A bicycle odometer (whivc:+lp-z/d,7 d .vso ** p8spiwbi7jv q7circch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wit8bi-.zovvpss:ic lp piv*w *ddq7rj7/c+,c 7 h 24-inch wheels?

Suppose a disk rotates at con tgj)d+5pk.oz9x j *jfstant 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 whj*jj )ktpdo5f9g+z. xich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described nlgoaod )7 c5wc)b jiz)k376v by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater j7.9ax le w08rhajz ,ew; -4c2iuow nctorque 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 han;yjvfm sw,72oi d q4+pds behind your head than when your arms are stretche j7oqi;+2pvwf4,my dsd out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets ah)4zkel4g2 z ftxd:s x+i1 l5ft 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 sprocket? W+gz5z2 :hxxfsf41kl d)4etil hy?

Mammals that depend on being able to run fast have s ib64+uw0mhdr oqe/ )ulender lower legs with flesh and muscle concentrated high, close to the body (Fi /erm4+) ohu6d0b uqiwg. 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 lal jz ;m:oi-ax7sxv 54ong, narrow beam?

If the net force on a system is zero, is the net tog lci, *pxg-e78fuj 0yw7w-cc rque also zero? If the net torque on a system is zero, is the net force zero?pw8elc c j,w* i-fyuxcg-g 707

Two inclines have the same height but make different angles with the horizonus; z*f y wcl7g:8e9;-wnbm sdebkb: 8tal. The same steel ball is rolled down each in8y wb ml8g;z;nc -:b fue :esk7sw9*dbcline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rfrc-md g seci k:i-(+1est) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic enr(ism:i fk-d -+1c cgeergy at the bottom?

A sphere and a cylinder have ti.gv04ms 7z d )m-ekzuhe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom fi-i 70dmeg)mvz ukz 4s.rst? 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 e-nf c zi8:c9pzsu55 kmoving or rotating objects eventually slow down fiz55ceuk p9nz-c8:s and stop. Explain.

If there were a great migration of7kgilj ucapi32t24:hao, f- ledq* c) people toward the Earth’s equator, how would this affect the length of the:fiol3e-aj *kqp 7gt hdu2la4cc2) i, day?

Can the diver of Fig. 8–29 do a somerst kos *jqn,:u4kxk0sx 5tf *u1ault without having any initial rotation when sht 4skutksk**15oun0xqf j:,xe leaves the board?

The moment of inertia of a rotating solid disk avr9;o2xzibkaa57 n ha/ +eup+bout an axis through its center of mass is aizbo5uavxn+ k9pa 27 e+r/;h $\frac{1}{2}WR^2$ (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 rot:nht ykf k; xaj j3tebi42t:87ating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? fk;nx8 k2t:: 3tj7ai4yhtjebExplain.

Two spheres look identical and have the same mass. However, one ispu6rw 5z)x kltgz:j9y 9th7 .w hollow and the other is solid. Dz)9h rpwltxgzuw.657:ytj9 kescribe an experiment to determine which is which.

The angular velocity of a wheel rimat )v8 yv0/q0 o 76cz9lmphfotating on a horizontal axle points west. In what direction is the linear 0m0zl)/f 6i8ocyv9aq7tmp vh 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 stand in76it,gb mfd v)/m(ping on the edge of a large freely rotating turntable. What happens if you walk toward g/,fimv )m(6n pbdt7 ithe center?

A shortstop may leap into the air to catch l0n1 v27 ilefsv*y4izvjg8q,a ball 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 and1i4lgfjv07e8 * n2l qvvz,isy legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of c2h ,1lyvvl) qvg3tba* onservation of angular momentum, discuss why a helicopter must ht1hv 2a *qy )lgv,v3blave 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 raqhvmde0)y:(yq e- l5 .chb7vcdians: (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 amazi) xmyj on772i7uwn9hqng coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) ofm7hjw7yq n2 9x) on7iu 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 Eartuy)af9pz pbebk.*,q+ h. The beam diverges at an *9 )p y bq,ezfauk.p+bangle $\theta$ (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 t1 w n)b* naa0 t-twjirkyd8))hhe motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelet1bk))jnh )i-*8r 0wwdaty a nration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the4mvt-*ksob( a7smd(u9bwad . ball makes btd49o(a7km- dm sb( .wv u*sa15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels y/qw p57432m ldeuby z44gc 2mbugbe: 8.0 km. How many revolutions do the72 be3z ld4b:becgwquypg u4y/ 542mm wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25fk 0il5vr 4j( -d3rjzv00 rpm. Calculate its angular velocity in-dzj jl4v30( vif r5rk $rad/s$ $\omega$ =    $rad/sec$
(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 makexnvvq/ s /pkpt11 5x(bs one complete revolution in 4.0 s (Fig. 8–38). (a) What is the lv (v/x 51p/t1nbxqkps inear 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 vev8lon +j 8-q6fhyw azyw+ 0q bj):0mjglocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointmnmtr1;p/uy(0y vuw /
(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 r/ w.tqt)ek9p1 e/u uwcotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,009 uu.etcww/ptkq /) e10 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceler-3e6/3bidx)wj vzz h,wx o.: upgll-f ates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determinex -3woh lv/jip l )uz63f,ze-.b d:xgw
(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 radiusg 2g8e ed; ysn,eb/ue- $R_1$ 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 q riwej (o+dg*q1 d*4j e12htg x8fgkquu*(w 7Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation2f ( qgqh1*ux14+ qeritwe*owduj(8k7*dg gj 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 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 1- a1h- z7sgj/anp 1cq sz9fwqfrom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? aq 1/zhq z p s7wf1--nag1cj9s   $rev$

  An automobile engine slows down from *ix8 rc5ea/p erd0g2m 4500 rpm 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 stresses of flying highspeed jeuhzk2fpf7,yh; 5 ;zi nts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revo,; fk2nzz;5fy upi7hhlutions 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 frt2s/nhk*n gy1 om 240 rpm to 360 rpm in 6.5 s. How far will a point on the edgekn1s/nyt *gh 2 of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runninit0/e-cvk/s ax l o-y/g at 850rev/min It turns 1500 revol l-tvek-/s c y/xaoi/0utions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutionyez1ii 4z1p-z s as the car reduces its speed uniformly from 95kp4ey1 -zzii1z m/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

  The tires of a car make 60ugiw3shu lf:bb :8 +t5 revolutions as the car reduces its speed uniformly from 95km/h to t:sl3+u8bw0ugf: h ib45km/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 ridingo j26a elqym4qm+g)8 c a bike puts all her weight on each pedal when climbing a hill. The pedaeql+)j 6m2y48aq o mgcls 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. p*jfnh0v;*m8( a- qq0e fpvx yWhat is the magnitude of the torque ifx0yqh qnff; *0vm aj *-pp8ve( 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 F l67.apc wuulahg f:*t5a: s.dig. 8–39. Assume that a fria apds.ullch. guw:6f* t5 a7:ction 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 rodar2 elg .xvp,7 which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releasedx,ple g arv27.. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening 56loi -f9cme gto a torque of 38 -efgm6c9i o5l$m \cdot N$ 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 ofdtdh 8dd7.(tg /mf 4ew radius 0.648 m when the axis of rotation is through its(g7dt dew.t8mdd 4hf/ center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6, zyl hxcp9k6p2(oq *d-,wyx: +qtbme 6.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of thqxypd*mlkeq 9c ,w:z +-6pb(ht,xy2o e hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on tzu +1yxmi,p yzuoikg2c9m 3.+he end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculatmx c3uigz+,2i pyum k1.+ zyo9e
(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 rls*wjl .*. )or sm3;:kexrx jfotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay i.*krsj* 3.s: jxrflx; w)olems 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 obje0 9wa0(k0+gnrxji8 s s p,pr,y1t jpjbeof4n+cts shown in Fig. 8–43 abr (n,fasw x+bt1yj4 +i09pjr e0g ,po8skp0njout
(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 wik-9+h 2e aeohhose 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 cylindricalhs l;z)by q-r) satellite spinning at the correct rate, engineers firbszq rl-;yh) )e four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is 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 uf,b*p ) g)54as pszgltniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcul )5 pfg apsl,*)tsgzb4ate
(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 my(6n)jqa )iqq:9cqz6w)hzt from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and.; o6jhs9ueo:s(aog m4 jtn/u/yn)vx 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 frict(a/gjj)ey / 9osvx4n.tumo6;ns u :hoional 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 evennaax du9 wue:5 z7y3zx(o2y r0tually brought uniformly to rest by a frictyo9x(0yz ax2 75:3earunz udwional 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 balu7dv-q a*xuu7 pe )i;rl 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 th d40vl9b) jl8 sq(qsbvobu+h+ e action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The balolubqvb+v9 s ds8j b 4l+(0)hql 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 b,we--nq7cpc7uza 7 a ou6(rn oe considered a long thin rod, as shown in Fig. 8–467qn-6np7a cuozu wr 7- eca,(o.
(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 t8 . 9lt(huq.t9mds kzof 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 full turns (ej2 ,c0q gqxt/hd r7*crevolutions) anqdj2 ecx*,g q r7t0h/cd releases it 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 of inertia7+)e w6d+rd. ,jtz dn- rgwmsg 7trsp6 of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque nylr.5 jh 72 e;5fzisbof 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 without slippih41xwb,kve k;-w;jv fng down a lane at 3.3f ;w4xbhkvk1 j-w,ev; $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect tc: qc q1ofc0+//zfmrgtm r r(+o the Sun as the sum of two te/ftq+/c0r+r1 r : czcfqmmo(grms,
(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 radiuej2 fm :ud8ot4i1 ((h x;mveavs 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 solid cylinde;avidh(1v e:f u8o 4e2mmj(t xr.    J

  A sphere of radius 20.0 cm anx0c i6 b. /igf4kk+udbd mass 1.80 kg starts from rest and rolls without slipping down 64b0d +kiifu/gb. kc xa 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 vertically8- )c oxx6tmnp on its tip. It starts to fall and its lower end does not slip. What-cx8onxp tm)6 will be the 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 rotating a +-f+5afabr2)c ,nqefu:eo bon the end of a thin string in a circle of radius 1.10 m at ab+2e r,)cf na-+5 obaa:e quffn angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifor) o9 *ebkp0teny vq-h2m cylindrical grinding wheel of radius 18 cm woye9 -v )p0bnhqe t2k*hen 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 rotatidqn mhx3 wj1 r7f:ag4.ng at a rate of 1.3rev/s If he raises his arms to a horizonta: n3r1aq.7mfg wh4xj dl 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 inertia3d4c-z0 mml*poi3b uy by a factor of about 3.5 when changing from the straight bz0 m3yicuo mpd*l-34 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 ra nimsn00eriy7,1 i p4pte of 1.0 rev every 2.0 s to a final m4p7np ri0,1sy 0eniirate 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 venih7d sd4w6 m,rtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the d,hmi 7sdw46nwheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figu37+g9 zdyzd tere 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 tha1anu)z 45,p) wxqbu6khnhpm+u p/u2e 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 cylindlx /vxr*g*k-a c z,fu4nu6b,p rical disk of moment of inertia I is dropped onto an identical disk u4/nurcp*6 bxv,g *zk-fa,x l rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atc jdc- xdm(sz)v9uz( - 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 kg6z dv,p n1e gl.;,jhhm stands at the center of a rotating merry-go-round platform of radius ,1nz h;lmve,g.h dj6p3.0 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-go-round is rotating freely with an angular veloci;x12s.7nrzeutz dt c*0-9smgy+nbit7p 7ohf ty of 0.8 t ni7ft hp byu7-cz dtg0m1ezx*7ss.r2;9+ n o$rad/s$ 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 evengiwgpnjn+s w2q 31h(:tually collapses into a white dwarf, losing about 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 momewn(ipwgj 31 :sg+n h2qntum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve wvl50 fsey;k 4kinds 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 (c 068m7styb nmyfj xb: ccq2ghb,c37$ 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, t7oq7nxxlzrv0z f-+- qhat can rotate freely without friction. The moment of inertia of the person plusv 7qrx0z +qxz l-7-ofn 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-m diameter merry-go-round turn- aytpis 26f-ntable that is mounted on frictionless bearings and has pnit6y- -fas2a 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 grouwt 8guw)td5+s(rh:vn nd with the end of the rope lying on the top edge of the spool. Ah5w8d:s wt+(nvu r)t g 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?

  The Moon orbits the Earth such that the same side always faces the Earth. Den56n2 wunoz.z termine the ratio of the Moon’.zno526n zn wus 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 accelerates from rest at a r2uij +q: dy(swate 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 cylinx6 f49e*cn bqs ,l, gx/urd w.vt62a-rhzke6m der of radius 0.20 m acquires a rota.9dg6- qw6x*z r vm2f bncts /k e,r,46ealuhxtional rate of from rest 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 solid cylindrical disks9b,nzihe b4 pvf.n2x0, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it re p x9bv,20inf4.nzebhaches 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 tah8 *f kklnu,4he 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, vh)p9) e1ckxmbut 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 thath cxvme9k 1p)) 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 j fvek5o 9(./na vzu7kin 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 bfneakv( 95vzu7o/k .j e 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 md0q znorlor d xm 6v,x2s85+(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 speeqyz7g na6ji 5 56qlf )aroy0s3d 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 (i.e., we ignore frictit5yx*e4 ciubi5r08 oi;ur:s oon) 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 gravityi5c4ux y bor5;:esi* 80uriot 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 vertically on the floor -p9huyvk w7e6 bn39fv, and we wann 9vbfw 93p vyk6heu-7t to exert a 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 traveling with speed ;eed ifke mzeg+n+5 0)8ktmb.v=4.2m/s on a flat road is making a turn with a radius km;eie z8bne.fm0++g5e) d kt The 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 1.5 m and begins )w)oh w*) y.0m ;k0fbzkrqwhcwhirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm aftr) 0bc)m .wf0;hhw o )*zkkwyqer 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 her6htof/)qt) j 0ej ea3a arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds fjoh) )efqta06jate3/ or a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consisrx8 4 n.mk.zmbfhs67uts of two cylindrical plates, of marusx b mh7.8mfn4k 6.zss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 l5 ue8 tez9jm1xooped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must dr 9xmee1u8tz 5jop 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 nh-:h,f mg:fedo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifor 6qe d)knrc6u3 f1(mjymly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration 6kfd1(uc n m)3 y6rejqof 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