A bicycle odometer (which measures distance traveled) is attached near pm w5bzst 8yd8 4ano080rb6y dthe wheel hub and is designed for 27-inchzwa588mt6sby dp 08yro4n db0 wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on th ) b;si4gxx8 ugvk5sqq9g ;v0ke rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly,85u9sg x vkgs)g 0qv;x;k4ibq 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 vk;e-nlg0pwxv 7sqa)r w ) ,u2-biv*hbalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater t m2j*yz4 n(wojgqm3 i2orque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your 4rad6p+.bprgpsl dl/is;- )j 0jauq:head than when your arms are stretched out in front of you? A l)lss;0 - 4 d gjd6+ipr:pqbjr.a/apudiagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at t( nyg;/xf h.qfhe 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 harxf hq;.y(/gfnder 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 5e dafd*2d479 sv /umdek 9;okdi6bplflesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotat /d*9 d2645ibuae dd plkefs9o;vm7d kional dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nh3zcqnfuv3 9f) 0uu .harrow beam?

If the net force on a system is zero, is the net torquq xcw gruw5w(lh9z 154e also zero? If the net tor1 9g 55wr l(z4qhwxwucque on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with t0 bw7w w4(dejxhe horizontal. The same steel ball is rolled down each incline. On which incline will the spw(d4ew 7jxb0 weed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclicyg cloh.2+5 lne. One sl.yho+c l cg52phere 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 bottom?

A sphere and a cylinder have the same radius and the same mass. They sta g iyfq, -7fymzoq)3e,rt from rest at the top of an incline. Which reaches the bottom first?yg)-ozmy7eqif f 3,,q Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movebdi : xfrxe1v4 p-b-0ing or rotating objects eventually sl-0ed bbxv-:iex p r41fow down and stop. Explain.

If there were a great migration of people toward the Eart2xsuknoj ,rdt3u3j)s pi-4)y h’s equator, how would this affect the lex-4n o),ip3ru jjuts 2k3sd)yngth of the day?

Can the diver of Fig. 8–29 do a somersault without having an(z.p u bn g9ai3 c+hfgvd2uv-(y initial rotation when she leaves the board?p9dg-z n(v h(buc. a2gfu3vi+

The moment of inertia of a rotati8rc qw.f,(pkr dxn)lp,i i.(g:. mzqfng solid disk about an axis through its center omq,, lgq)8(rp :.pi dw .(izfk.xfcnrf 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 wtz2go g*8melooejm; t3r; );-kg mass in each outstretched hand. ljt2 em;grozo e;*t3 g;o8mw )If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and w)+ui7.g d rrox:jea6have the same mass. However, one is hollow and the other is solid. Describe an exp)g.j7:xreia6 u+orwd eriment to determine which is which.

The angular velocity of a wheel rotating on a ,q t8b4i8u kj kwh1d4zhorizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points4 18zk8u4hijbk qd ,tw east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a 7;uxisz+vd. 7 y iyy0a: o8nqxlarge freely rotating turntable. What happens if you wal a0vqyix8yx7;7o:+.s d izyu nk toward the center?

A shortstop may leap into the air to catch2 fciz7za 2ws+ ytcb4+ 0fc(vv a ball and throw it quickly. As he throws the ball, the upper part of hi7+fic2fta(cc4yz0 z+ 2wvs vbs body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law oj:(ufsgn cu/kf ymy ec;25d;w2jv-je ;y(,rkf conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stabl yv;2e;wjdu k g( ;smuy5-c(/c ey,:rj2nfj fke.

  Express the following angles in radiao9hq/ 7 w6o amnrg5.mons: (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 coe4( gn dp geg4:*zk4b3yv+ njgincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) ofge(nvg4g *+ ygb3z d4kpe4 nj: the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fro58,4e l eg pe8vzdcfp5m Earth. The beam diverges at an angle e cpevf,5 8pe84g5ldz $\theta$ (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( ze5ecv y0l4dxz-l8 /b/ldte a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is8bcvel5/x z t- ldzel0dy4(e/ the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flotvj9 wxje w7ga+ /8;)r ggtj*kor 3.5 m to another child. If the ball makes 15.0 revolutions, what is itsg8 )gkjg+x/evt97trjj;aw w* diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutimg x9x )nx;o 6in7fzh1ons do the wheels make?;19 xogximn6xhz 7)nf    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250) a/b hrfto 4ztc+cn:-m)a6la0 rpm. Calculate its angular velocitoc / zt:6blmn+a-cfa4)tr) hay 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 complete revolution in 4.0 s (Fig. 8–3 qd4 au(ikywgx5u+0) h8). (a) Whatx(u+g dqk4iy 05u ah)w 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 vexablsj6( j6yz qy0q :)locity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedb83)6 xtuwy; u z:svk7sdc :vv 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 cm from rtubp,;w qz)c)8q *bn the axis of rot8p* ;,z)qunbbrtw)qcation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm 6.rw4p*0 kuxctc( b jito 280 rpm in 4.0 s. Detrj kc60 t(wcp i*.bx4uermine
(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 (vvhmz: .uqioxq,r* 9$R_1$ 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 thv:pys 3(r9bzr8p 4yt/w8oze* ,ouiobe Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated f8y zsb(o ,vwr bo:yzpt *iupor/9 e834rom no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from resggt1t :ki dx-w8t xgz05+v8gqt to 15,000 rpm in 220 s. Through how many revvdt kg- z8wxg+ q:it580txg1golutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 12000 08 daixnv 3aypp7l5a eb-w* s8c,*l0yyxjrw rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the s8 rd v.st80c k +um/ g-;*hcq;6ttjaqdfv+leutresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min tt+a8 -t*vvgq d djf k/u;;+.mqc u86s lrtch0eo 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 accel8dms- cs;sji* erates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travi8 s;-s*scjd meled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It 3ytv55vik d 2 ad-eeb:turns 1500 revolutions before it comes to a :tb5de235v a-e idvky 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 redupi1jc8ct68eob yu .0 jces its speed uniformly fro8yc1it p0 j.jo8cb6eum 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

  The tires of a car make 65 revolutions a:9h1 etaxz0yn82p 5 +ikqdkourxdt *:s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diaxqpkdxtz :d 0+1r*o8 ahut:ki5 e 9n2ymeter 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 pq kwwxk*(1ne;z: /bidedal when climbing a hill. The pedals rotate in a circle of radius / qdwz*w; : xk1enibk(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 p b.*o rve02 df0kzl3y55 N on the end of a door 74 cm wide. What is the magnitude of the toodz0.e3vyl k p20fb*rrque 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 6m w66/2idz7a0ys fb czfdlv1v98lg w about the axle of the wheel shown in Fig. 8–39. Assume that a fric7 fsw6vc/ fl dvd zbz890wl2 61miagy6tion 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 hodrq6a7d-4 n ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held q4no dd-7rh6ain 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 reque 7qq*i-w) tkmire tightening to a torque of 38m) q*tekw i7-q $m \cdot N$ 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 u+ tv)ib124o wuj nfodh*14kh sphere of radius 0.648 m when the axis of rotation is through i)khhdb u1 no4+1jf*t2o4wvuits center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle k q 4 it4.mks(yoj-cd,wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25kmt i, c(od-sy. k4qj4 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 rotl 0ml3-.ra l56rzc ub de1v,zqated in a horizontal circle of rcraz5 bu6d l0zrq,mlv 13-le.adius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a ;yq;d) znk9s(8clw qf bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betw;wq;nq(f ykc9 8dlz)seen 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 inerj,fx go-6h mnk 4kg1(f-wz-ubtia of the array of point objects shown in Fig. 8–43 6x-wgu 4jkgon k-(bf- fmh1,z about
(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 atvq8- iouh3i*coms 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 saodl(r4u 3xnp( tellite spinning at the correct rate, engineers fire four tangential rockets as3((polrux nd4 shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylindec 1pnai930xa9dgg+ aa) 1+lnv4efr ixr with a radius of 8.50 cm and a mass of 0.580 k90+f4inx g1 cl93d)vin1epr +aagaa xg. 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, accelera+o1tkp0hyf)y vbqgi:fvt k,6/ 4,r zlting 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-driven m jzme+ 8.jmd2h+vw5: uk * /lzpjbmirv-6x3j rerry-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 uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass 5j vm:jkr +iu eb8rz./d2j + -wxlzphmm *6jv3of 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 9+y uey5pe cx4tl8fc /7zwk*wd t p9a.10,300 rpm is shut off and is eventually brought uniformly to resufp/cetacx4t+8w7zpe95 w d9y*l k.yt 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;j/5/.bi-; wcbld 9 q juszafl 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-ke)hg,7nxuhcz jqm)kc( :/qp- g ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps mus,)uqz nm)g:7e cchh/ (kqxjp -cle, 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;8n/ll oo y o7rsu*xw/. 8–46.o 7u*xno ;/so8wlr/ly
(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 wa36nl.c wtc *i-cu6h 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 withiqa y5.23xi4ywl w9 edcn four full turns (revolutions) and releases it at a speed of 285cq wx 4a9.lyd yiw23e $m/s$ 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 inerti:bij-jz vyp1 d04oa f8a of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of quuq+cg.. h+uu d,pu 7280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg luw6m)s ntp 0+and radius 9.0 cm rolls without slipping down a lanep l+wts nu6)m0 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 sum of 3a6n v24kmoxntw xnv3 n6k24amoo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass vq5 shm . emt2x -n:mpu b+(mk11ipg.npd2w6lof 1640 kg and a radius of 7.50 m. How much net work is required to aw2 1-iupmnb:5h1x2m.me6qnmt l. pd p+gk( svccelerate 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 cm and mass 1.80 kg starts from rest ax s1foar+)/kkrj6gvux,pu x5: ay)n* - aqd )und rolls without slipping dow5ox n6x+),1srv j aur)-uk fk/ayx*paq:du )gn 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 qp:l;d8 -y ddnto fall and its lower end does not slip. What will be the speed o-d8n;dlqd :yp f 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 t;)a1dm:o wtt +o/odeo:rr;6e xzefb4he end of a thin string in a circle of radius 1.10) ;owo:or1b; 64mro/xtdaf:ezd ee t+ m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular284z6lyopw db momentum of a 2.8-kg uniform cylindrical grinding wheel of2o6w8p4lzd b y radius 18 cm 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 side, on a platform that is rotating at a ratv3i4 k6:ii4 ogydch,z bmt ;v9e of 1.3rev/s If he raises his arms to a horizontal 9tivmbzvh6,;4cd4iy: k3io g 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 2jrp 78( ;xxa74j -f4c.sea pqflad szshown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from tp4x4x -;7.(pz2sf8 q7rcaja d ajs lefhe 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 8p 7fcysh.e9z8r e u:bher spin rotation rate from an initial rate of 1.0 rev every 2rbhp9e8 7 c:y8z f.use.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 rotzzlz,8y2e 9u 5map l4la3x;jq/fg:lyating 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 0.40l3 a2zz qe lly8xazlf;9gy ,j4 /:p5um 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 1 l5an x,to)5 tehwwh,figure 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 m;wqd eb4y*ls ag 1x;o xq,;9f8c4.:yut xdzybomentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk onu,0yzx;c nf, f moment of inertia I is dropped onto an identical dincuy,0; ,znfxsk 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.iz0 t e2hf9yg04xmt th8k vn0+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-round platforp1 98rr3wxhmlm of radwpmr 8l19r3hxius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freelym 6.y srwf;7lu-txgr, with an angular velocity of 0.8m,; -xr6u r gy7s.wflt $rad/s$ 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 intomdii/vcz9wv5z w 5* 7sw,wb+d a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round w i*,wdzvsd/zm57c59+wi vwbto 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 involvesdy;w,c sbo))mr/ z ywa:,pf8o*ebm- winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass iff qj5 8o4s-p$ 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 . ab,vor(cif:can rotate freely without friction. The moment of inertia of thcv,arb f.: oi(e 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 *;ok q8oh qgbs9uu22mat the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless beari2qqok8 go *hu;u9 smb2ngs 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 oyijh h/ n.zn-5f 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 with5i n-hy. zn/jhout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same s fern 3+ oxxud.: 3.rb9ssip-hide always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to irbs eois+:.xufx3r.9dp n- 3hts 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 )evxo5sueha p6;qj0ha*e.r (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 ine4y8bdj m5a,si47 anv the shape 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 the torque delivered by 48vyabaj4s 5nd m7, iethe motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg:6hdi g ,th/hn and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conser ih,/thh 6g:dnvation 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 ang0fcpt zv6)2 uxular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, nww ; g,ao/zhjpokhs9.z2mg5 4-tp3ebut with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that . 4-9w3eo/m5s hg2najhppgo w,tzz; kthe lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a :)c ll7,iadphq(5fv z low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1ph:5(,fvc izllq7 )ad 400 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 e3 h+l:dus899w)hwy u ig2ekzyqk0l9 an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface ,fkmjd*vr 0ohlj21gqkb0nu( i o*q3/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 pivotsmo2eyo( wt+/ 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. Ayms t/wo2(e+ ot 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 center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M ha4mlu8x2g i fy xhu5/,ws radius R. It is standing vertically on the floor, and we want to exert a horizontal force 52x ux84h,iwuy f m/lgF at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.5ebbaxx. )b g42m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal fbg xb4x.a5 eb)orce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 la,o5orv/n y 2 u,j6jxdength 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 torqu5j2 yrj6,oovn xu/,dae come from?    $m \cdot N$

  Model a figure skater’s body as a solid cy p rlg*v uzk6)xfzksp;*to u4j,(u9i *, fykv;linder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agt*(*,zgu xuuf;fi 94 p kls;)r pkjy*v,vok6zainst her body.   

  You are designing a clutch assembly which consists of two cylindrical plipf dv/ , 1h:f);betnkates, of maskf; eb/)fth: dv ,p1nis $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tfykhah69zrz5 j;f x7-w ,n06yo,upmh rack of Fig. 8–58. What is the minifkjz 5ym yzhhuxo,96 ;a6p7 -0hf,nrwmum 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, butrkg a )f/zaxleh. 73om fi 35/)z;kfpp do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif.3ixsi0 nwzqbd 7rw5 4qv8u.mormly from 90km/h to 60km/h The tires have ai53 0qqd4m..w r8suiwz7bx vn 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