo spacecraft put themselves into a slow rotation to distribute the g tlxzn;dl)jbw3jk)dt4/n + -Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thr2n k(-;r)rln xge8qk ; 3brnspough how many revolutions did it turn in this time?)krk;n8en-s ;n2pqrbrgx l3(    $rev$

  An automobile engine slows down from 4500 rpmjdf /8m:wj z2)fov-t o to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stressess 8ll w+a gb lvrkc7o,e-36b6a of flying highspeed jets in a whirling “hum 8lvb3rs k -+baglac766w eo,lan 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 rpm 99h c*r2txna8ipzqo) in 6.5 s. How far will a point on the edge of the wheel have tra8hacxi9)2qprztno * 9veled in this time?    m

  A cooling fan is turned off when it is r;jn l.fhc(q1d unning at 850rev/min It turns 1500 revolutions before it comes to a stop djf;h.(cn1lq .
(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 sw*s 9; ;hn7v2ar)n;riww szj65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.zr*)2ww a; rhn;nj9is7 vs;s w
(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 0c+y92v6ift 8/x 7b lwsq t)z9tdjybythe car reduces its speed uniformly frombcy/lt +8bq )xf0iydsw7 2yvj99t tz6 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing 0qh jhfj w-koayv .;46p+j6 qnltc9e3 a hill. The pedals rotate in a circl-he046o.w+l6c qqf yn9aj3; jjk hptve 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 cmk n+vt:zqhat* fistf09:)ac* wide. What is the magnitude of the torqvf aizctthk9* a):t*: f+q0snue 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 torqura ;ewkfi,v c.*m69 swe about the axle of the wheel shown in Fig. 8–39. Assume that a frr9.awk wmf6 *vcs ;i,eiction 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 mass;;dqa 1qr/gqa,) krxfless 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 oqkrdxq1)aa/,g; q;fr n this system.

  The bolts on the cylinder head of an engine require tightening to a torqmi he1;/mo j2que of 38 hmme1ij;q2 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-kg sphere of radius 0.648 m when thzi ,1*wr 1 k/u nlvm6cf(z4+ uz8esgxje axis of rotation is through itslci*z6gmu ef(1z/+ ku,81j vw4rxszn center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim wzkk 3z8z/hc: and cz 3wz8kkz h:/tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the fut;tg t6p7sh+ss - qg(i6eq+cy *y 9cend of a thin, light rod is rotated in a horizontal circle of radius 1.2 me9+hi gsqs-7yq*(su6tt c f+ytp g;c6. 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 d dbatf7.tt1uv*gqpk+.. o4u constant angular speed (Fig. 8–42). The friction forc do. 1p+tvt td4*buk.agf.u7qe 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 o:ut a8yypv *f5 3a1navf the array of point objects shown in Fig. 8–43 abon* 1pa8f vav5yt:a3uyut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms iza qbl ym1/ +/5(i7ll/zfcfo 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, unifo,f blii z2lb, mri.o gi*ue07*rm cylindrical satellite 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 a radius of 4.0 m, what imb u0eoig bl,f7iz *2*.ilir,s 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 8.5g r x, jt sppgt0,g0au1y+q9-o0 cm and a mass of 0.5 p,gr 0+gp0y1 9,jaxg-o tstuq80 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 restjx9:8e-/lf nsp znu7k 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 ablebe w o*: 1nhuqco.0t(z9 ;qxka 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 (z*:t( uk ;qoeh9n01 w.oxaqbceach with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought t+ kjk vw+f)0lg. gvr-uniformly to rest by aw0krt -ljkfv g)+ v+g. 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 accelerates a 3.6-kg ball ajz:rlbqlb1: + t 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the formeq74 b,i.ix:epfj9uearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The balije4x um.b9fe7 q:,i pl 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 canfgy2t 4-,pt qi be considered a long thin rod, as shown in Fig. 8–46.q,t4if -tpy g2
(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 masses mj3j97r26d6v2na1mafm+p4f-ds q 8g xanqv l, $m_1$ 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 fromj fqyiw) dupt50k 0e;i;rxr1* rest within four full turns (revolutions) and releases it at a speed of 2jp0 yq5dxi1urek; )tr*iw0; f8 $m/s$ 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 c7ywl(ckmv+ / rlx/ 2bof 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 engineu.fyobhn( gzb:s(6 3f 2rbi 9h develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls withogj0lqn rc()lo kt-, p-ut slipping down a lane at 3.l,o) 0(-cgkjpqrnt l- 3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sunug9dws4ssaa,z. d bm7; it3qba3 dw 66 as the sum of two za,3b b;w6si 64gsdtd q7.uwd3am9sa 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 m. How mu /..xfdp0in1lbj t-plch net work is requ0xld nf.il/bp1 .t j-pired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withikc/zzy2q27-xwr-mu b v7n -tout slipwk yzr7q-m22c/i z- -vx7nub tping 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 bala.1:.ypnuapc pn0ea- z nced vertically on its tip. It starts to fall and its lower end does not slip. What will be t pa.: p.1y0uc-nezpnahe 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-kg ball .aovznvu/ym t4u1r*orn90o . rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 19u /rv4nto.v oan0yrum.oz*1 0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrica-6d 7yi,2:umetd ryts l grinding wheel of radius t -dd6m yy:,ret2 ius718 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 that is rotating at 09o7hv9efa*gc /p tc;c7673l7pmk r o xztmjaa rate of 1.3rev/s If he raises his ar ox c7j* 3apkmg;cl9ot/a67z0rv p77m f9eh ctms to a horizontal 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 her moment of vfm3p( vcvp hi2 7u a4.aade;(inertia by a factor of about 3.5 when changing from the straightfavda ieh74mv3cap( 2u( p.v; 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 initial rate )i 8c7jv 8crzyj-3fsdma c4. iof 1.0 rev every 2m7c3-y8casc)z.rdif i4 8j vj.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 vertical axis:f7ocic00 r pl 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 cel:r occf00i p7nter 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 ofmnaj 4/ 0 8)wt;kmosly a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Easln: t8 r/s-/eb*etw n 5rnvn.rth
(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 an1 *xy c.u4y ./jtusqmz identical disk rotating at angular speejs c um.yqz1t.4/ y*uxd $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2a3g9x 6e(m qmi 2x1gmy.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotatinggyc/u6 i oonh3+etce)22yy/ v merry-go-round platform of radius 3.0 m and m/ct)y+iye y e/3u oo ngc22vh6oment 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 freeod0-zf/ g6l.dx bktwm9 -ayp*ly with an angular velocity of 0.8 dwla* z kyf6/-xpd.g9-0bo mt$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half i4axcg/ulfmj,/pq t r3og11 )y ts mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be? gut1,mx r /qlfy)oj4cg31/pa(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 i:snft.jni7j nc(z;6 ( e:a biufuk5+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 t2o ivf dwh.e cds4)jdjzu2d+ ,t 0c.,$ 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 wist 3xsz6g)n yy jf( b-voq+nr-4rhw.2thout friction. The moment of inertia of the person plus the platforz)yvh6j ( y+4q.nwtrn f2-rxob gs-3s m 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 1 2bz(ptkkse4t ,nj,o-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mometz2jskto n(bk ,1pe,4nt 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 r3-1t b rivrpk6ope 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 without slipping. What length of r -tr1ib6k prv3ope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the E xs+lu7j+ hth/,nc4h 8e4oljks2l )p garth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital ghlp, t8lj nh/h4c) l s+xs47+e2kuojangular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rexb41*hi)24pb8b qno sfjfoj/st 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 in the shape of a uniform cylinder of radius 0.205nj8wk8 .kkq rhs:he4* (pxa g m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelern:ke88 hq.xwhgkk4(a* sr5j pation. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg a+a9, b 0nox:zoxadjga2k6 4n gnd diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diamx4a,+n06 2:o9ogazabxj kgdneter 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 )* lrye0ceu,5l wunkw()bj. x angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with )/ikmfmi k j2 fkva5n5,j f7h:mass 8.0 times as great, were 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 procejvh)k ,752 faf/ 5:imifkkmjnss, 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 lowy)nc7z1zx k 7d-pollution automobile 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 mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.y x7c)z7dk1nz ”
(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 illustrate3 rnfpwd f1)rqg3x5- *me d+dbs 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 hof k* q7oq-8xd 32dakhjrizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (b jr8rhekf/*3i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then rerh f*rjkeb 3/8leased. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standi2 1ju l9w22x;i ep:pqhm(v mvang vertically on the floor, and we want to exert a horizontal force F at its axle so that i; h: (2 av19ujxq2 2mwmeplvpit will 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 )/ pw7 1c my7xacr/asiturn with a radius The forces acting on the cyclist and cycle are the normal forcea ca1csx/w 7)my/pi 7r $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 whir 0sn3wxan/q*z oqmbnm) ;:o7 kling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm aftewqo:) m3n snmqx0nk;z *b7 ao/r 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 herrk)x31+ xvibe: )kd ko arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms ke) b+ 1vk o3drxixk):held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical3*- 94jgelsmomt. fzn plates, of mass jeoz9 t-s4g m *.n3mlf$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rb+ idc+a:1f;vh 6kk tthe 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 r ;61ci:+kv bk+adhtrf each the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buizig,io4.lb.c5 / w yat do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as :,urepyc wa8y6 e*7 wuthe car reduces its speed uniformly from 90km/h to 6*ea ywru 6u7w 8:cep,y0km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s