A bicycle odometer (which measures distance traveled) is attached +ym(se, ej6*-z +oodu nrdm3e near the wheel hub and is designed for 27-inch wheels. What happenseesod r nu-+j,+mz(ome3*d y6 if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant a)g0,m:dmuc tgj6 td l8ngular velocity. 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 of either component of lggumm0jt)8:6cdd t , linear acceleration change?

Could a nonrigid body be described by al q*6poj,il1y m xl0*zf:a g2m single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater t k-h)a0tg;zlrorque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your heq k 32xpe x543 yti450ck*ejql/txp coad than when your arms are stretched out in front of you? A diagram m2*/lc 4 qj34tc30ypk xeq 5piotkx5exay help you to answer this.

A 21-speed bicycle has sxyd, 3 jih0)sjeven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket ory dhxs ij0,)j3 a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs- ru.f s0xqn 2t3nm2xc with flesh and muscle concentrated high, close to the body (Fig. 8–34). On surnt q nx.2-fm0 2c3xthe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) caroc. ,; iyxu2xaoggz)dp6l;b/ry a long, narrow beam?

If the net force on a system is zero, is the net torque alsomh /:q6wju q/tbq70ee/- yqzw zero? If the net torqueq6t qemzqq 7euhwy-0/ w/bj :/ on a system is zero, is the net force zero?

Two inclines have the samyfr(rspea *r,p vd(;1 e height but make different angles with the horizontal. The same steel ball(rdre,a(1r *pyvpfs ; is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an injo ;mau p*vqgt cn*4 er*2;i8ucline. 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? Which has the greater tot;* peu*vit8g;2 *4n rmucq ajoal kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They s ipbs 8 8 bb5rd6mq64g99.uusoeez9oxtart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which z89gd.rm s pib6qs ou6b954o9eu8exbhas the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentume,f468ueq u mu and angular momentum are conserved. Yet most moving or rotating objects eventually8ue4 efuum q,6 slow down and stop. Explain.

If there were a great migration of people toward the Eosz71 + g6yr dzg ,*n)r:bxlx; ue1qnvarth’s equator, how would this affebry z*;6,n1g x)qxzsuo evl+d: 7 gn1rct the length of the day?

Can the diver of Fig. 8–29 do a somersault withoupo7jb 0yeh0e(k6yw, s*wuc; tt having any initial rotation when she leaves the boardw6;o0c,w0h ye p sybtk e*ju7(?

The moment of inertia of a rotati.xx jy8r:22g9i oncrfvp-td f;b1 /yi ng solid disk about an axis through its center of mass fn c rpxb9tx:g2i -jr./vodif8yy 12; 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 rotatqybc+7rx-je +ing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, wil -q bx7+jeryc+l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howb6g.sxo2fg-: n jrdh9ever, one is hollow and the other is solid. Describgrjs:.ob xd- f gnh296e an experiment to determine which is which.

The angular velocity of a wheel rotating onm-)nk 2t li1fube-b v3 a horizontal axle points west. In what direction ib1-e3 2 tf )kunvlmbi-s the linear velocity 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 -3yu.qqc/ iqj the edge of a large freely rotating turntable. What happens if you walk toward the cy. 3juqiq- /qcenter?

A shortstop may leap into the air to catch a ball and throw it quickly. As he r3 5cqg*ecu(b throws the ball, the upper part of his body rotates. If you look quickly you will notice that he(qc 35gucr*bis hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discu3dycevj 9ox1b*e5 j-rm vr .)mss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the secondv 1rm3j.y5 jed)m* c-e9 rbxov propeller can operate to keep the helicopter stable.

  Express the following ajbhdjp)6f c./0 am k4engles 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 because of an amazing coincidence*0ihra )3kpmx . Calculate, using the information inside the Front Cover, the ang mxhakp i3)r*0ular 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 fron b; h.csr pj3)anghps :s8.4im Earth. The beam diverges at an angls8ngj .i)asbpr n :cs; .34hphe $\theta$ (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 au ln+ lr*o 3q5k-.bjzz rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades+3n qu zlo-zb*rkl.j5 slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothen 9a8 5a k/d2gqwkshei39tu0er child. If the ball makes 15.0 revo9nqa93kdwi8 tgh5s/ 2ke 0au elutions, what is its diameter?    m

  A bicycle with tires 68 *, d3tvtzrjrb.le 3u; cm in diameter travels 8.0 km. How many revolutions do the wheels.d3l3 j tbrue *tz;,rv make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate irh,5/o6afnyu rb ht4.j)d,j v hnuk/ :ts angulayv 4u . 6d,/n5jra jhk ubf:/rthn),hor velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution inr cfl9g0jr81 nh )xi (d6oag:j 4.0 s (Fig. 8–38). (a) What is the linear speed of a chilhj lg1 0:id jor) a68(xfn9rcgd 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) 4yaqs tow uoqpzj*cop9f7gne/ +) .8 its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe ovq.;fs0pk( zed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the p6d/28g+ gs5gjdnqpwaxis of rotation is to experience an accelg w+5dqn6j /sp2pd gg8eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm tourfnqy9) /m t9 280 rpm in 4.0 s. )u9n/tq r fy9mDetermine
(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 5n aq .6bf)fc evl x9tg hj0u*wp+9xj+$R_1$ 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 sgs+t7wfm: is. pacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as i t+mfs.w sg7: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 rpm in 2sg 5x/ vw*;zy /ccuv;g20 s. Through how many revolutions did it turn in this t*zs5/w u v;c;v/cggy xime?    $rev$

  An automobile engine slows down from 4500 rpm to bmy(r58 7 s na8b/hbia,k* e*oqv2wd myaz/w21200 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 stressesg:h x)vx: keg5 of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachi vkx:5x)egg :hng 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 hjz7i8*gnz+ e 6.5 s. How far will a point on the edge of the wheel have travelnig h7j+*ez 8zed in this time?    m

  A cooling fan is turned off when u6nzvbeq7 f8u3i 7kv;it is running at 850rev/min It turns 1500 revoluun3e76i vuqf; vb7kz 8tions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fru f,a w30zuo 6zbi2*hkom 95km/h to 45kuaukof*zbh6 2 z,wi03m/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifowf0j:j9 y.op96o uazc rmly from 95km/h to 45km/h The tires have a diameteru69wya oj.op 9z0jf c: 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 b x3*ejw)/ sha w-.lmtfike puts all her weight on each pedal when climbing a hill. The pedals rotate in a m)ewls.f3twh a j-/x*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 on the end of a door 74 cm wide. Wy6gl8hev 8tc) 44k 8wb3lva cshat is the magnitude of the torque if the 6vac scwk 848v4by lltge38h)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 in Fig. 8–3oyf.xem)/a0g tss )ulval)ru0oa g *+.)vo 9 c9. Assume lg+.uof 0a/ets m0u) vvr9og)a*s)x ol)ac y .that a 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 ab)y6s a4 -kklittached to the ends of a massless rod which pivots as shown in )ak-46kls by iFig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tipjxsl/h /ond bgjrbc9m,3e6,ps5 22 f ghtening to a torque of 38/hsrgp/l2,p 5bes92f,6j b d3nocxjm $m \cdot N$ 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 spheidg/ wy10d 8jwre of radius 0.648 m when the axis of rotation is through its cent/1w0 dw8yj diger.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclej)ul83gj7 dup,al:r d wheel 66.7 cm in diameter. The rim and tire have a g l8ua:p ,lr3j7)ujd dcombined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on1kfgwaxx;h:w+/k,d qu w0s 9j the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calchx/ k ju1;sx,w+:kagd9qw f0wulate
(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 whb -8fu1,b3rrad aujl rp f-/v,eel rotating at constant angular speed (Fig. 8–42). The friction forcej a--blu8 r,d/r,r1ab f3ufv p 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 point object sw3faze;. 4kts shown in Fig. 8–43e3.;ka t 4zfsw 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 oxkxcm o(b01e66w +focy ygen 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 cylindri p4 sngyxh uh-go8((i3cal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fy(n8 g-sg43x (p hiuhoig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is 1b;pwdcw9k: kg m6 (v rd6m3j.xlqyz0a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcul 06:wj dkmc; 1dbrlz69 3(x.gkpm vywqate
(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 f6 d)jpxk1 uvb1:fqhsq13)v eerom 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 tange: qy g4hjvkb.ud7)xyyr.cm ), ntially on a small hand-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 children (each with a mass of 25 kg) sit opyjg.ru:)7 cx y4h, d ymkqv).bposite 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,zd9bs7d:vl nus(e ,6d300 rpm is shut off and is eventually brought uniformly to,7 sud6nd(:zl bev9sd 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 accele 57f v)ggqc/xs9jf6xirates 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 ball is thrown 4c 9u,5+bd lr pd9owebsolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig.du9w,p 9+erdl5c4b o b 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 considere384mldf 2m1 ilp2 rdlqd a long thin rod, as shown in Fig. 8–46iqp2m d4fl2 lml1 r38d.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses,c:x:gs * x cno,tbz/ e9*:yzll $m_1$ 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 acceleratesgp. l,ou) ihz+ the hammer from rest within four full turns (revolup+ h )ig.ozu,ltions) 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 inertia of vw6ma+9xb, j vk un.:v$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque 1 p cfrj*6g7vb36iyn6f m0 .uecr 9hqjof 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 3;c;xc6sffkjej y4w/7.3 kg and radius 9.0 cm rolls without slipping down a lane atexf;/jk;4fwc c3s 6jy 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thtr tx)zk y9ano,3 v/v/e sum of two terms,n3y t //,k) zvtvo9rax
(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 h5-pgt8ifc(x-e,s i r3yhw e8 and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume gy3(i xfsee- h-pw5tc8h 8ir,it is a solid cylinder.    J

  A sphere of radius 20.0 cm and makhnc 4s 5+ g),3rqb riws3pgo9ss 1.80 kg starts from rest and rolls without slip5g ih3)sr n s+g4bo,wc9k 3prqping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced s*bx0 fc.aqi2 yo4x27 nha- evvertically 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 q*o27y0 hxv-sa f4a2e.b nxci pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bpf(vz6nmb5yuu;2 f0 qall rotating on the end of a thin string in a circle of radius 1.10 m at an angular spev0y( ;2ufb6um 5zpfnq ed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindingtoak cpsl(. (4m;1otv wheel of radius 18 cm when rotating a.c1l4;p(s(otktm oavt 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rateuxhn l8ti05t 8 of 1.3rev/s If he raises his arms to a horizontal position, 8ni80 tht 5xulFig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of iner.5arcig cbs(3 tia by a factor of about 3.5 when changing from the straight positi5ccbg. 3( riason to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an ini 1av pg; nsy4bdx55vr 8aq*wb9tial rate of 1.0 rev every 2.0 s to a fi bvbs *4qady5vn9gx8;w5a1prnal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its c-zfajx 9; i a*s0,hikubgc03oenter 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 thi sa *j -ai;9z0xgu o,b3f0hkce 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 h1 ga 8dfnc03p6xwt e3spinning 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 momentuh (p.r ;oj7nbpm 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 droppedzrjsby,,i,lgv+58h9 mp s*u:b 0dejr onto an identicsvs0zhjr gb8i,m5djubr*9y + , :,elp al 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 2.pp*g y)*.zmzh/6l ry zg/b52eabt, f a4 $rev/s$ 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 9o c ,ezn7wdxit6g28q center of a rotating merry-go-round platform of gdi x2ec7 ,zt9n6ow8q radius 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 rotating2 yvefaxxb8-x*7x4m iys2e g4 freely with an angular velocity of 0.8y2y2 xxbee8mfxa4 vx 4-si* g7 $rad/s$ 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 wsdk.8- kf3pdmass in the process, and winding ukkfwp.-8 dsd3p 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 to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 12l * degdb4)qvs++bk(z(8 hnez0 $km/h$ 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 u3,8a s gzv4ga$ 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 platfteeta 7w6r(1borm, initially at rest, that can rotate freely without friction. The moment of iner1wtee6( rbt7atia 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 edge of a 6.5-nl ssy0)tk; ;;,q4m* qrgjzagm diameter merry-go-round turntable that is moultr;0 msyg kg4z,;q*;)nq asjnted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rop.+ rc n;:x0h7s9d c*km/ jca2ppiief ae 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. 8–5fp: *;h jekcn +ir0 m/dxs7.a2ac9icp0. 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 sam.ri2tp:-eh 0iy cdjg5e side 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 l-gty0j : iph.5d2r ceiatter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest zo /fn cd+8dh/eih a/: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. cz7cia0p y c-n4ojw5.20 m acquires a rotational rate ofp5 0 cow.nc4ijc-a7zy from rest over 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, ea:;kk5nl n1 5l,ytptwzz* z8a 0ngl:r sch of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m.8lp n tgs*5n 1nzy5 wkk :tll;az,:0rz 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 ep*29h 8dio1yj 268gmlsori j 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, b(c7j s2n1sozq m 3dc5yut with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it occ72s dzj3qm5s y1(nwere 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 automobile is for it to use ene zw /;(tz(wcm uh/m:gx /*xkf1fpiu 2rrgy stored in(wzg(z /ur/ *fpw;u2 1i fkm/h:mxx tc a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to 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 al(4fdn0 : 6jfsyf2nn qpjn u 0;f.yr+nn $H_2O$ 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 ati r5a8y4pi 5lp 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 ignore frictiam :bj w,(c:iq53;mxzkfzfxo z:6z +jon) 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 6c:i3(abxx; zfz w, q+kjz mmjof:z:5 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 vertm0, hkodi6- ; m53wnne)nu ilaically on the floor, and we want to exert a horih n0,l ieondmu-)n3;5mk6aw izontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speecypnqs: o.+n rm :o(*a6 , egoeabq8h8d v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclio.s8gomp cyaen q(ear :o+*q,nh8 :b6st 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.re/ ,r.2wyp pw50-kg rock into 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 is the torque required to achieve tw,.ry p/wer2p his feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body asgaqrzt *6v.p2j zjbe k1u1 9z* 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 bu*qrk 1 9gvj.6 pt2z1ezj*zatightly against her body.   

  You are designing a clutch assembly which consists of two cylinsjdp2 (p p4ww1drical plates, of(pw dwjs1pp42 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 of Fig. 8–58 40te hyl*fv4p. What is the minimum value of the vertical height h that the marble must drop if it is to reach the f *pv0t4 y4elhhighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumej11 lsmwcx2k2 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car rgyq sz)k41n n.educes its speed uniformly from 90km/h ns k1)4yg.qnzto 60km/h The tires have 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