A bicycle odometer (which mea fo 7a2:rle kx9l++x3h:wiychsures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a b:oh c:7fi 29wyk +3x+elrhlxa icycle with 24-inch wheels?

Suppose a disk rotates at constant an14hfdgtif: 8imgkj (-gular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angulaf8dfj(gg miiht4 k: 1-r velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body beg c szxy5g51hp3 5-m.hph4sed described by a single value of the angular velocity $\omega$ Explain.

Can a small force everj; nz qzd+3blky/v*b54 h1zf -) mbitb exert a greater torque 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 behi rlq*(w2k4rxct2u vs(:ut9mknd your head than when your arms are stretched out in fr 2t4w9ku q(cu2tr : xlrs(*kmvont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at t .nme51xnm;a khe pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket ornam.xek5n1; m a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legw;5lqzk8 yo)p s with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mas5q ); lkpw8zoys is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longnic4ggtlllm .bdyg f-x5*gw9;98u+ t , narrow beam?

If the net force on a system is zero, is5k ,zrei; a8l g8l1htztp)2og the net torque also zero? If the net torque on a system is zero, p8 1,zkli2)8 ga gor ;5ltztheis the net force zero?

Two inclines have the same height but makek fb)7tyaup3e 0zf s:. different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball a ) bff3 y7uz0etks:p.at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclk2l -cf-kfgf+;xa:*, yo tokaine. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has thg-o 2oca* ffkk tlx,a+-;yfk :e greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sa:i:oh5 mjmlkv97vco+ 9c6 dnnuf,b6s-vfy( eme 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 energy at the bottom? Which h6:dc5v9iv6mu vfjb9l :ne+c,f n s(y7o -kmo has the greater rotational KE?

We claim that momentum ;8k .yed1nfacvsz.r ; and angular momentum are conserved. Yet most moving or rotating objects eventually slow down .s;af crzde vy ;1n.k8and stop. Explain.

If there were a great migration of people toward 4yvu k33q4kuvm dkhr5/ qc8i7the Earth’s equator, how would this affect the length of u kmq qcv7rd4vkh u5i/4383yk the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatlm h2d u*4b7w;bcc .mlion when she leaves the7b.clum4l2* c;hwmdb board?

The moment of inertia of a rotating solid disk about an axis through its centeyns obo68dl.w,r h(+i9;gyrj ca wj;3r of masjr9b 6l3g;o;yy.saj ori (w8 nwdh +,cs is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding anuhz.; cs9k2g 2-kg mass in each outstretched hand. If you suddenly drop the mas2u. k hzcs9gn;ses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hor eu2v c.6oap-llow and the other is solid. Describe an ex ae6p.uvc2 o-rperiment to determine which is which.

The angular velocity of a wheel rotating on a vl/2:f*lu,1t kj tztyhorizontal axle points west. In what direction is the lint2* yv:lu1zj ,fk/l ttear 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 the edge of a large freely rotatin )ite1q5;p tyfg turntable. What happens if you walk toward the center) tifep;tq51 y?

A shortstop may leap into the air to catcnu+8 nznn/x v2h 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 rotate in the opposite direction/znv2n+nux 8n (Fig. 8–36). Explain.

On the basis of the law of conservation of angulag u mhvj:knuf o69l b+1(8jls)u+yq;e r momentum, discuss why a helicopter must have more than one rotor (or fsmu(ug qj l9luk+;v j:+6ne) 1boy 8hpropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i9 rsw(yty e3 ny/:*a5cheo qy*n 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 ng9tm;a4g7z n because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, gz;9nt4ngm7a as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam d9(t:irc( dp2- pvrqt wiverges at an an-t (cprrvw(2 pdt9qi :gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mot+ge a6mten a3t)lp)w5or is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? lew m5tgaa+6 )tpe n)3    $rad/s^2$

  A child rolls a ball on an/*q 4f yuchq5 level floor 3.5 m to another child. If the ball makes 15.0 revolutionuhy4qn/*fq 5cs, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. 6a;wru8 kg(wr. +i upyHow many revolutions do the wheuwu8ywair+r.g ( k p;6els make?    $rev$

  (a) A grinding wheel 0.35 m in di0xrn:q7i6obm ee7nu: ameter rotates at 2500 rpm. Calculate its angular b0x r6eonum i :7n7:qevelocity 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 revn-vg)qs9 /iwv68xkwy ,iq dqyiklpg6t 6 386 polution in 4.0 s (Fig. 8–38). (a) What is the lpy6- v6inq, gkytq/ 8)6qgvwk3pi8lxw9di s6inear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earthnjuo:c 0gg5 9j (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ojh)hggo:p 4u ,f 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 cm1 i,zg:n2 fp7jlk,ytj from the axis of rotation is to experience an accelerat,yjzg jt21lknf, i:p 7ion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpdd8 /;5ohr3kukmju ze5e v18a m to 280 rp5v;a 3kru1mkhuj85dd ez8e/om in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius9/ t zzun,6ohz9+xlg:ldge.z.g zu e : $R_1$ 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 qio )8zp*kluc hci7+/bvh9h9themselves 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 revolut vch78*uklco/h9p9b) +i zihqion every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,/y:ddweetcv6b h . +z5ar bb5h 7x/)nm000 rpm in 220 s. Through how many revolutions did it turn):xve m /t/d5da6y+rb75 .cbwhbnzhe in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcucco7 iej(zy b+x0j5-9cnbeh .late
(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 rsj6 )-ak rtcpedv ,6 pw0ly;8zsx9d.bwg1v8a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before r wxdv6.s88zjc pys,e wv;0bgpt-akdr1l )9 6reaching 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 inzo7 svas.trhcw1 z ;0yn. mv9- 6.5 s. How far will a point on the edge ooz. c.;17r9 whmvnty-s0sazvf the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running fft* : 8.bukklat 850rev/min It turns 1500 revolutions before it comest: f8 kfl.b*uk 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 speedf)cd 2x/- qyfm uniformly from 95km/h to 45km/h The tires q/mfy)cd2-x fhave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frj dxu(q z;;ec5*8a ,,mv unosfom 95km/h to 45km/h The tires have a diameter oszmn;qo5,*x v,f aej; uc d8u(f 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 pedal when cln n0wm6 sx-5n7wvx4u pimbing a hill. The pedals rotate in a circle of radius6x- s7w xvwpun540mnn 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 u+m7oe 4 g/a,qozs 95lqxiv/e cm wide. What is the mag4 oieoe5/xa,7 uq9gls/+qvmznitude 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 about the axle of the89thqh9gs xd2 wheel shown in Fig. 8–39. Assume that a friction torque of 0. s htqx2g9h98d4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached toh+6o7n/3gvwz ; 7l g s*xwqmds the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cal /dx *wos+s7 z;6wgln 7mqghv3culate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqaekpno(ek x fs om+.-9vwc.4*ue of 38 n( cf.kas+ko9eo4wxp* e- m. v$m \cdot N$ 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 sphere of radj 3dlheqdxf4 41tpa7s *h8*leius 0.648 m when the axis of rotation is through its center llsdeht8e4 p3*1* haq7fd4xj .    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheejtoffltn x) s.-3j5:ql 66.7 cm in diameter. The rim and tire have a combined mass of 1.xo3.tfql) 5ts-jfn:j 25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod i :tfe.bp1il b0s rotated in a horizontal cil: fbpb 0e1ti.rcle 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 potter’s wheel rotatbq;lm/ncl1u .1ete0 w ing at constant angular speed (Fig. 8–42). ulne m0tbw;11e.lq/c The friction 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 point +j (3) tmxnskyobjects shown in Fig. 8–43t( ) 3ksmyxn+j 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 oeqg39d ggndcf7*;v;z6 6 nn aeqc73rcxygen 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 atlcsbum; jh/ e b),alxo05k+ s5 the correct 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 sxl5s o ak/meh b,+u5l );jcb0is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform 5v :i 8q:me9r5d diafocylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calc::rm5 v9eodidq fia 58ulate
(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, acceleiub ) eot0)2* if1rfdr2xr1 he 2o48gnfzo-mgrating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on admg.sq( o y0bm4d48k1tq :y wx small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 (ywtyqxqd8:10 .ks 4gmb4dm o 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 10,300 rpm is jr-slw 3d,e +pshut off and is eventually brought uniformly to rest by a frictional torque of 1,-r lp3js+dwe.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-kg ball atd.ksy-a1462 j4yd3m xh jrite 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 bal:h9 0uc (k3ss,cfcay hl is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The balhssf uh(:9,a k03ycccl is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long n +-oz pyz2x;h vcnrj6/aa v+)thin rod, as shown in Fig. 8–462)o/6 vnpcny jzv-++ara;zxh .
(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 maq+2b q18sszkr sses, $m_1$ 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 restpmz : dp*uuvxhd+*82x within four full turns (revolutionsxp +puv*:m*8uh x zd2d) 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 o+hvd ic**t:c tf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a- nr+8zw rtdkf-*0ffl)6)a0fm k 8hfmxbswc0 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 and radius 9.0 cm rolls without slip0q* -ne sgkzj9ping down a lane at 3.0k nq-9jesg* z3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asndlr0jz ,:;js the sum of twdsl;zj 0, :jrno terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much + -gd:pue2 oeil+i*zaon a /,pnet work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assumi,ia o:l2p/g+depu z+ eo n-*ae it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg startduyr( 4d8b-vt0a6th w,+c iyes from rest and rolls without slippi d4taev( y8,+c hwyrt60-dibu ng 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 hv gosm fr*x*07n/ut6vertically 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 u hmg6fxnvo/* ts0r 7*hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tyv6s2uvsq+ 4f he end of a thin string in a circle of radius 1.10 m at an angular speed of 2yuv6 +4svsfq 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg u 1+f7m x:iuhn: fln 2iiw+)z d.t4sdlhniform cylindrical grinding wheel of radius 18 cm when rotating ai1wuiln4 )n7hlt+xsf z.m :fhi 2 :dd+t 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 rat 7elp0+ pry:,bqt:wb0z 59q ul:f rovpe of 1.3rev790+rbly rvpuzq ftlwq, b pp0e::5o:/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertiat92hm0)vd*s1 ;w nsn5 qb pnko5mzh; d by a factor of about 3.5 when changing from the stv 0*1nqnw29m 5pz)btnd5;sk;m dh sohraight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin u)1uj3lwno2efm h 13z ; b(qvy.x/dyzrotation rate from an initial rate of 1.0 rez /y ewbvmhx;(luq zud3)12 yj13o.fnv 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 rotw0+i.ool4 op jating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto thw .o+ 0o4pjlioe 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 spinnnsrqs24 c yp71r3tm1 wing at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ear+ v hs1nmk+7 0etygm1g b,;ojsth
(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 d tmqw*ga1m 2q6xc5 bvi qo4-i2isk of moment of inertia I is dropped onto an identical disk rotating at angular speed gbt 52 2qq6 14 *mciqmawvxio-$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at )h.t4.vs/yi gt xp-/jbj.stt 2.4 $rev/s$ 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+) kef,;ur9i1mxbdgb of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9bbg 9 +e),mrf ;uk1xdi20 $kg \cdot m^2$ 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 fren)dd :-cy)azvrwz-+m 82tb uiely with an angular velocity of 0.8 ) c :rb zzi u8+dwvt-ad)ym-2n$rad/s$ 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 ik* qho y4-x3epnto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angu3x* 4 eoy-pkhqlar 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 wbu:p5kux*: xo inds 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 mm rx)w9 8mo(pbm4.zr$ 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 freely witb)klp5igx z*qd5 v 8x)hout f5*zbgpq58kv ) )xxldi riction. 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 edge of a 6.5-m dio7mmwm3. 6c9uk 4ofemu6/dsuameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1s 3mu67/.uo umk6w49efomc dm700 $kg \cdot m^2$ 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 lyqs3 idzb,feze*+;hxr iw/r //ing on the top edge of the spool. A person grabs the end of the rope and w birxz+he,;/d 3//iq zsw *erfalks 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 Earth. Determf78ic kcgw x2pt6n5 q (a u)hmao7iq-7ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting (wh -q7 q2cfpi76ca)o kxtui8m75agn the Earth.)    $\times10^{ -6}$

  A cyclist acceleratesfqbzsr-w v ;g-8xl0* m from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unixd, *c*se0lb m(ov0hr f4*ec 4 itry8uform cylinder of radius 0.20 m acquires a r e 4vohm4b(rfe*d8uc cx lir*t*0,y0s otational rate of 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, each of mass 0.050 kge . do+k3:c6vjnwxhz1 and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 hwdzc.:o6xk3e+ 1vn jm. 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 sp51 up ojae2lg dip) ogg6*-ab:eed 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 8.0 times as v j5y4:jzf(ye .8)tp /ukknrs 2bu8bngreat, were rotating at a sp(8un y58puy b/jfkb:4tz).sv knj r2eeed 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-pollufj(6evf0 +hmu:z l9 i y (w7g;mmb3ekjtion 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 and mass 240 kg, and should zj: 6gw jfmh;mivl7bm ke+u 30y9((febe 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 ai 8au70o eifi,n $H_2O$ 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 horizonta+w b1v+cw8,ich n5zycct+n s9 l surface 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 ignore fr(5s qhh2oa 7uciction) about u 5h sc7h(q2oaa 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: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stand4,ro rn5xyrjri d g:5,ing vertically on the floor, and we want to exert a horizontal force F at its axle so th45xy,jodrgrrr 5:, niat it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speeiw6ctk qv i xf;7-,dq7d v=4.2m/s on a flat road is making a turn with a radius The ftq, vi7fx6;7kqcd wi- orces 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-krplbfsa0zf+p;6j xg l9t y ;-;g 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. W9 p6sg;l;0ta;zpxyffjl+ b r -hat is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and h *- ulmm)u1 pl8gb/pjner arms as thin rods, making reasonable estimates for t-bm)p8 l*gpjn1m/ulu he 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 assemblysaefh+re6a/nw3c 1g(m. rn x 3 which consists of two cylindrical plates, ofe ra1encs3hn .m( rwxgf36 +/a 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 r+ybr1i/paf0a2v tn(vj b j+ 3rpcz7k(ough track of Fig. 8–58. What is thjtbfvj1 7+pryk ra((+ n0ac2/bz3vi pe 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 track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not as b )a2oz 33cr zx4pje9spbhm9*sume $r\ll R$ h =    (R-r)

  The tires of a car make 3eb /b ofmo0j2d+hfq(85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angulhfj2( o0 f3bmd /o+bqear 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