A bicycle odometer (which measures distance traveled) is)-,g vb +o*vzdsyxdf-q 2gw8z attached near the wheel hub and is desiggbdd ),vvq z*osgyx--8f w z2+ned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ,nv)5 9qudyxevelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point h v n,5exd)quy9ave 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 valu /54l thf;aamme of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a l/qnt )hp2rbj( arger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands bp0q4c,o ,epl jehind your head than when your arms are stretched out in front of you? A dia qe0pcp j,4lo,gram may help you to answer this.

A 21-speed bicycle has seven sprockets at the reag6 hm)h9da bx,r 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 frogd) bmxa,9h 6hnt sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fles sb0xu-2xi lm5i56a noh and muscle concentrated high, close to the body (Fig. 8–34). On the basis o2ib6u5s50amn xo-ix lf rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walker5hmt7rg 1k,bxs (Fig. 8–35) carry a long, narrow beam?

If the net force on h0m7( aeoe 3baa system is zero, is the net torque also zero? If the net torque on a system is zero, is the net force70aoa3mbee( h zero?

Two inclines have the same height but make different angbdfw 5fo-,4n:mn 9bkg les with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greatb nnb5mf,go9k4-: d fwer? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ib wt6nb6xa haid8)z 9oom kk;n62mdr,m6 lo*0ncline. 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 i2obl6daz89) w km6n0ho*okx, 6;rtnm adb m6 kinetic energy at the bottom?

A sphere and a cylinderr yhby5g nb0.) have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bo0b5y.yg) bhnr ttom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conser+hb - cd2re4i7* gswodved. Yet most moving or rotating objectss ge 2c+drdbw4h- o7*i eventually slow down and stop. Explain.

If there were a great migration of pe5ti ,f+d7f (4ohg rz)z;zeco uople toward the Earth’s equator, how would this affect the length of the day?4zdiz 5 co u(zhe+,7otr)fg f;

Can the diver of Fig. 8–29 do a sfvvv1 ; ,n.p9dr5ls ecomersault without having any initial rotation when she lea.crv9 ,v;5 vd efpls1nves the board?

The moment of inertia of a rotating solid disk about an axis through its ce4u s50ni7x:xknfkywp0tvv s8d5+ 8cxnter of+fxv v5 k:wn74d5tuicxs 8xyp nk800s mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass i x.u*iat*d op*n each outstreti xp**tdua*o. ched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol 3 48(bjc wyns9xnq4xnlow and the other is solid. Describe an 4n83 (4qcwsxjnbx9yn experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points2in7q4z 2(u5.x3kt ee vq:+hcgd rzhz 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 the top o4zqgq53u :z tex.2idv2n7cer h( zk+hf the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of az:xm qkh;k en:*:tno . large freely rotating turntable. What happens if you walk towanzk:; e*:hq tnmx:.k ord the center?

A shortstop may leap into the air to catch a ball s4unm,lon m*zmb0, ;z)j-divwmf l47and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will noticel-u),mm0b 4noi,lmwnv d4*7fz ;zs jm that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a n r4 b; 8pdvjc: r nff9su4+j.z2+p,gugia,iihelicopter mu9bria4ji u.8gjpf+2 4r:;ns+u z vn pd, figc,st have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i(5m,)r r4 tyrhhkfwott:ey 1wsru508n radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidslgw)i/x1ii6k -7j2 - a sbfimence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Ms/i l- fa x -2kjwi7b)sm61igioon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The b+7n9b7guxl udeam diverges at an ang 7x9 u7bdlu+ngle $\theta$ (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.crrk1 03a9ijn When the motor is turned off during operation, the blades slow to rest in 3.0r9 injrk1c0 3a s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floqs8yd3c jya :yht6r)1 or 3.5 m to another child. If the ball makes 15.0 revolutij8a1st 6: rq )y3dychyons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travelszi*e*nru p8jre660tl 8.0 km. How many revolutions do the wheels make?6ri6nr8eeujl * *0tpz    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Caov,u8gs/1jafycf 7l0lculate its angull78 , vojy fsga/fcu01ar velocity 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 complm2ll xu9r +(paete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fp 29r(+am llxurom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity o2hgj)0 l.v(rw pk d00-qdumf9hv v:je f 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 point. ia4 bfv57:0xqjk*y b*kp rgh
(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 cm-fzg ppay;/q / from the axis of rotation is to experience an acceleration of 100,000p/paf q/ -;yzg $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cenp mtlew+*acv;b)i qf38y2v/ ,(glga uter from 130 rpm to 280 rpm in )q+8aly( g lvtv/em 2c3;*i, wbga puf4.0 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 radiup 3,tbahgnwp:qvx 6oa;9 8wb8s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboarws 9 (tle*i4jfd 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 in stj9i*w4( felterval. 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 *o+:lip ( +m* ohfcwcfuniformly from rest to 15,000 rpm in 220 s. Through how many revofi*lph*w(m : f+occo +lutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C vs-ig:+ihsbf*b 1fs /alculate
(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 m4 hcl kpi blb0c9rid5m.i2.4 the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1. lrp2i40h4b5iimk.c9 cb. mdl0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 29)qin nw.nhlm,:/bf4nn ,stf40 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traqb4. f hwnlf:9,n,/nnnm)itsveled in this time?    m

  A cooling fan is turned off wvd2)z ;le- 8/ebpo1huehq.n phen it is running at 850rev/min It turns 1500 revolutepbqvl-) ;.18 eenz/u odh p2hions 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 revolutions lah ihd6.rpe2am9 a29as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter oadhleia p69h a29m2.rf 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 revo(a f d;get smbjjc.)+,lutions as the car reduces its speed uniformly fromefas;( d)jtbj ,c.g +m 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all hert6u 5 8d zj+lhy8 c2lcfiz-zm1 weight on each pedal when climbing a hill. The pedals rotate in a circle of d6iu- m8cljc52l+y1ftzzh 8zradius 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 osj9pxql4qlly .2 y2pl-rjv8 6vvn/a:f 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force 9lnp 4.-l:jqpylvqjx2l2 s a8y6v /v ris exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Asul9c,ztk8ovl): brm 5 c,ub d*sume that a frictio:,zuo ) tvl9mkblc 5c,d8*u brn 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 aaw9l *iw 0c)5i-soeoo massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magcwe9)*0i ao 5 liowos-nitude and direction of the net torque on this system.

  The bolts on the cylinder head of-e9 ajd(do b+fr+ qk9p an engine require tightening to a torque of 38 deb+99rfo+jdp k -(qa $m \cdot N$ 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 oxjm wb:ck6m+jz : e5w3f inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. mjk 5w3w j6:x+ce:mzb   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whe sqe4r2glbv) :el 66.7 cm in diameter. The rim and q)br4g:sl e 2vtire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ballij tpyof9x x)bh.j 849 on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculat )pft9x 4j89b joyi.hxe
(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 constant angular s ks)t pc1xohk 24f-sh+peed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N t4h k )2+hk1xoc sp-stfotal.
(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 iowg.y3e: oz7wfr0a:h5z d: g enertia of the array of point objects shown in Fig. 8–43 rhz :oe 5::7ggoy fe wz.w0da3about
(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 two0krqac-/gs o5 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 thj4xgnm: y-46a shx vv:e correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and aayv6j 4x-:hnxmv:s4g 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 radi yxkje)jj.ap/o- 8c:swcb-7 r4 x1ddhus of 8.50 cm and a mass of 0.580 ) -k rye-d: c jja8xxpdoj.shw1c7/4b kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest bu;y q::zoen1zl( *oato 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 and1ep 06v/- 5d*9auqrrr uqd-aci c lxt3 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 tw 6-pi5*a9l1v-0 qu cxcd /u dtaqr3erro 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 is sl8z dwun;s4q(8fc 5c/ z tlap8hut off and is eventually brought uniformly to rest;wl pa8 su5dc8z/84nzl (qc ft by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 ac kpvj e25,wh0q4f;e gacelerates 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 actiontz5xc q .;wl6ob5(f bxjon., n of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uni.j5 6o; (xlbc5,zf qonx nb.twformly 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 shownk 7 q r;qtwtrlep578vb1d*q 9q in Fig. 8–4 t7p9tq1bkq d rq;e7l* 8qr5wv6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses, ok9ra g2 p69ry$m_1$ 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 bw 1-gtoq)as-g8 d3xb four full turns (revolutions) and releases it at a speed ooqxwb381 dt-s ga-b)gf 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 momqq)nkk : r7s*ient of 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 torquej: c,:uxx b/wt of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wis.vm8l- *d af g+)caucthout slipping down a lanedcf+ *8.s -uam)l acgv at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sumicu0lbop4) 4+*4dlpa vwm ; vs of two tla+p;mvu)4 cwpb*il0 svdo 44 erms,
(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 mabu7kvumt69-t aj 1lna1k;q 5oss of 1640 kg and a radius of 7.50 m. How much net work ial1 v5u7b9tut;kjnk m-aq6 1 os required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 n0vqk qr9c /u31kx,jb cm and mass 1.80 kg starts from rest and rolls without slipping down a 30.0 ,rv/qu nq1b39xkc0 jk $^{\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 staoispby3 :z td*:y-0zl rts to fall and its lower end does not slip. What will be the spi-3t bz0o*ly: zydp:seed 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 on the end l /aeiga1 h,2 rnc8/dlof a thin string in a cirr n ,ce1aa li/28glh/dcle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu/9pc+adq.k f:xyz g iz)6nl4mlar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 raq96)4fz:xzm +p k cngylid ./pm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate of,l o0-juj8gca 8xwn9w 1.3rev/s If he raises hijnw0ocl ga 8j8 9ux-,ws 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 one shown in Fig. 8–29) can reduce her mome xitei 9/ 3qnk7cvg-,hnt of inertia by ct ik en3- 9xi,/qhgv7a 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from awp*hbr8b(s 5dkn4q o7n initial rate of 1.0 rev every 2.0 s tord (bp*7 bshnqo45k8 w a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertc.w*pnah c7 k9ical 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 ow. 9ch*acn7 pkf the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinninjrpkj(mn+cef54 q */fg 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 off2k83vaqz. ea 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 cylindr p3;9 ;oz oqv.-jbz9qn9zfjw/6z sbxtical disk of moment of inertia I is dropped onto an identical disk rotat qz6 t-;z 9pb3b j9.zoqosfzv;xn9w/j ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns aes:b uis ifjc3:(*i.at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at theo1inr v +sd9hohu f+.4r (uvxi4tno35e0yu;j center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia nrhdiy+4+4uf jox5v13h euo.vit u0 ns; (o 9r920 $kg \cdot m^2$ 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 anguxi9h*mgyp.h.u. jn 5a+s mlh;lar velocity of 0.8 ;hgup m yh.im*+n9alh x5..sj$rad/s$ 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 collapb/dhcggn t y)3-3 fuqxy7j,6sses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius.fx)u6g bj7-dtyh /,yqncs33g Assuming the lost mass carries away no angular momentum, 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 winds in excess w o)c+wgy* 9cddi0r xcep 6xf(** uoj2of 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 k9dqa86 j: mje$ 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 rot.ptszq86welh e:5b w)bg1g -s,r na d6ate freely without friction. The moment of inertia of sw s-dlwz,b.pt 16e8a gr 6 gq5:hn)ebthe 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 ep,e4pqgt28r v6 n /f g(5i*kxsu ga/xqdge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings anp *skeg6qi,f q( va ugx24/ng58rp t/xd 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 groun ug 9 xoq;vwys (j)fs/-ufj)1id with 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. Whu )qwfj g;yuso(si /v9 )jf-x1at 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 3b .eokw*eq e1l7gpq1same side always faces the Earth. Determine the ratio of the Moon’7geek1loeb 1w*pq3q. s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 zakzk: 8l- 5rim/$s^2$ 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 acquire/+imgr nm*5 yz(ux+y is a rotational rate of from rest over a 6.0-s interval at constant angului/+*z xynr5i(+g ymmar acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mas8yz;rbf91oe m s 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0. z1ymo8ebr;f9 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 yk(1rjg ru5(.y84ky uk iush8 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 m(c;h9vc moh8 our 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-qua;8cv (h9hmmcorters 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 automobile is for it tof sauy-. .bh+2h dcc;z)+wbc s use energy stored in a .wbc zy. dhccauhb-s sf+)2+ ;heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(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 an / rrw/msy2u k.$H_2O$ 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 speev (ep2im cc5o-2jsi j+d 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. f2i+/il41a7epbqnoffyl,mx b-9w7ze., 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 lin+f 4l2p,7oae-x b lfzbyf qm/n71 i9iwear 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 v/5 pc5pe 9sltg, twy,1oimpy,ertically on the floor, and we want to exert a horizontal force F at its axle so that it wil,1,wm 5e9s/,pp yl tiytcgpo5l climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is maklw r2;1u jsbo)o 8k3sqing a turn with a radius The forces acting lk bs)j o;1u23wsq8r oon 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 intb* mwpo+2ee w7d8.)vz+gnkpm o a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the tork 7w2vopen db)m .g++wmp e8*zque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin8)zg vchbunjq4 p26a+gku ,a2 rods, making reasonable estimates for the dimensions.u)p g+8 jvgzak2ucha b64qn ,2 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 of two cylindrical zxa:dn s.s..f k 58r5w gn/y 4*lzufxkplates, oflk kfrws 5:x5d ng4x*8.auzszfy./ n. mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track s+4kjtvlfzjlh d4t*mfe10 6 i+*fx( uof Fig. 8–58. What is the minimum value of the vertical height h td 4z4mj6htuf*l1j( te0 sfix*f l+k+vhat the marble 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 ra ixg e8+b):vnbz n3)k9r ;8bxb+ rzf$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions aqxc5 yxod7(tww /x/lcfkw(92s the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) 5w/c lxy/d c2k7wx9(fxw( tqoWhat 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