A bicycle odometer (which measures distance traveled) is attached n-6od1ru7e5eg up1 qj pear the wheel hub and is designed for 27-inch wheels. What happens if yopeg-1ruu j1do5 e7p6q u use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vej*))/on1umdo flan ,kh y d0f8locity. Does a point 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 o))0k d*1/flnano8 mu ojf,dhyf either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular +f 1wqtvq 1+krvelocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger for/ f1l/tfwh0v kce? 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 withkpm-j l-vn4)igi wlboz(+k(na2g :qx+o 0bx ; your hands behind your head than when your arms are stretchem2l gko:(-+wb0 oan b gx)-q(n +lp4j;vikizxd out in front of you? A diagram may help you to answer this.

A 21-speed bicycle ha24 8rojmeujgj5 ec3i- s seven sprockets at the 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? I-er25u mjj8 e4 c3jgion which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slende3vc2wg: v z2str lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is adva2twg:23 vs cvzntageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?3zqdb asj+*5m

If the net force on a system is zero, is sl5 n9:/ptpl bthe net torque also zero? If the net torque on a system is zero, is the ne 9tnbpp/l l5s:t force zero?

Two inclines have the same klnt2/e3h +)xopan0 iheight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bot2xa3 /0t ln+nk)hpie otom be greater? Explain.

Two solid spheres simultaneously start rola2-cv hyow51sling (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 h- 2h1w 5oycavsas the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radiuss///d)wghz v* y9fc6 uj(tq yh and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic enesucth( vdy/j qz)*yf 6wh/9g/rgy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are lajr cvs3 i445conserved. Yet most moving or rotating objects even4ljvr ca 345situally slow down and stop. Explain.

If there were a great migration of people toward teqlum0/ ml90b797o sy wnnmej sr*,9mhe Earth’s equator, how would this affect the length of the dajbs,qn797n /9 e0r ummlolmw *0em 9syy?

Can the diver of Fig. 8–29 do a kxwu+ o yv;g +zjjck8m9t5 .;msomersault without having any initial rotation when she leaves th u9;yj 5o+tvmgjc +8z.w xkk;me board?

The moment of inertia of a rotating solid disk aboude ytvu918: rnt an axis through its center of masu1ne dv9t r8:ys 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 holdiygh-b zj )co64zi r9+yng a 2-kg mass in each outstretched hand. If you suddenly drop the 46)j cobygr-yhi+ 9zz masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and lxs1 v:2y-t,p7yt roa the other is solid.p2y,-r x7tsal:y 1ovt Describe an experiment to determine which is which.

The angular velocity of a wheel : 0tucol d0 +0y+vgihdrotating on a horizontal axle points west. In what direction is the linear velocity of a point on the t00 t0i:y vlg+hdcdou+ op 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 freexazwd l 9gy /-:vgy+o5ly rotating turntable. What happens if you wal dvo 59ax:y/l g+-zywgk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. A8 r8iqzd)l1r /i/dah .ol gjoqss;e2 0s he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fiaod1 rqjl)r/8hd2 sq0g/;8s eoil zi .g. 8–36). Explain.

On the basis of the law of conservation of angular momentum, gkyo t8m37ff:+gwex 7discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the gm7:owx7fyk+t8g e3fsecond propeller can operate to keep the helicopter stable.

  Express the following angles in rar9:l9ne m1tzbdians: (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 coincidd0gai+.aly. fence. Calculate, using the information insi. .a0+lifd aygde 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 directed at the Moon, 380,000 km from Earth. The uvq li el m7os /7/:/(foaq:cxbeam diverges at an angle al7 qseo qcxf 7oml/:/(i:u/v$\theta$ (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 gzva3 v89yhbmab. q9rf l99l+of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accyl 3z hqlmv ag9b98.b+9fav9releration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chpvqk):ymk 6 :wild. If the ball makes 1wpk): qv:k y6m5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 z0xg.ynd7yl + km. How many revolutions do the wheels make? 0.n+ y zxg7yld   $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. 9s nknxf,7 6phbc,)vrCalculate its angular veloc6nk ,9sp7 vh )cx,nrfbity 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 onn umx .uy+qy)d-tqkm .;rcokho5 .+;de complete revolution in 4.0 s (Fig. 8–38). (a) What is .u .d);t q;r+5m. odcy u+nhyqok-xkmthe 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 th r dg)3otk.t;( 6cyckde Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp+o pxh8 1f-ko,q c4oeqeed 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 9de s5*ei(lw0g0tcxqcm from the axis of rotation is to experience an acceleration of 100,0d9c elsqx *gewi t5(0000 $g’s$?    $rpm$

  A 70-cm-diameter wheel ap32tb n n19qtxlb47gd7efk g1 ccelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determt pg17 nx9f3gt2 4ek7b1dlbnqine
(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 j0-9tgv,7n8 iezkc zi $R_1$ 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 spacecra,+:mw cxn6i hdft put themselves into a h6 n:dci,w+ xmslow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. 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 8.vj3r8ryo hl to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?h83 rj8rovl .y    $rev$

  An automobile engine slows down from 45p+ lcs32e1 (lvhkicz000 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspee6220(grzch (nbak xfmd jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed ca 22(khzxb(m gf6nr0.
(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 unifkv03f(h m v+yoormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled i 3hofvm0 +y(vkn this time?    m

  A cooling fan is turned off when it isl ( o+ocyi907szi gru dz52wh; running at 850rev/min It turns 1500 revolutions befgiyr o0( wlz97duco25ih;zs +ore 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 twycf+-r( gc0 its speed uniformly from 95km/h ryf+0c gcw-(t to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as t3 ee48wx +kurd/rgbgy k )p8f(he car reduces its speed uniformly from 95km/h to 45km/h /xg8gpw)r(4u+edb rk yk83fe The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bikew(sj ) zl8(tfx z99tghi: s.oe puts all her weight on each pedal when climbing a hill. The(: j8s9 oi9s.w x()t lfzhtgze pedals rotate 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 ofm) 5,(lwdwrsu 55 N on the end of a door 74 cm wide. What is the magnitudr,)(wld 5uwsme of the 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 azwk -ryw3cwmg z-8bmh)w 1q4*bout the axle of the wheel shown in Fig. 8–39. Assume that a friction wzyw w 4-b zq-8g)w3rk1mc*mh torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of 17zh4j jn*i7t bah5 womass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–toz i7h5nw4b71ja *jh40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requir (tov0g;3:i ddv6j,h hp(9o vbrmjaq, e tightening to a torque of 38 ot h0mb6apqgv;v9rdiv ,( ohd:jj3( , $m \cdot N$ 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 inertiaop c4,yix 9 3me(q0zcy of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is througpc3yc90 e,iy (mqzxo4h its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle qsm97g :6cv z9iz a4pye/zq ) 0eb-fxqwheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kem 7fb s:c-4z/qga9qp qxzv z)9e06i yg. 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 horizont eyel4( a-gg8ja; wm6cal circle of radius 1.2 m. Cal eag e(lja-6gcw; 84ymculate
(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 potte94b4mj6( nkm vpabx6or’s wheel rotating at constant angular speed (Fig. 8–42). The nmk6o b464x( mjapv9bfriction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of pointpwzt2(: e fit /yc)7s1+bctim objects shown in Fig. 8–43 ab( /)ft7tbpiw +2i:ccym e1zs tout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass isd s1lmcu;oer 3(3ss9r $5.3 \times10^{ -26}$ 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 corre-,p5tmw g0m4lh:5xn jaca q v+ct rate, 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 reach 32 rp5 m+maxpqhjntac,v wg5l -40:m 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 and a ma4 jxt7.inw-cn 4hcxq cg-4q q/ss of 0.580 kg. Calcuinnqq hcwc- tx/j4.q -g74c x4late
(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 to 3 ji9yv1q-2 bya$rev/s$ 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 sma f7x, a9gfjmql*5l 1bill hand-driven merry-go-round and is able to accelerate it from rest to a f q7al ,l95fbm*gij1xfrequency 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 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 10,30+crd y-i4s *je(8dbun +-p jtw0 rpm is shut off and is eventually brought uniforwbij8rs(*u+dt yd+ je -p-c n4mly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6(bmb4i 0kqsx3ohk ; 8w-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 foru:)rh.wob aze; 2uz2hearm, which rotates about the elbow joint under the action of the triceps muscle, F ;abz2): hr.uohezu w2ig. 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 jl;9 ) aaeyn8qyojv,,long thin rod, as shown in Fig. 8–46. ,vaeo 9y ql)a8jj;y,n
(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 twr7+yfgv*g fm jibuk; )o0+zc 7o 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 four full t + qfebkk7giq v)*l;jonf)6 v)urns (revolutions) and releas6kfef+ lg;*i kb7jov))n )qvq es 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 momb mco)v19c4zv ent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tospppvegm46l;sr 9:8mi7g ll) rque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and raq hgilyx*ngwugf jrz z/s1vj*3 .6 q45;cl3r7dius 9.0 cm rolls without slipping down a lane y4sl *7lg c;/rwgj1 g .hqqvzjr 6n33fxu *i5zat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the oap)z/ 43 q(5 u.s pqr++dgtlvpm-drcSun as the sum of two (+3p/u54.sdpq -a)rrmd + ct vgz pqolterms,
(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. Howmslgp3q./mj e0;y)w3 hrb i 3bvg+-lv much net work is req. gb+hl -glqi/ 30)j3rvym;ve bp wsm3uired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withoulg4, kpe 2-ljht slipping down a 3k-el2g 4jp hl,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 balanced vertically on its tr o ql a/y,ikbd*)x;:*hjp;tu ip. It starts to fall and its lower end does noipohjr* ;lqa */u tbydk,;x:)t slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of auzfv2tdmvxv f*c/ d s+*i*+h : thin string in a circle of radius 1.10 m at an angular sp2i/xd:v*sfvd+ tz c mf+uvh **eed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical- cagojqmaq0-.)/rkp-pq6 ii grinding wheel of radius 18 cm when rotating ai/q)cj p---qk0 am.6rgiqpoat 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 pl8k++9zih*k- ubu3e0nco2z gddbk9 rhatform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizonta*9dzbkci+kuhg9- b 3rekn u+8 z hd2o0l 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.3fxt6b8x xq-(4lyx rc i3jdwr moment of inertia by a factor of about 3.5 when changing from the straight position to the tucl 8.d(xxxqi-cx 3ry3bj64ft w k 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 heemo6f s;wra *cg5et+e58,*fhbauk 3lr spin rotation rate from an initial rate of 1.0 rev every 2.0 s to*;eh*+fm5cos8 t56 gekfeb , u 3wrlaa a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through ic*vrtvi5( zkce )( 3-aqgk7ja ts 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 po7 kjk)v3aqcv5zag(t-(i e* rctter 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 figure l s4u.0wusud5 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 the2c q+ gz+xhd+.:t l timv-ua3z 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 l)cze/jqo:e, I is dropped onto an identical diqe: ez/ )lc,josk 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 2.4kp0bo aaiy, 9+ $rev/s$ 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-roh2;f/6uzq uhson+st 3und platform of radius 3.0 m and momentszt;/suh632fh ou+nq 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 free7uk6pum.oq9xza1 rr, ly with an angular velocity of 0rp6u,rqu x.z1 o9m a7k.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 w9le3c;vecc rig q3h (9a on8a1hite 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 3ahccoigr ;39 891cneqve al(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 excrpt-.mf zk1* f 6)muymess 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 to4 b,c * 9s;md 5rpth.trksf5$ 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 30qj-d/*ocphq dk *kz,k n kl5platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person kq/l3h** z-,pn q5k0kdkcod j 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 edge of a 6.5-m diameter z *4seby7ofr*wg 6)y xmerry-go-round turntable that is mounted on frictionless bearings and has o7f eswy*r)xg4b z*6y 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 wfdo31/ i z,zsend of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds 3foidw z,/zs1from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always f )l c0g7br7s32(ydvx qnay4x4)xbf odaces the Earth. Determine the ratio of the Moon’s sl7a(r) xv y2 nb7x)0gbddc4yx4f3 soqpin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/ptc1r83 fubn2e:.nn; xpgwe 9xz9a n+ $s^2$ 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+ tbcc*l6fta 5 the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest ove5tc lc+ab*6 ftr a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg e 91 izo; c)wqan5 ;u1ojo;frband diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg ic r5qn9ob ; azeo1uf1;o;wj)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 rear wh:y6t91px lpl( xbc4a+jit 3 *fbjj+molu46hteel ($\omega_R$) 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 manop1 1;by5pzd ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If itpy o15z1bdnp ; 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-pollutiou77;ry:aj im wn automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m anwau 7:m;7jy rid 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 r),jbnfok+.zc*p kaxy)g .* c 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 at uf( ygiw3i*0x+ k*cazspeed 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 uejt nu3 :i43rignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the 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: ni rut3u e:43jSee Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the ipi1/* + jw) q)v:k5bjliptjlfloor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step hasli) twk5ppi/ jj:v)1j+ lb*iq height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn withoi-ynn/c(ntmjs itp d 9 u:-9(* ag1gd a radiuso dn -d i9gp/jag(:m(t- n*91tuyicns The forces acting on 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 i(mzfcq 72el*urknl (3nto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What i(ql c3zr efml2uk* 7n(s the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bjwkvj5y2+ugq;lfj o*6aw . z9 ody 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 sa 9 lz.5o vq62*wujf+kwgjjy ;pinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which qq8nrzl) dl*-j ;(bfkconsists of two cylindrical plates, of mq)-drfb l(zn qjl *8k;ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of0 cc4azoebe,(v2cf;f2alx)3v w w-e g 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 lac cf;lvx-c24 oeegwf b3a zv e02w(),eaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but sq(a70knc 4xpdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducez z1i ph1 (uwq3ub74xms its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acc qbz w17u1 u43hpi(zxmeleration 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