A bicycle odometer (which measures distance traveled) is attached neariiut*8s8i .j d the wheel hub and is designed for 27-inch wheels. What happens if you u d8*.8uiji tisse it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poinbi.lhg4xj27pp 5z 9lh t on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration changh 24 ph 9x57zjibgp.lle?

Could a nonrigid bodym/;z l y7o;efo be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explai *fqzd)j(b -bqh 0xn5kn.

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 you .leqacr (:ws8r hands behind your head than when your arms are stretched out in fron sa8w(ceq: rl.t of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thuf i .3pu(8o c9lipi1 ru+pg5he rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a larg( cuipp +oghi u5l19rpu 3i8.fe front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fle;5rbu9.j u0fbq otn2x sgu h-;sh and muscle concentx9bfrn;q osb;2tu0gj u h.u-5rated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long y:0jsps+6fr 6 +ojpva zg i:4t4ulwd8, narrow beam?

If the net force on a system is zero, is the net torque also zero.1 a/l ki+tark8pjr1h? If the net torque on a 1ka8r+ipjahrl /t .1k system is zero, is the net force zero?

Two inclines have the same height but make different angles withd xo 5gup hrl9+3xwd-4 the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of 4+xwxd h95o3u dlpgr- the ball at the bottom be greater? Explain.

Two solid spheres simultke8b: l ;dhcxl jh.07zy dfe1-aneously start rolling (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 there? Whidy ledx; zck: 1l7hj-ebh08f.ch has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the samenybs**j4p3c ogw *q*j z) pbm7 radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which ha) jw* sp z cmqb7*yjpog4b3*n*s 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 moving+h rip1hvz/+f or rotating objects eventually hirzfp1 v/++hslow down and stop. Explain.

If there were a great migration of people toward the Earth’sw9rfivax//1z dep .h(+vxa+y equator, how would this a1 phxa/rza+9/y(x. vf+v ediwffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rag)3kbw6 pp 18qj l8hcotation whe h 6pa8jk81qlbp3w) cgn she leaves the board?

The moment of inertia of a rotating solid disk b yuas:/t-6kjs o.ox. about an axis through its center of mas. a:- sutby 6k/.sxjoos 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-ki qxhcd6tl6- .g mass in each outstretched hand. If you suddenly drop the masses, willq x. lhct-di66 your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sayf23ia,nwiu, 3 3heclx u2+lu 5bu*iame mass. However, one is hollow and the other is solid. Describe an experb3u 3,y5 uxnl aai,eu*cw32u i+fhl2iiment to determine which is which.

The angular velocity of a wheel rotating on a hori 67l*h;f bmtm64bmva xzontal axle points west. In what direction is the linear velocity of a point on the top of the whee6bv mm7a; l*t4mfbx6hl? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable)*l8duc0 ct se. What happens itdl )c0 *cus8ef you walk toward the center?

A shortstop may leap into the air to catch a ixg +pgxf +fxf17:gi5)gjyi 6ball and throw it quickly. As he throws the ball, the upper part of his body rotatgfi 6+gx7y x5 :gffij+g i1)xpes. 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 of conservatiory s 54 qsi-vw/e4ps*rn of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discusw54s si*yr /sq4vpr-es one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radian68 n.f x.d4yepdr3ed ds: (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 coinwc1ya)2qb .g fcidence. Calculate, using the information inside2wqfacg. y)b1 the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directebi*. d* ss-* lnvy-plzd at the Moon, 380,000 km from Earth. The beam diverges a. y ils*-n-pdvzs* bl*t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When 2 tp 6f327 faysze,rajthe motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angu2 z jff,326ast7r ypaelar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anow2 hc8d;ghpmyaau.-d4yx 1x/ther child. If the ball makes 15.0 revolutions, whx 2 w 4;pu-ya ah.gdc8hmdy1/xat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolg jsj.tk10r. kutions do thjjs0gkk .tr .1e wheels make?    $rev$

  (a) A grinding wheel 0.35 m muzqp2(vf yvud, f07-in diameter rotates at 2500 rpm. Calculate its angular ve(zfud v70v q ym2pu,f-locity 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 ini+im)k 3lxax ; 4.0 s (Fig. 8–38). (a) Whaxak ilx+) i3;mt 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 o68mzzaa( d1z(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 pointftwfols,sl-euj*u6p50li:um v z -6 9
(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 cend (h atvntj(f)9z.-m axe,dq4trifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration )( 9xm.ajda(t,th -nfzd4veqof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centep++ :hs8ou 5o8wxx6 x1xl(z xg kw sl0(jwtw(xr from 130 rpm to 280 rpm in 4.0x(o+1:k0ts5 ( lxz lpwsh(+8ujx o xwwx8wxg6 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 5 u ld;pt931uywjdjhr5ir+kkwzh1q ug* q*,/$R_1$ 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 amtkz03ki9exngi( n(0 u:r6xa board the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, the3gxt 96kna0ize0m kx:ir n((uy accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rv.c7.xa;z svy p+ua1k :6qlampm in 220 s. Through how many revolutions did it turn in this tim.vpa6v lsy ;a7 +zm1ux: qcak.e?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 r/jij7w 5gnsr4pm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying higck(cw )gi0agah.*a g(6m cl o5hspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaco5mcga wci (0g( .lg*c) ka6hahing its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 36xhsg :cu ywi tdd(ptt,) 48a34(0xxaz0 rpm in 6.5 s. How far will a point on the edge of the wheel have tra t4g( s80duxdh xttzy,(ap 4xca)3w:iveled in this time?    m

  A cooling fan is turned off when it is runni*v:/qgx )t1kvzke1 bt (fab ;nng at 850rev/min It turns 1500 revolutions befo a b:1nebv/g);(xkvz *k tfqt1re 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 suhxi:.y 8; ) onmo 3lwba4epb/y4ap3qpeed uniformly from 95km/h to 45km/h The tires havh. o3ob)ul3 :4awmx8y pq a;i/n4ep bye 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 uniformly fv 6nk-(4jvmz:1ql; 4vsc xqjarom 95km/h to 45km/h The tires have(4v1vcalj6-qkqv:jsz mx ; n4 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 weigluqjlfe14. zrlj l :;yy;1 0dpht on each pedal when climbing a hill. The pedals rotay;le1 ljr 1:0.pd4jzu f; qllyte in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N x/)zys wtxkv/4iu 77 a.wg(vpon the end of a door 74 cm wide. What is the magnitude of the 7x//iu)g .akz7wsp x v(wty4v torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown inphn9 ot.b5xgv dhb , g ;toh+0)ip.d-t Fig. 8–39. Assume that a friction ngp 9bt -;vod+d.p05hth go t)h .bix,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 1f0 vplbn4t3 9i:f xdnkzi42tto the ends of a massless rod which pivots as shown in Fig. 8–4nbx2t4f ii:pzkd4tn13vl 9 0f0. 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 engu8-fml4o co 1ius kwe.7z rszp4s 95x)ine require tightening to a torque of 38 1uki z9meox -u.s7 o4)85rf zsl4cwps$m \cdot N$ 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-kq- g./t;8vz (kpi anaag sphere of radius 0.648 m when the axis of rotation isaq .ntkzi8p/ -( ;avag through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. aa5; 7tk bsbg5by0tg2The rim and tire hav 2b0g5 g5kytbab s;7ate a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizo w-3vab :nvpj:ntal circle of radius 1.2 m-nap :vb3vw:j. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angul jc 66na8:knaz cla o;8mh(*qgar speed (Fig. 8–42). The friction force between her hands and the clay is 1. :al8nzh(g nqo6kac c86;mj*a5 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 i tu2582qng vtsnertia of the array of point objects shown in Fig. 8–43 abvqu58tn2 tg 2sout
(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 a7m +2 ;q*bl pjhtwl,igtoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engra riywds9cuc35 -:x-1 l2 vtkineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is9r-d wrc2-3ik 1 cystval5u :x to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8w x+tc 51vyt kj8dn(0n.50 cm and a mass of 0.580 kg. Calcul8vc kwj1n0n5yx( dt+tate
(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 toq46dn 2intdoi fp98qqwr4/ p- 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-g 18dqlwt3od1 7 tr23bfhm*(pdk t5wm driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two chiob1wd mgt3 k(w*d875lm tdf3 rq p2h1tldren (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 shut off and is eventually b fu-uot xue*xl1*i/, mtuv4 p+3 0yjoba)fq1jrought uniformly to rest b t 1lbup/0+3e4uuvxj juqtmxo 1*if-a*),yfoy 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 acceleratetv zdqvg7 (/s b3k)dqy0mv 0aa4e ()ngs 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 bak 0na*( plajf6ll is thrown solely by the action of the forearm, which rotates about the elbow joint under the a jfala*(0 6knpction of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considere5 vdu .qxc38jtd a long thin rod, as shown in Fig. 8–46q.t53xdj v uc8.
(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 ch2fntek6p3o6 onsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from resajla1 387 r0k2c wdloet within four full turns (revoc8031alaorl jk d2w7elutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of in(2 wahsws2nc -3ghid(ertia 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 2)vcp( xv rye4jk8a,e(80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radi;7 s zvvi)bf;y.i/mzvo2/uklus 9.0 cm rolls without slipping down a lane atmbvz /l7zs;y v)o 2ui/v .fki; 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the surn0a3+vd4 9 pvf et 8sit.t 1yshx*db5m of two f*dtb8he r. ix91s vn0543y vtsatpd+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 of 1640 kg and a radius of 7.50 m. How much net 8 d1gj cd3mv1y8k, t aor;a*gm1qz/zw work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assum8d/v ado1;yzt8,wj11gmmq3 akc * rgze it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stahykg:4h:gp ;arts from rest and rolls without slipping down a g;h phg4y:ak: 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 to fall and its)6 2yc2a:r g3yl-(sxsqj k mxn lower end does not slip. What will be the sp3ml)6g(s-2 k2xx nqjry:syac eed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular /fr,rmg1 u5bgdi q8u9momentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.1dfm, / rg58guu9bqr1i0 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr2- ytbt8xi- fvw3f( hvical grinding wheel of radius 1ttb(h38fw yv-iv -x2f8 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 plmcf7 0bdq zs(nh)v (2fatform that is rotating at a rate of 1.3rev/s If he raises his avbqs)(fdmfn 27 h(z0 crms 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 he.o, 0y , te:gwiyfk7- vkzfsam 40bf+dr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations ,s tmz wve-a7odyfi0, .k 4+0yb:kfgfin 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 rateg6/ p5 ah- yx)2vsq0m2oo;n 9r dmaflo from an initial rate of 1.0 rev every 2.0 s to a final raoa;godn 6l5 2fyqarm-o vx/p0)hm29ste 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 rotata- qvg7*y1tkk t7k:.hhjs3rm ing around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk 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 chks7ath .k: gj1m7*v ryk3- qtlay sticks to it?    $rev/s$

  (a) What is the angular momentum of a fig gfzq;e158t5tczj0urure 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 Ev/refu0eb pwb977s- e v y +h6vu/ob9marth
(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 inertia I is dropped onto an 60bzy h(qrud)1i4ceiidentical disk rour)1(ihcq0yib6d e z 4tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at :r,u5uk7e( 6b t.d ktxaal q:t2.4 $rev/s$ 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 ce(cfn+*ns aht, )jig 71dko 6qwnter of a rotating merry-go-round platform of radc(j7i n) +qao6*tgh, k 1wdnfsius 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 w9 d.:yel vuw;rotating freely with an angular velocity of 0y l9:uwv; w.ed.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 white dwarf, losing about half its mc w3ozdi f:7-h5bwr,oass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries a 3 rwwboh7io-c:5f ,zdway 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 ofv( bja7t +x.xz 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 rj4qnc.p3bofv 1 xjpbcb3-; / v gpm86$ 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 roupf-q hcr96630vf8h r2perbctate freely without friction. The moment of inertia of the person plus the platform isr3r qccpu09er vhpbf8h - 2f66 $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-mkd 2-o gd69mpu diameter merry-go-round turntable that ukmp2ddo 9g6 -is mounted 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 thesgs ti6f) 5vbi8ts81+ (3 sd(ojng prh end of the rope lying on the top edge of the spo8 v1)(s tss8 d5o 6trih(gijgpsf +bn3ol. 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 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sid9 ;uy.p o6+8mdvuzddc le4.hrw /d(yae always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat theru.+d/wdp4; ddzy6.cya ul(e 9v8m oh Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates /dqrxvxd4x.w+y8 8b vfrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acqumf9* 3ri5vqpz v cw;6cires a rotational rate of from rest over a 6.0-s intervar59 v6vficq w zmpc3;*l at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylixxsljj /2s0j 6n,;f9+elgtri ndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) nj +,i6lgls9stx2 xjf;0/re jthin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it 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 rear7 b*5u s12 blvcy9 xy,qk7iq;efdtqu( 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.fg-2j;vpb g9 of1xedy 6c) )m*fsc iu 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, whafsoc f1g-p6d c ;*2euy.xbf))gjmi9vt 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 to use energy stored ino hjp -94fmqhuv4qi -1/zb5qc a heavy rotating flywheel. Suppose such a car has jhqqzup4hv1 bm -qo-i /fc594a 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 sienqjn 02;;uan $H_2O$ 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 surfacgm: /6 tg:qdgl1* wkfne 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., w (ae(mz ypq 10)mn n1daidi;7ke ignore friction) about a hinge attached to a wall(d1ka in e ()7 imd;zp01qnyma, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. Ijfm/d 2c6 a.tx+,zvo4ia rw5 mt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that a ,mo adz j6cf2i.m+xt4 5rwv/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=4.2m/s on a flat road is 4qax5qe dt: v- ;g6kefmaking a turn with a radius The forces acting :g4fvkte56x q ed -q;aon the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sliho .lmy n)5xd1nv:9t 1(bmofkng 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 achie:(kt )od 5fm.no9yx 1bvm 1lhnve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skateibce d7789)9yvpzx tc r’s body as a solid cylinder 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 against her body zdc itvp9c7 b)7y8ex9.   

  You are designing a cl;)-i3ub6,g j(rkk khrieipxjh h* 9z8 utch assembly which consists of two cylindrical plates, ofbkz6)hjjki 3ergrph- ( *h8,9;ii u xk 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 of3cum6))jjutx ohs /,kr uj 6y5 Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point jjyc umu) huos6x6k)r,3j/5t of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumo2kh9 ) j.fpdfe $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduc2 q *ly;uh7c 1ypq+lxlqj6qk ny;p*f-es its speed uniformly from 90km/h to 60km/h The tires ha-+xyn pj;qy7q;lc*h1ku l qy*f p6 2lqve 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