A bicycle odometer (which measures distance traveled) is attached near the wheelo6 5w pc d.0,kwf9tuhiy(5c9t embd/y hubdu5c/.bh9yy( ,i6 5t pk0w fdc wtem9o and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doe -aj4,kuddb7 ws a point on the rim have radial and/or tangential accel-wbd kd4u, ja7eration? If the disk’s angular velocity 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 ano)zy zb1+;*tne - vvza-6ygy agular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thj9 f39hd/my1dcj0 kp dan 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 behind your head thae; u.d ujtuc7eedp2 0 dbe204sq*d5 uin when your arms are stretched out in front of you? A diagram may help yod2ie4scpee u;70d. uu2 j*5bdq d eu0tu to answer this.

A 21-speed bicycle has seven sprockq;n/e l 78abw.foupb 4ets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a smalp. nal;q7w/8ob uefb4 l front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have05 z+wst drg3y slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis w 3ry +t50gsdzof rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8tv7h8zd hhtvs6s /6,1upr gt 3–35) carry a long, narrow beam?

If the net force on a syrz 1,as/i(w13 od;rbh bsrp 0istem is zero, is the net torque also zero? If the net torqueirbs1da/(1zb;psi , r3ow r0h on a system is zero, is the net force zero?

Two inclines have the same heighx)xd9h vc-ak*t but make different angles with the horizontal. The xc9)-ahv*d xk same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ic4.(78,i ;tu fhtsch7b eg nqzncline. 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 energy at the 4ngsfh (8cbehcutqz 7.,t;7ibottom?

A sphere and a cylinder have the n.s ia3l,-amht s-qz: same radius and the same mass. They start from rest at the top of an incline. Which 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 Ksz-tiaam:,q- .3 nshl E?

We claim that momentum a8 wr* azw+p 9lb/xhnd-nd angular momentum are conserved. Yet most moving or rotating objects eventually sl*8hzrb/ pwn+dw9- a lxow down and stop. Explain.

If there were a great migration of people toward tn.s((pu e0o :r*ukv,cg hw3lrhe Earth’s equator, how would this affect thev(o*,0ewg(pur.nhl u3 r sck: length of the day?

Can the diver of Fig. 8–29 do a sgfaem 6g9,6pz8g h7vnz )ygb)omersault without having any initial rotation when she leaves the board 68e,67p zgfhb ng az)gmg9vy)?

The moment of inertia of a rotating solid disk about an axis through its center5hyp n6,gnxn 9d8uvpg.e6hrp7ya. (n of mass ise78ng g ( 6nh 9ypr, .v.aupxn5h6nypd $\frac{1}{2}WR^2$ (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 m+68/*n9)cvsz: k dqyp )-wevxz2eujt4umo l zass in each outstretched hand. If you suddenly drop the masses, will v+: z6co-* yztw u 8psjen)z9vmq4d kx) e2ul/your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical7n(-t sk hrxu/i;o5c/f mlpe. and have the same mass. However, one is hollow and the other is solid. Describe. ktf;i e/5(r/lp7m -conu xhs an experiment to determine which is which.

The angular velocity of a wheel rotato,xpn 2(0x)/e g uwqk )cwu0);qe mqtying on a horizontal 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 acceleration of this point at w2mgytqq 0)euc (o,) kp u/0 wxqx;e)nthe top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largx)2cdq9gz: ze;/ xedk.4 mmr he freely rotating turntable. What ha zdx m9qr.kx:dgzm 4e;/ehc)2ppens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he t71kry x4xi)t2a3b kq4ggwno)hrows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opxtnr)bw 7qgk 1g4i y)k o24a3xposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular moment*tb0jt a8zr:d qg0 0hsum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secoq * t g8:h0jdt00srabznd propeller can operate to keep the helicopter stable.

  Express the following angles in radians:a.1ytmcj ,w/sdjzo5/+hy2fw ,o5a kc (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. Calcue44x,1fnn ciflate, using the information inside the Front Cover, the angular diameei n4x1f4nf, cters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mfpwio -6-c.w+ grdz5amt - 7neoon, 380,000 km from Earth. The beam diverges at imw6onf ac rgt 5p- .d7+z-we-an 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 motor is tur;4yma( zdm .ix.hq:bhned off during operation, the blades slow to rest in 3.0 s. What is the angular accelerid 4hy .(mah;mbq.:xzation 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 .d+v mgw n(3yp15.0 revolutions, whag(+ny3dp.m wv t is its diameter?    m

  A bicycle with tires 68 cm in diahsq5 kwmlqy8 ..l)rr/meter travels 8.0 km. How many revolutions do the wh )/lqw .kq8.sr ylmr5heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25 r8o m(h4k;sa5f b)hpr00 rpm. Calculate its angular velo) bmk8hoa5(frp;shr 4 city in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one comp.w.vm8 5 sj k+nmtm-bnlete revolution in 4.0 s (Fig. 8–38). (a) What ismb.n+-t5.km 8 svwjmn 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) ia.x qzr bd8of: wii)d n(fd;((n its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed vv/n/+8ys dtc vcns7;of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 hj/u5vugfxbt+1(b jr. iw n*8hc *jh5cm from the axis of rotation is to experience an acceleration of 100,00 uc* 5nf /j(j+* u.b5gjx8h ht1rbvhiw0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centm8e 9w8evio, izcmn2/er from 130 rpm to 280 r,2eom /9ev 8zcwmnii8pm 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 radiufxx( lc*f -nd3*)k shms $R_1$ 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, jc5jm i(ugb at739f;ogx:ql:astronauts 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 interval. The so(g u q mf::gb;3alt9cj7x j5ipacecraft 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 uniformmdc.b* qky;qz ;1q/3wt6 0d a5f0gz- eycztvaly from rest to 15,000 rpm in 220 s. Through how mzk0 az q/vmdz 3*t0q ;y6qc-yc.fgw5te1d ;ba any revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpkijt+1ct 2/ xs9cdmq50 d jyt/m 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 , ppdkcbxy vd34xzh6, a2- 1csflying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn t 1-v36dczp ,xkdb2p hacsx ,y4hrough 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 rp..rc nn0 .rdcbyxx4 r,m in 6.5 s. How far wilx. xccb, 0ynr r.r.4dnl a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns p*;(bl q; 5fd;dj7tkylvlk f t2;llo91500 revolutions before it comes to a stop; ;d;ll5lj9;lft7 fk tqlv(d 2 ykp*ob.
(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 mj:y2fqv 8l j2a5jej:g ake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameterjq2f8v :j ljea gyj2:5 of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces itsne++;eskwl2f speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.l se+ek;nf+w2
(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 wvj)wx yc/* c/zhen climbing a hill. The pedalx/* v/zcwy )jcs 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 a/ d/tf 0+bbhu74 cm wide. What is the magnitude of theufbth a b+/0d/ torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the a *r5 t2cwf f9mgak4fe0xle of the wheel shown in Fig. 8–39. Assume that a f*90gf4atefwkrcm5f 2 riction 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 masslesx x qv/ h4ywm;wpxk(fq),kb+1s 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 torqu h w4y/ k+x,)qmxwxb 1q(v;pfke on this system.

  The bolts on the cylinder head of an engine r6uax5l+cn,ms1jq , b require tightening to a torque of 38x +la5,bnqr1jc,6 u ms $m \cdot N$ 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 rad k musreyl52r(c12 ep;ius 0.648 m when the axis 1muke2 c2r5;respy (l of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whez-/*pxb,61of5) fr ng hlgnr vel 66.7 cm in diameter. The rim and tire havg* r/ -p6,f5zx v)hgb ronfn1le 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 end of a thin, light rod is rotated in a horizontalvxkdrh,9i fbw q69 ;;x circle of radius 1.2 m9 v;x6kw,riq bf;x 9hd. 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 potwjy/dv.o0k g boga* )ijlc1; 7ter’s wheel rotating at constant angular speed (Fig. 8–42). The friction;a1/ g0cjo.) ikw lgbjvo d7y* force 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 inertiaejosf q+ie)-hrf+ 9k- of the array of point objects shown in Fig. 8–43 r -kf9fjh+)qeei-+ osabout
(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 consissjgzz,4 u+r3g ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the colsv5r;p3nf q 8rrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required ;3p8rnv lf5sq 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 wzh(uqqgk 6b ,-ith a radius of 8.50 cm and a mass of 0.580 kkh qz(q g6,b-ug. 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 swih7esw8bq:r r- ngs a bat, accelerating it 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 tar33 bphj5f, 25so:1snzr uojr ngentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uni:rnpzosj uo33125 r ,rjs5 hfbform 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 frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm gxd/ewm u9chh ,(79mkyqkh8 6is shut off and is eventually brought uniformly to rest by a frictional togck hmykud8/ 69h x,(9hw7qe mrque 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 at 4g)lnpd cgjpz 5rbpc, 54)3, dh;cemgu .c og17 $m/s^2$ 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 a4gt04 q,d w7slu8pha by the action of the forearm, which rotat g dl4pathas7 w8q40u,es about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–4b-14 yfq thy+o6-4bhtf+1yoy q.
(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 rn xgs v;4hs, *:wsrr1two 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 e (u88wpb .b/ed(m s+li:avz cfull turns (revolutiondvpzb l (+ amcs/e8e8(w:.bius) and 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 inerti3.a auor 3p e3(za+5 uwzic1pga 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 deve8ir kt.d gx zs h;x*ma+ks:)9n9j3 qmtts q-0elops 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 rcv(brm822 z;g-qr-hw -zoje qolls without slipping down a lane ato8qr-zmhv2 -wq -jg2 rcz(;be 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respu3s4jqv9x/zmu 1y3wn ect to the Sun as the sum of two te3jq/93z1wn 4uyxm s uvrms,
(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. 6rnb.gufpecf/ *on(a3 3z ;mq How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? fr3e(n bpoma/g;cun3 .zf *q6Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg7sve waj2uyre1 5u+kpc3y846ls uo,w starts from rest and rolls without slir7 kyc p, 6s8owjwv42 l1u e3ya5u+uespping 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 e r)b8c.8ns toto fall and its lower end dos 8nbctore ).8es not slip. What 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.c:u1q jmszc)1c0zk q2 210-kg ball rotating on the end of a thin string in a circle : k)c1qq2z0jczus1c m of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular 0g /).3c uzcy6gy nqfnmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm60 ynfgn)qz/ y3cc. ug 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 sid+tvn8mdc0p 4wal n.v2 e, on a platform that is rotating at a rate of 1.3rev/s If he raisevc 2tam+nlw 8d4pv.n 0s his arms 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 onelg6c7 v+:ahc4kkm 0 mzwkoex *+n9k4xsx xs 41 shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If shgoxc7 4 m*kse+hakzs9v46 kw0+kn1xx 4l xmc:e 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 swnla-w :-vaaq1 y0 y7es r5tx6pin rotation rate from an initial rate of 1.0 rev every 2.0 s to a y:a 1w r6wex0-l ava-s5t7yqnfinal 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 rotat/so4ph x6m 1kprmy4bnjh t 5uc6 c-/d+ing around a vertical 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 oxndkhs4u6jmyt+m c6 p1b c//5p- rh4of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skq6s/n7 8 im*u; rk1kp lqas*njater 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 of7mpzfh 5a )ss4 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 oujfcqm. v4t6:f moment of inertia I is dropped onto an identic ftj vu4:q6m.cal disk rotating 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,e3zyb8v*yw;n9klo s.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 th )p1jk. ocwuy23b apx0;dmf8 te center of a rotating merry-go-round platform of radius 3.0 m and moment of inertiydm.x2o 0c;pfa 3tbj) kp8 w1ua 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 xi :leyiuz 87(cg1ma8 velocity of 0.8(izm uyl871ecixa:8g $rad/s$ 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 aboutpg,xi/ ifmi d8u 29(8jqp7nau 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 momentum, what would the Sun’s new rotation rate b(m8 fpp 8djinaxu,i 9ig7/u2qe?(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 exc /iek htbu426hfmg0 +sess 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 ,cu3p(+cvyko5+h lrq- 7 tisy$ 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, initiallyn 4 . 0dqmng-q7eshso0 at rest, that can rotate freely without friction. The momenmsq7e04 .on0dg hq s-nt of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge onw.q.xrvkll:r-70 u 6uk5 tw sf a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momen :5xw0s ktlu u .6v7lwn.qr-rkt 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 yp:t w:f,q xwp6 s6g/bzxwm 2+the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping.6 g/ys::fp6wqbt wwxpz +2,xm 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);qev5zxr .kjq qa (z7v21 ofs. Determine the ratio of the Moon’s spin angular momentum (about 7z zq(xv.1 q5veo)jq sra2k;fits 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 fnoj+ ;fkk:l2qrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the s9sacw6 ukii5x*m47hh hape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate tu4i6 c9h7k5s hm*w xaihe torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mqns0m ,i9x ke9ass 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 reachq9 0m xk,sni9ees 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 theq.qn7n yhukray 70mw(k- ,bt1 rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8 9.eftj aklsm80fo vj+e+ 4pw-.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitatiomje8 4pw t.-+s va9j0lfef+ok nal collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autom()kr*hegg )d mobile is for it to use energy stored in a heavy rotating flywh g mk)derh*()geel. 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.”
(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 15o1t,s5dj+ymte xta ywy4 , qan $H_2O$ 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) n q3q30r(q5eobhwu7g is rolling 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 length L can pivonyjy0wn6cc/ 2 t freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the cent/cn20 n6w jyycer of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stand/w il whsr)a4 au3vvnu/dto-9- ;* iuiing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb ;hiw-ru /i)ds - vuotaln/3 iav 9u4*wa 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 roadt 7rg k9f*t)6i cbcc+o is making a turn with a radius The forces acting on thec b*g9fkt c)c t6+r7oi 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 begit 3evuf 9:c8sh9yuk)c9 k cdk8ov a-,ins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does sik3oct8ae9 yf,cd:kkc) v-8uu v99h the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin r-;le *qcf 7ku.hq c0cfods, making re07ch .fkq* -lu; cqcefasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of8j5f b,fpss /y two cylindrical plates, of masy ,5jsf/s p8bfs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rw +qpu3/(edttolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach theqtewu3/p d +(t 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 as31emfix( 2nmbsume $r\ll R$ h =    (R-r)

  The tires of a car make e,4e-hbr kxq *kl ++otd*c6ha85 revolutions as the car reduces its speed uniformly from 90km/h*-++klr hh q x c,kdotb*aee64 to 60km/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