A bicycle odometer (which measures distance traveled)ms2x1 9mbbh(2pv7 wno is attached near the wheel hub and is designed for 27-in sm2(7p 2o9hmbvw xn1bch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velzb q-sy4p1q: qti 1f3mocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocm -p1b34 tzsqq :1yfiqity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocityy exq/x : b-yzpoi5u9/ $\omega$ Explain.

Can a small force ever exert a greater tork0a/ 6mnnobq: d85rg rque than 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 to do a sit-up with your hands behin8fc.oekd.-kb y1sm w;j 6kt+fd your head than when your arms are stretched out ym w-s.;+ e.jkckfkd1o6b8f t in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel j lv1jeg11*rbjm-vd: and three at the pedal cranks. In which gear is it harder to pedal, a small revb 1mjjl1er1g* :v jd-ar sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ableq-hegg/p(5p p t;tb: p to run fast have slender lower legs with flesh and muscle concentrated gh5b( p/ep-tp p gq:t;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, narr+(auqd0l 0kj.i76 ezbd2z d- p plasw(ow beam?

If the net force on a system is zero, is thau96eqnoq, f .jz* +ike net torque also zero? If the net tor. qji un ok6*ezf9,qa+que on a system is zero, is the net force zero?

Two inclines have the same heigh7),fj 9d )ot-brezw(b 5b-ioj6gct r et but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thzi) (r7 ojtr6 ebw5b)-e jgfob dt9-,ce bottom be greater? Explain.

Two solid spheres simultaneously start rollingaoivxn)hc vb 7p23r ,1 (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? W1 ,viponh2avxb )3rc7hich has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sj +f3qy8vfk bl6p.pz p(jp;k ;ame mass. They start from rest at the top of an incline. Whichk8ypp(.j fjf b;lz6 qkvp3; +p reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum andflgrt: fgz2+9kuq 77 g angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop.+7gzqff7l 29rg:u tk g Explain.

If there were a great migration ofnxfe/ mujr.:-pn/r0 r people toward the Earth’s equator, how would this affect the length of th peur rx/m.jrn0/-n:fe day?

Can the diver of Fig. 8–29 do a somersault without haxd1 hkv t1cd4. yz wm-(:)btrqving any initial rotation when she leaves the board? hbdx4k) (z:-t 1vycrwdq.m1t

The moment of inertia of a rotating sei5sjn-dxwf6kw ,7h /a* 3wtr olid disk about an axis through its center of 7/x6h ,nkf3-wi*jd tr5 eswaw 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 mt tp3xlz x0yf2 : b-:ggt)+gnstool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same0gn : -b 3xt: gm+p)gxy2ztftl? Explain.

Two spheres look identical and have thewd89dqk(jns5c my m 9r5qk4p; same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is w5rq4kqsjc npw(m 995k ;ddy8m hich.

The angular velocity of a wheel rotating on a horizontal ax,,5afgx;jv ty le points west. In what direction is the linear velocity of a point on the top of the wheel? Ifvaxf,;gjt y5, 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 7hcf/s ew(u5anad-u : large freely rotating turntable. What happens if you walk tod7u-nfwea:5( /a uch sward the center?

A shortstop may leap into the air to catch a ba. iqtdkv k 8.tm/lkq,jbii6 9.ll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (8i6qv.ijk/ tb,ldk mt9 ki. .qFig. 8–36). Explain.

On the basis of the law of conservatios0ka ndx9khbs z.h)lql24/x 1n of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller .sl dka h4) shnqx2/0bzxlk 19can operate to keep the helicopter stable.

  Express the following angles in radians: (ap yw;lb4cl6. dgygew 9h-+u/i) 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. 2bgu5*ki z1m oCalculate, using the information inside thei*zb mgku125o 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 diverg;fqox ljr +t+*es at an j+ lx+qoft;r*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. When the m4 3,urcgdq0 fxec5kde*hb-l,g/( opfotor is turned off during operation, the blades slow to rest in 3.0 s. What is td4fb5 dq3 p*l,fkuoxg /e cg,rc h-0e(he angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 mwvzr.1 j6x q/t to another child. If the ball makes 15.0 rw6xtjq .r1/v zevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels dcbo.1f , gsj)8.0 km. How many revolutions do the wheelsbc d,s1 og.)jf make?    $rev$

  (a) A grinding wheel 0.35 m in diam b 9c0vuu8ewbugs,,e4eter rotates at 2500 rpm. Calculate its angular velocity is w,gu8b0u4eebu 9c,vn $rad/s$ $\omega$ =    $rad/sec$
(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 re:bta0 h ;q2b8kybvo w,volution in 4.0 s (Fig. 8–38). (a) What is the w;b,b8a hkt2b vq0o:y 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 th m4cl-:-e2q9yrjheg hx j5xl2 e 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 of a pointfm*aeu.1i6 fhormh- 5
(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 /kx3/p : 5pjp emr:tpew5+gie rotate if a particle 7.0 cm from the axis of rotation is to pt3::k+/5mxepp jwrp/5i ge eexperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accele 275sdcufep2brates uniformly about its center from 130 rpm to 28cdu 2s5eb7f2p 0 rpm 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 radius x8fq*g a u4l59od5rghzi( (xn$R_1$ 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, astronodc+me)kn2l fj,mgil( a - 7h+auts aboard 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 i7 +c() fh aimmgln+ekd-2o, jlnterval. 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 acceleru2dyiv zwoa g/c85//mates uniformly from rest to 15,000 rpm in 220 s. Through how many revoluto2zc/85 auvw/gi/y dmions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm 1pfjhq9xs er 1 ft,;2m*;eh+rv)iptv 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 highspfk:zjt :v;t*16hi kqoeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throug1* :toktfijk6vh:qz ;h 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 r voaxt5)j,( cspm to 360 rpm in 6.5 s. How far os(tx5,cja v) will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runnin:d/:5jdzn af xg at 850rev/min It turns 1500 revolutions before it cod z:/ad:x f5njmes 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 65m4m c632bo avs revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Theamm3 4scv62 bo 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 t;fd;)*vdt 5dftykg6 tduun0 )he car reduces its speed uniformly from 95km/h to 45km/h The tires vuu;6 tdfg)0ny*)dktdtf5 d; 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 aixm0tm c9 y9o7ll her weight on each pedal when climbing a hill. Ttmxo9c9im7 0 yhe 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. Whtn et1od0c+ntrk i/ ;(at is the magnitude of the ton t k0td;/n(io1 tc+errque 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. 89baa: l,n lkma7c:r( )r ajiv7–39. Assume that a frn aari7,a)r 9: vb(l7kal:cjmiction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless k 4wi9;,voe8y; c:wt5jc gnur rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posiwj g erwik4yu5t;c:c8n;v,o 9tion 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 tvq(yla k+.kev q(xyn2ox,f6/ ightening to a torque of 38 olxeq(q(2y6afvk , nx +.ykv/$m \cdot N$ 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-kgkau 8 :yawdm7 ufd.6/h sphere of radius 0.648 m when the axis of rotation is through itdh aua: 7 86ufkd/ywm.s center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Tt6levz. t 8q9ehe rim and tire have a combined mass of 1.25 kg. T9lt6e zve8. tqhe mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizon;/ a)ze r6kz2p.b*f bubi ,vhrtal circle 2);,/bier r6 zzvubaf p*bhk .of radius 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 co; ;5h4vykh hdgnstant angular speed (Fig. 8–42). The friction force between her hands and the clay ishydkg;h4h v5 ; 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects mb, 92tbrfxs7 shown in Fig. 8–43 about t27b9s ,fmrxb
(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 atomld7d1r 9w* chjs 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 cylindri5de* xhzc v;di (p2a9tcal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satelli5a92(pd;*tdxv chz eite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass zc0ore24 o(2txn co+jof 0.580 kg. Calcueo jco+tx z (4on20rc2late
(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 rest2rc /5lx )hlce7qnmn,g76n mq 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-r,dk0 xnc fb0+mound 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 acceleratiobm xknc+0 fd0,n, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpa sgpzww4 as6j;hm((f8,wzx4 m is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2s4(f ;4hzj wwmx(wa6,gsa8pz $m \cdot N$ 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 accabq.nm o k5o3v*)lzy4 elerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, i k2+z q)0am0dbvn;qt which rotates about the elbow joint under the action of thea00nki) dzmb; qqt2v+ triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, akjp cauu,-ey 6/pd+ o.x fco-.s shown in Fig. 8–4,u.do/+exkucfjc poy .ap-6-6.
(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 q duq9v7gs1u5s r;v;l 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 fr qbmmlir 6g(yjv 2r-kok2j838a s n0,vom rest within four full turns (revolutions) and releases 2-8 y 3,8kv(mgo jvbi rk0r smqa26jnlit 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 has a moment lkrr*,9 f + wt(+ppkk ui(2icsof 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 tor/94tx:foqox yy q5(by3y(a skque 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 m2dy..r qd s;xand radius 9.0 cm rolls without slipping down a lane at 3.3 ..;sryddx2qm $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respe ep*m847zntrzc25z ds ct to the Sun as the sum of two m*p24sr87 etdc zz5zn 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 in ko fv11p k531 n lofmknbt ol,uqpk:;248o(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 it is a soqt1,f1 k2n(io:k l pf nlm3pvn1;bk8oo4 u5 kolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sta:2kv:5f ljao / k4kr,minoqu:rts from rest and rolls without slip:kok ,o5njr:v4k/im u q2: lafping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall anka: 18sqsec r5.jsgq2d its lower end does not slip. What will be the speed of the upper end of e5skga81rc q ss .:2jqthe 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 rota n .42uu+4 mjeyt9:qvf mxiun,/wa,wnting on the end of a thin string in a circ,+ ux4.nft n9iu qjy2n, a4ewv :/umwmle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grin;3 zv lq kyi*-*ef9 2k bpinc8rw2eb(pding wheel of radius 18 cm when rot*2r(ece b;nkliv3 w 2 p9yzq8fb-*k ipating 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 t q+g benpwx*10hat is rotating at a rate of 1.3rev/s If he raises his ag qnb0+p *ewx1rms 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) cl -8bm)1;thc r :6fibod 1 qdpxih8b:ban reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her x8 d -tbrd m8:pi;bc:liqhh 161bb )foangular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation oobo yr5 .9-mpyt2z 0crate from an initial rate of 1.0 rev every 2.0 s to a fi-o 90y zb2rc5op.tymo nal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its .boymiw0al/hj d7 x0 kg 4mg h7t:1r(ncenter 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 sha.hd7x:mt a/jgn0 lwo iy0 1 (h4bg7rkmped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater skco7v*gs7 .3hj 1m spt2:lcrafiel4 3pinning 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 1)9,bnq uyxj 6p*qvu zmomentum 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 g) h1 9ieayv5umls 4n2an identical disk rotatglsuyv41 ) mi a52he9ning at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 d. ouuamt(d,p;masoj1ge 7,v il0a oz-r):;e $rev/s$ 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 thehn):fa chanx5 gz lwuz54 755q center of a rotating merry-go-round platform of rag5wz4 )5alahq xn: 7n55fz hcudius 3.0 m and 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-wj b dy d;p4bw/897nb;egq9f fgo-round is rotating freely with an angular velocity of 0.8 ;bq8bgw dw9bedn/yp74f9 jf; $rad/s$ 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)hdxxr7*;k f+iy5d 3rt kdih8 its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s nei d;rx k+h)f8ddrhi7k3*t5x y w rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exces9*on u;lr;llf s 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 d:dtfhg bjs q0 8k4:*m$ 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 frk fchw.-, g + :amacqhlx*q1j k8-;bjzeely without friction. The moment of inertia of the person plu+xk. ha,gq 8h;bfjcl a-:qmzjwc1 -k*s 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 et mxi7t 56n /i/)dqavbdge of a 6.5-m diameter merry-go-round turntable tit txdnb/vm5)76/q ai hat is 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 e pk7x2mhg8b8 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 without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?kmpexbg 82 7h8

  The Moon orbits the Earth such that the same side at;9xt1r* fvq*w2zg x*rqg m* always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angularrrzf;m2* *gq*x*w vt19 axtg q momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rat 3w((adxy f:hle 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 0.20 m acquires 6ab f 6wa-g w*hw q-4ovhz1ci;3v8avm a rotational rate of from 1qah6 w ;6fm-w hov4 i*gvwv8a z-cba3rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s zu4o p ypvu;tg6gt2 4h0+l)k:oibkc 4olid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin soliz+bih 4tpk4u 4;g2 o:pvtou)kgy6c 0ld cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel (5mmr vh7 mvx16k7vmp 2$\omega_R$) 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 xat12 qrkkx(1i*e5lp Sun, but with 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 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 momenp115rt e2 *qikaxl(kx tum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy +jf wk/7d bui5stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flyfb 5d +ik/7wujwheel 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 illustrates 14zbo,dkfsc v d56*-irsv:la 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 rollin6 +tmxg6b e1wm mq5aw8g on a horizontal 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 lengthen; w4d njldav,-2r5c 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, dete5 vj2-a,l;4dcnnr w edrmine (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 standing verzyjpm6 /h1v ,ltically on the floor, and we want to exert am 1ylpzv6jh,/ horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelinsb1g5evnrq8 06g6zo iyf. 6ipdq6rsl)o (- cug with speed v=4.2m/s on a flat road is making a turn with a radius Th 6ugd0 16zebr.lsy6) nfoq5r8soqgciip6 -(ve forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1a -ttwl jy993qab 3z5r.5 m and begins whirling the rock in a nearly horizontal cj a-q z3bt5 9rl39wtyaircle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a soliuz8qf*m;)wox bes j0/d cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held ;msxoj/ef) 0wqb8*z u tightly against her body.   

  You are designing a clutch assembly which consists of twoiwew.xw4caa t lxy 77t976)wj cylindrical plates, of mass 7w w.7)wj exyx9a ict6a4 7wtl$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough tra5ivp5-;4-7l rt6jdq8ecvd e esl: bg4y2ecjw ck of Fig. 8–58. What is the minimum value of the vertical height h that the marblel;svje e 45dg rlwdpc:6ic572- ej-4teqb8v y must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume :t lm4 4h3ww*uyt: p6tuk g.dd$r\ll R$ h =    (R-r)

  The tires of a car makslc)k g xkbhzo(-nx)p z6o-21kso49gk6r 1 rie 85 revolutions 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 )nk-k(kz)ob1 o2z k6hs9o-pr6gs 1ix4xgr clangular 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