A bicycle odometer (which y4; b6 hrlw*uqaha1wxr :dz31 measures distance traveled) is attached near the wheel hub and is desigh ruh3xw;wrla1yqd*a :b164 zned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velock a*;zmihp-e2 b/)plt5u aem dht /90eity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial9m2tpahkh l* b ;)edei5pm-z/a/teu0 and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descruwp9aw:zk25+p 6b(-ymvwc xsibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largq mnas )w92k:fvzm5j7 xhm k4;er 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 yo -z 8k1p6cwqkeur head than when your arms are stretched ow68cek kzq1p -ut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheeeik9ci5 -um.om14 dpjl 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 gmdmu ei-9o.45ck ipj 1ear 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 w3u,e qh qb r+n.xcfi7z-w b3k)ith flesh and muscle concentrated high, close to the body (Fighx,7 ni)f qz+q b3u crkwbe.3-. 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 loxs rdb xsme8z1*b )mkc(a;a0yg(, qj3ng, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the 5 p z(q 1duwc 4/bxcaakp4d+h8net torque on a c 48zc5d1 (4udxpp+q ka/bhw asystem is zero, is the net force zero?

Two inclines have the same height but make different angles wita/6n i9 nknn8eh the horizontal. The same steel ball is rolled down each incline. On which inclniankn8 6n/e9ine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)q179s gcmm li5/ gwr .lsn7s1c down an incline. One sphere has twice the radius and twice the mass on1rgs7c5 q g17lms/i .c9ws mlf 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 yqir) )60lz wthave the same radius and the same mass. They start from rest at the trzi )0lyq)t6 wop of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momzt;qfehze03(1akhp 8 rmo)o , entum are conserved. Yet most moving or roto1, z8p m(fh r3the;qek o)0azating objects eventually slow down and stop. Explain.

If there were a great migration of people toward th*o ,4dief uti.e Earth’s equator, how would this affect the lenu.4ieto,fi* d gth of the day?

Can the diver of Fig. 8–29 do a somersault wita3 f(g nu3 7y*vvkyfk-hout having any initial rotation w3 vvf ykk7(-ugf n*3yahen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cente3sbh2 m c-;ojqr of mass i2mqsjb3h- ;c os $\frac{1}{2}WR^2$ (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 rotatinmv,8hm0v i9ln n;*cm fg stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase,m;lf8 09 m *hmncnviv, decrease, or stay the same? Explain.

Two spheres look identicalsc,y *w2y 1vqs and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whicv *2s,wyqc1 ysh is which.

The angular velocity of a wheel rotating on a horizontal axle point gq ;qkca8 1synw)v)0rs west. In what direction is the linear velocitygq1;qnwrsv) 0a8 )ykc of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What*sc; vxc-wk/hqrz +7j / jtxml/i1j-g happens i*st+ hqwvzjxijr /mlk 7;/-cc -x /j1gf you walk toward the center?

A shortstop may leap into thet 5udt b8ysy:j3. cn5o air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs 3ntd.5y5 oj:stuyb8 c rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, : cczd480h3 w,-mvq*+q zkvzxyuz an 1discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways4z kdm cz:* zxyhv w1,-anqq8v+u30zc the second propeller can operate to keep the helicopter stable.

  Express the following angleqbbb fo:ok ,qbhx c/;r2a ,d+9s in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth bewgx9gkx 3 n8jc dx v6 t3,rz3qb18m6sdcause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as sx 6d w1qscgk8tx 98zxn3 dgm3v,b6 r3jeen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from c )zco3jcxzq q,3s5/bEarth. The beam diverges at az3bcq3csz ,x )q5/jcon 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 blenpn+:vb4 5/ ozrxmjkw, der rotate at a rate of 6500 rpm. When the motor is turned off during j / v ,wbr+54:kmxnpzooperation, the blades slow to rest in 3.0 s. What is the angular 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 bd::y:u yczocn +ck *my )q9/fkn9kxa1 3may b:all makes 15.0 revolutions,ad: uzfo9a b9y:1n: yxqycm*ym/ +kkc cn)k3: what is its diameter?    m

  A bicycle with tires 68 cm in dia2arqrh;y;j- u/:yig3l fa v.x meter travels 8.0 km. How many revolutions do the whry.v;f jxai:/r- ;23 ual qygheels make?    $rev$

  (a) A grinding wheel 0.35 m in diac(xf3i+:g p3apxrl4 mmeter rotates at 2500 rpm. Calculate its angular velocity i+3(34f xpig lrpm cax:n $rad/s$ $\omega$ =    $rad/sec$
(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. x)8qc(u- sz*l1 hp,oc hf/us p8–38). (a) What is the linear s/u1 8-olzscu fc phs, p)(x*hqpeed 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 its orbij:bf y3gt+i 4lt around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point gh we; aj)4bnyxq3*n9wz6 ibs0+- oku
(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 parb,hu31wh5h1e / *se/wwp n zowtmj5k:ticle 7.0 cm from the axis of rotation is to exper/ ukw*j/ n1hw :b,eo3e5h1 5mswzhtpw ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly ab(d ete5: id)tyout its center from 130 rpm to 280 rpm in 4.0 s. Deti det( tey5:)dermine
(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 radiu lwt-bv.3 fe)f kqcy;0s $R_1$ 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 spacecraft put themselvesujm hdpj3 n(z *h,/)nz 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-jjh(nzn/,dpzu *3 hm) 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 15,000wza8 3 11jqupgr3zwj 2 rpm in 220 s. Through how many revolutions did it pz23gqawwj31 zu j1 8rturn in this time?    $rev$

  An automobile engine slows down from 4500 rpm )q3wlhaq7x( cto 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 the stresses of flying highspeed jets in a whirling 2s(4nh rz n1pa“human centrifuge,” which tak 2pz1 ahnsr(4nes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5*, tvmo:-,omg,15 tgit b h/k3ad isbu s. How */-i,5smi,to3au kbgog h:m vb1,ttdfar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/hyyy/w jij6:o15 /rqumjdc. )min It turns 1500 revolutions before it d o/16yjqjh) uyr.5ciy/w mj: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 revolutioo d7az oi)o.hx1,ucdd s ;)3flns as the car reduces its speed uniformly from 95km/h to 45km/h Thfslc )aooi d)z3;1u,dd o7.xh e 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 asmqw65mvq 9ka7md w9k8 the car reduces its speed uniformly from 95km/h to 45km/h The tires have a daqk vww9m5d967 8q mmkiameter 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 yuw)xm d hc/myvw:50h(/ by* tpedal when climbing a hill. The pedals rotate in a circle of rax / hm wy0* ty:d(yu)c5b/mvwhdius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcet )ih3bj-- :flpl bqld9sk+60gblos2 of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exerted 0l 9qtllsgj 3-s 6db kl ihfp:b-2)+bo
(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 shg8 or2 dvaoq5*own in Fig. 8–39. Assume that a8*5agd2 vo rqo friction 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 t88up .;2s9k;i fbqi -k 0t m3ygnkgyfso the ends of a massless rod which pivo8sfkg g;-.0yk ys2 pi8 bqtk n3fi;9muts as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of:k4 nf9tttfj ) an engine require tightening to a torque of 38 )jtknt:4f9ft $m \cdot N$ 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 ynd -/ ke8u8aasphere of radius 0.648 m when the axis of rotation is 8e yaa/-k8nd uthrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7em 5i-5wzw/io l96lkoq9 (y .gntx*le cm in diameter. The rim and tire have a combined maiowqnze5 5.9 (ek/- l ilxgtyo 6mw*9lss 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 ,6qyit;xq )w06ozoib )v (ozathin, light rod is rotated in a horizontal c 6i y6xz;, v)abtqqoiw0 z)o(oircle of radius 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 bowl on a uoyo9snp 79l a3dle8w/ nbq im3./i1cpotter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands anp/w78c9da mlbos3 o 1 /ne9 iy.niqul3d 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 of the array of point objects shown74s szr9ocub7 o hw(f weqm;*8 in Fig. 8–4377 (z hof*r8;cu9 4eqboww mss 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 dhqmd ln)l ,hme7ycum8dv ++ ie1 5e93oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rateec msc;( ph()w, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reaw m cpc(s(he);ch 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm ans:.uyydk4tw0y15 b nad a mass of 0.580 kg tysn 1y4wdak.0buy: 5. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fysll0e7-/nx la kewi 5rzw5,r) w ih/-rom 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-dx n; ,)vpsyxs6 ri;aq,kv ,29m iwvgm-riven 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 children (each with a masg2;xx,,;vvsm iwm a ,ip knv)r9s6yq- s 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 at6oy 89xo t6lxj 10,300 rpm is shut off and is eventually brought uniformly to rest by a8lo6o 96xx tyj 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 3.6u;mqin 47l;c u-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whic58itv vp topyd)p1,xpn ;5,rqh rotates about the elbow joint under the action of the tro51qdpn5xp8p,)vi t,;vyp triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered+3a n-vg; ;wubom ek+(roe;mt ibj7.k a long thin rod, as shown in Fig. 8–46be ;m g;mj+a(k;uw-+ .voi 3ok7rbtne .
(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 consf15mzwssq-a.jca:, dpij7 smf 8x 00 rists 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 rest within fouuad6t 7p /.mtkgm(fk4ykw4qb58g ; afr full turns (revolutions) and releases it at a speedf7;58q/pm6ag (4kudtb.yw kkat m 4f g 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 inertj ,g5cu (9 ;gm(vhq+cvqsq m9yia 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 develofv y+1k+, *ku( wp.csja+yfgpps a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg atq*;xb ye.c8 ,/eufngnd radius 9.0 cm rolls without slipping down a lane at 3e.bf u/xy ne;ct* 8g,q.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as 4dq7ge3 0txzv+hku ;pthe sum of two te7zt4e h+ux 0d3 vp;gqkrms,
(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 xjkck- 713kl xwork is required to accelerate it from resxxj-1c k l7kk3t 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 and rolls withoa*ba1omo 0b(r us u-n1ut slipping down a 301u u *nbobas1-(amor 0.0 $^{\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 balancenb.sjcf 5;vurl u 8krf9 (-+nad vertically on its tip. It starts to fall and its lower 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 env r;u +b snukacf5rnlj98-f(.ergy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tl 2pe 5o ub 2s*sgnrv1.2-qfhlhe end of a thin string in a circle5nlu*vl 12.pso heqb2rfs -2g of radius 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-kg uniform cylindric6k)u)kiyuhz (craf5s:c )9 alal grinding wheel of radius 18 cm when ak zc: ul9(yk))cs)f hr5a6uirotating 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 platformzpp .rfs *9qc*4,o.k a c/ogbf that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positio.s* p p.4b,9g*qrff oac ck/zon, 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 shi; w+lvjuy- 0qkc3 on,own in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 +w0jqlv,uic ;nyo k-3 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can ;8lh2ayhfvq5 w43 ls5 /vfgirincrease her spin rotation rate from an initial rate of 1.0 rwg vllv;34h2 f hqr5/5 s8aifyev every 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 axis3)w8q0b4du iay tky8s 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 sy8wy tb)d8 kai3 qu04a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a; 1s sn*fzuaf. 5 )s2lknq)frqm*: bhat 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 momengu kk; 1u :ii ,kh4, 7a5caqr+l;jbg)rth5h uctum 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 of moment of inertia I is drp7( l3nun3j4x xl-1a ap8mr vropped onto an identical1 (xp3l3r4 ular7 njxapm-8v n disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 26eq8+g0*yi v0qq1,5 93fc oegrjz trqre oo;v.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 rotatingzd : s0t8bp1a:dp8o mg merry-go-round platform of rat p:m8oa zd0pgb1d8 s:dius 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 frg,y 5laz8zq0yt0q t/s eely with an angular velocity of 0.zzyt0s 8g/ qaql,0t y58 $rad/s$ 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, ,b f d: ;nzy40a5jayrhlosing about half its mass in the process, and winding up with a radius 1.0% of its existingd, a yhnfra jby4;z:05 radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve5f; be- xe6jde 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 e1 j/ 3n0e(wjvlur j/d$ 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 rotate fre:b ndkrc*a(:px;:ssi m1 r1kbely without friction. sxd1pn:sbck bri (1*ra:mk ;:The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edgeyhf4 ) iu m249vzq7ewk of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertq4y 7zvw2m f9uh)k4 eiia 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 lyinglb/:/w5q +ajzx 1womr on the top edge owar :j+51l wzm b//xoqf 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 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earp9 xzisq:t7 sf)btl30s)r g0wth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treatl )txf:0zb)s ss 3p07qt9w rig the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratey7 )m1:6c:t9(psp 4cn a dyssdhatl;o 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 t-f7kb *6ilwl2d wu+t4uyeg2radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered 4 2wku6y7+wug-f tltdl bi*e2 by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg an, p hn9+syhl7sd diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the lilsh,s+n9 y7phnear 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 angulaq,)2ecvz rmo -w/n9m hr 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, but with mass 8.cucl v+r .yyr*+ .xba10 times as great, were rotating at a speec1+ycbr ar.lx.+ uvy*d 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 the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution au8:qtr-joian:r k9t 1jm ry31ztomobile is for it to use energy stored in a heavy rotating flywheel. Suppose 1t q:iy8o-3t9rm nrj ra :1kjzsuch a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrate2 iuqz.60da2jdlbez 8s 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 surfaceh(4 n jdmq : wzc-(dma)+ vgj5pg6m7tm 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 i sy)h5vs* molq95sqs -gnore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontaloyhsvsqs lqs*) 5-9m5ly 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. It is standing vertifng97 88 8zrxlbhp(ozcally on the floor, and we want to exert a horizontal force F at its axle so that it will87gp( ozl88f9x n rhbz 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 making a turn with a sk9pkopvu;ayhbc::9f5 u17 l radius The forces acting on the cyclist u 9uhsk7c1op f9:;y5l:kbavp and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m ang9edwn2dblh,c 2 cr+t*g3 p g,q46rrwd begins whirling the rock in a nearly horizontal circle above his head, acceleratingp ,g,hq962we+g*crcrdbd3n w2rtl 4g 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 skaterqkjg+ 8 vbf7+pqq03bc* 6 mhts’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for7fbhk*m bgqt8c jqq3 +0+v6sp the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyli czwthzr-2 -glk5 ;6a7xs1xztndrical plates, of mr; t6khx a 7z-5 xt-z2zsw1glcass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tracjk g -fb tq6x;0w 8sint05fa2ck of 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 of the loop without leaving the tqw k5si8;ja- b0nxc2f0gtf 6track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dr-sv wv0 hza xvu1+s/,o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uni 8m:jjq4a 2pszqx.8d: q)w w-o9ltaujformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) 4jq xuj:plj8q w-o a29q. swzat)8 dm: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