A bicycle odometer (which measures distangx2+m g1ws arr2m3 oyg,u; yp.ce traveled) is attached near the wheel hub and is designed fousr2w.mr; gmyx +p 3agg ,2oy1r 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant ang j 3eqnq7aq32lular velocity. Does a point on the rim have radial and/or qq 37qja2nle3 tangential acceleration? 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 sinjw f2tm(jt+yazo+8k yh 62e s(i.3p oogle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largers 2*j, *nbwjyx 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 headn0v4/l+49q lnzv hh:5muenon than when your arms are stretche9oq4h 0v5nm4: +v/nnenlzhuld out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hasp(eh .44gdw+h0p:wo,4 vri qx 5gsg qu seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small reaeq4d,ggq s+4 (.whr igxh50vw p: opu4r sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flesh724y( )0cjl lq-v sxczmlt)ts and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is t) vq -mjlslcs0)c2(4z x7ytladvantageous.

Why do tightrope walkers (Fig. 8–35) kry9kjyw/cevp : v07v. q-e; fcarry a long, narrow beam?

If the net force on a system is zero, is the net torque alson(q h+n7 :cm-esat( dv zero? If the net torque on a system is ze at :nvq+m-n((d7shcero, is the net force zero?

Two inclines have the same 4+(vtek r(piv tve ;v*height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at th(t tvv+kp4*ir(v; e eve bottom be greater? Explain.

Two solid spheres simultaneously start rollkzoz7t7)ns: xing (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed tt):k7os nxz z7here? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius az wbg/5arr+/ wnd 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 /ga/+zbr r w5wgreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentuf/)scr;fopxq:h-64fy 5y et nm are conserved. Yet most moving or rotating obje h qytrs5 ;4p-6fxff:co)ey n/cts eventually slow down and stop. Explain.

If there were a great migration optqv/ 9wm6beb f3z78ef people toward the Earth’s equator, how would thisz7vp q8bbfte9w6em /3 affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without haviyvk/z6rwt 9:hc9c gamjph;+* ng any initial rotation whevm;azrt6ph/yk h+cj9g *wc :9n she leaves the board?

The moment of inertiauq cx69jtrxk/ 7b+ks6 of a rotating solid disk about an axis through its center oc 6+kkx76 rqj 9/bxstuf 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 in each outstret,vd4 vdz.5vbo;zdxolw9f +cwr1u;q, ched hand. If you suddenly drop the masses, will your angular velocityz cx ;;zo qulo+49w.v, wdv15,rdb dvf increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and m- /1qg(u uqcpxsp+1 uthe other is solid. Describe an experiment to determine which is whi ugm u1+p( p1u/xscq-qch.

The angular velocity of a whe m,/xsb+fl0 tnc5hn ,wel rotating 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 tangen+fm,nthxl5 s w 0,n/cbtial linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largeoxqu+2y pdli; hk -m,6 freely rotating turntable. What happens if you walk toward tho6q ix p+ml-k,yduh 2;e center?

A shortstop may leap -b:9/oupj apyk 6btu3a t5 uca*p- cw*into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look tty5b-juuw-up /ap*c6:ac *9a3 kp boquickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of anguly7p3u i+s7m(h laglqrphz.-e ;l k.3xar momentum, discuss why a helicopter must have more than one rotoru .ask 73pgly-7q+.ilhlx;(mh 3e pzr (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3g9(mu-g mz (i4ct9a lo0 $^{\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. Calculate, using the6gjg-i k7r gl+j.f n6w information inside the Front Cover, the anr6g6i j-kf gwln+.jg7 gular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0o;r 3/,d+l;dn bgulxm 00 km from Earth. The beam diverges at an an;/dxm;o, dulnl brg+3gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blendxjrk( qrb kr; p(68ve8er rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angula kxr;qp6kjr(e(vb8 8rr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mak6r+3 nhu ynic4es 15.0 revolutio+ u3nh ycrn4i6ns, what is its diameter?    m

  A bicycle with tires 68koa/)-, rgcqe cm in diameter travels 8.0 km. How many revolutions do the wheels ,o)a rekqcg-/ make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate4bfcy 3. gd98hgwz-g:md(cboqgu8v 4 its anguldz 98 o - (4mfh:ygd4 q3bcu.cwbvggg8ar 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 compwl6g3*r:f(ot9wbqj f lete revolution in 4.0 s (Fig. 8–38). (a3wbgqwto(9f6rj :f* l ) What is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in itso:xhx v* wkaa5)38qtc orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a prui he0c /io(4gcw7 1ioint
(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 from the kk ygd1ytug),y;5+hpaxis of rotation is to experience an a)kut5hyg y,gk ;yp+d1 cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratoz*hw z9uf npzr0m /58es uniformly about its center from 130 rpm to 280 rpm in 4.z*wu9 n0mz/p5fzr h8o 0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius d d8 y qt/3l9zomd)-ar$R_1$ 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 aboard the Apollo spac2et2 o +jdu2d bpk-z*iecraft 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 spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determ 2id-ptob+du*e 2jz2k ine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220*t55q5d1 me.fihcvx o5d)yx* pije :y s. Through how many revolutions did h.) q1t5 oc5* e v*xiiyx5edjpy5:fmdit turn in this time?    $rev$

  An automobile engine slows down from f/d3dk2ah *hc 0 me3ql4500 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 t)rv9d e uo(ni 7e01udvhe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befddurv)n ue 0 97evo(1iore 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 diaq8cv;2nb,z ib (zto2 *yp 3vtcmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travele;tnpoiyb, t38 c 2*zb 2(qcvzvd in this time?    m

  A cooling fan is turn-kdn ep ap*/nrc5n+5 bed off when it is running at 850rev/min It turns 1500 revolutions before -kpn5 5na+d*nec pb/r 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 as the car reduces its speed uniformlqzsi+k10fdb4gg07ze 3b :ceby from 9gzbkdb fs 7 g 4:3bcqz+e01i0e5km/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 65 revolutions as the car reduces its speed(wbg x/ pf1;dkjtl 1d 9b/7h;hyd:kei uniformly from 95km/h to 45km/(:bhpkf1/k w9;d/dyid l ;bjhxg7t e1h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pe + fhif nsmtre .fic:,e4r6v/2dal when climbing a hill. The pedals rotate in a circle of rac+vi/fr24,s rht6ieff em: .ndius 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 r) babz4 c41g6csn8u bdoor 74 cm wide. What is the magnitude of tar b n4)ub8gcz6 sb4c1he 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 ab+k:x( 2 rrkrneout the axle of the wheel shown in Fig. 8–39. Assume that a friction tork e :2+(rxrknrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of cop* 9.2wk/tb3dpll jmass m, are attached to the ends of a massless rod which pivots ap*3 tlk2wj/9.o cldp bs shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqu/ol0 cq(,:jkq oeqd(rz ,c3hd d:v d3pe of 38 cpjqdk0z cvo,r3 qq3 odeh,/l : dd:(($m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 mp4m8.ollv s3o when the axis of rotation is througvs3 48po.lomlh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cmg2iwazg2 es-t (rt)w2 in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of th(g2wt2at ) 2ziges-r we 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 horizonti qps.0xov q mt+rtce+)u*+ 2wa70zi cal circle of radius 1.2 m.+p0v.zscaxmqotuw2 i r7 i+ etq*c 0+) 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 on0k 0s;r+h;ls aa ,taxzhl kk78 a potter’s wheel rotating at constant angular spee ; hskk;8tx0 lhsrl0a,k aza7+d (Fig. 8–42). The friction 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 inertia ogwnt43t0eymfiu3pw)6 rv + *vf the array of point objects shown in Fig. 8–43 a w*)6 uvrptnymv4e0t 3wgi3+f bout
(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 whose rpz 7t+2 ns lhti/ygknw 5:3mvmi*0. btotal 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 bavq12t+qf(y5 fq.b cat the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satelliaf q+ fc(q5y 1t2vbqb.te has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cyl jjnm b*-t oxc(1 ff)6sw4p+uhxc2y,linder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate )uw,ffj6(s* jy+xcnlcp 14mbt2o- xh
(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 /t qs.g)*-ac/mtjh q zbat, 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 tangentially on a small hand- .yyvxrgvbe -aq) kb: zn:-9*cdriven 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 iyzc:erk- yv9.g ) b-nb:vqa*x s 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 frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 j5 v*(bli54mqyy)wks +8x seo rpm is shut off and is eventually brought uniforewl55+oj s)(y bk4vq ix m8*symly to rest 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 accelerates a 3.6-u1slt3 vjxza ydg e,8 bq:3fs+l-1xh3 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 tw s q08dal 95sd4c0wcehe action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated54cw sd9 8a ecl0sdqw0 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 rodg4go4uij: tn-57kb8o +jcp dr, as shown in Fig. 8–4:ic-go o7b utdjjg 5p4n+8kr4 6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masj 7b v526gc9fy(pxfarses, $m_1$ 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 turnsjk ;c4p+//yi5re nmzc 80gdaa (revolutions) an zn+ /4dyp85mcrk e gjai0a/;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 inertiab0vws.) )cijj3b,ird 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 t+)cuimdldfoy xx4f2ex +8 2a)orque 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 rocom s+xj0v 64ani 2.cblls without slipping down a lane at 3.3 .cj26a ovmi0 +4csnbx $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic evh62s*dhlm+j9gp px(, ;pf j c7i)im cnergy of the Earth with respect to the Sun as the sum of two termx s; ,mcl*v(difh+mp) hgji72pp69jc s,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much r1hyk5z sl; t2i, bs+knet work is required to accelerate it from rest to 5k,1syrh ;kzst lbi+2 a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.01bkqpbi:0 za 64ptgxxk(rq/ 5 cm and mass 1.80 kg starts from rest and rolls without slipping down a q kbp4g r0 5qpaxt6(kbi/z x:130.0 $^{\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. Itf /q ml :.xxyz l-).mypctd50d starts to fall and its lower 0m.ddxfy/.qlcx5 ml- ytz :p )end does 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 0.210-kg ball rotating on vgpthw(4/.j::gk0 sp. us klbthe end of a thin string in a circle of ratsk(u .gpsg40p:jl. vw/:k bhdius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kei q*bt, ry*2yg uniform cylindrical grinding wheel of radius 18 cm when rota *2iby,*q etryting 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 rwx8w1 yf(,my:z(dg za platform that is rotating at a rate of 1.3rev/s If he raises his arms to a hoy8(, dw : rwxz(fy1gmzrizontal 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 inertia hpbd21e od)r p;u-gvd/0p:d z5sr.bd by a factor of about 3.5 when changing from the straight position to the tuck p gdrb0)2z:d5;ero sp p-h1bpu/vd d.dosition. 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 rotx3s/bvtyg -a 7ation rate from an initial rate of 1.0 rev eta- bgv3sx/7yvery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a j8*n(y pdf wzxm u*1a1frequency of 1.5rev/s The wheel can be considered a unifoz1 da(8jw *pnu1ym *fxrm 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 wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a rcv60y-4sx b x v1o0 (gqu1pntfigure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth x2 ,kugl295y zi0geoh
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of in :76jsn 867e(/tieigg pbyzfuertia I is dropped onto an identical disi zju yi(fp6 s/:g6e8e7 7tbgnk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ohzb+ux8e1qgeb 0.q /f6 vfo0 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 the center of a rotating merry-go-rq)(azrrufmq1f7fn9* 5 6mjcx ound platform of radius 3.0 m and mom19xf56r) a*qrnm ju c(qzff m7ent 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 7ufd f,ie0*6q ;vhq0bwpys72l8kou hrotating freely with an angular velocity of 0 2e,qpb h8uvd70*;shyik wq7o60 ff ul.8 $rad/s$ 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 e;,h h9fd g58k d2*qrmtshti+;g0rhy white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lostr*f 5dh;0s,hethm;2r9y dgqh+gk8it 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 of 1d s ;0* ,2foczcyheqs.20 $km/h$ 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 ne*jt;:eq c+ 7fp3gpjnto 8pe, +h2gv$ 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, i hio0hu)u3(n nnitially at rest, that can rotate freely without friction. The moment of iner 0i ounnhu3()htia 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 stanu vz57w7mpt0/ r;6xmflhhgh 1ds at the edge of a 6.5-m diameter merry-go-round turntable that is mhu zp765xlvgf/hr0h1twm ;7mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on ther.q31up2tpi *rcql*s top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far dos*qcqp* 1p2ut 3l .rries the spool’s center of mass move?

  The Moon orbits the Earth such that the same*x xgx9)mk. vl side always faces the Earth. Determine the ratio of the Moon’s spin angularx*9k lm.g)xvx 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 )-inr2 dn 6mt)oe,3byxsd (mt m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder.3be-l dkm+0s pj7tcs of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant a 0bpe .ctd+ 3-jml7sksngular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylinu37 wz 1qhce2zdrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050we zqhzu7 c231 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the re96z yed7 3elvuar 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 8.0 times as greatn)bfloags,q9y),:wws y* 2dx , 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 rotations)la syg:wd9onb)qxf,y * 2w, 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 automobm*y5u0frs/ q uile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to t*u 0yqr /smf5uravel 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 aqlfs-zhm1ezecl .a( or y26qyo)o6c8m0 6u: cn $H_2O$ 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 horizon4l vtyro)4 gn1tal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igntq-tx7,j 5h.m/dykbf -i h) koore 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 accelf-hhkkq )o,y x-tt bm.dj57i/eration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticall:at8vpr ( vpks9l*x:hy on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it res(hl8:9 kpxrtva :v*psts (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spld7 08i3 ydztqqc) +pleed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist z+p03 q ddilq)y8 ctl7and 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 sli m nx15mb zrk:bbh).s:rwg: 5zng of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelera h ws5 bm).5rnb:z:x1rg bzk:mting it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as tp32qj :n,zce ghin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the pcjz g,q:2ne3angular 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 platesk9idb .e e*r3a, of madk 3.eeari *b9ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ofn ,ob4r1 c .-oj pjtl2cnv9si) Fig. 8–58. What is the minimum ov 29scnp- b.1jt)col4nr,i j value of the vertical height h that 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 t-wo8 f i0zr;edo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformx+:etzev )kx;ly from 90km/h to 60km/h The tires hav+)x:etxze;kv e 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