A bicycle odometer (which medw(fmq m)0b )oasures distance traveled) is attached near the wheel hod b)()0mfwmq ub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant a abt2wer3 q8uo0w. aag*9w o 7x;yod;kngular velocity. Does a point on the rim have radial and/or tangential acceleration? If the dis; ;97wwx*eaodraog3oq auw k 08t.by2k’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descri/ 542e+b0t wfq7 vwbfzdco m2jbed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exerzw kx0z/dp/x c /:exr*t 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 --r/diw 6fv o1z4yr +rkv v6laen1p/s sit-up with your hands behind your head than when yo+pf v/4-1r n w6lkez- ioy6v/darr 1svur arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thel 3d sq ilke3 ,*vlzbl7b8tj p518tmy, rear wheel and three at the pedal crzeb3 ,plsmb8t*3 8ldqlik vyj17l ,t5 anks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fastd5g+y 6rv2::kvy v kjn have slender lower legs with flesh and musk 6n rdv:5yvg+vj :2ykcle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nar 5(w2 kz(2b2ho+jyz5+sxrh xih nua4jrow beam?

If the net force on a system is zero, is the net torque also z1o0 x 9gy4sj1y:h3 2b xuxovzzero? If the net torque on a system is zex39: 1bo1 4zgx ozx0yjvus2h yro, is the net force zero?

Two inclines have the same height but make dif-vu9 m,f .yvbcferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will ,uvvf-y9mc.b the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incos c2i6vy/-jb/l1 ;ef8zrxll line. 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 s- lzrylo28v 16ice;b /x/jfltotal kinetic energy at the bottom?

A sphere and a cylinder haven ot 5 zytm7:xw/ )6-e26r fjrxuuej5k the same radius 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 hasn5mz-ujf)2kr 6o/exrxjutt w7e5 :y 6 the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.*v4ui orfai2ih hm..; Yet most moving or rotating objects.umirh2;*f hvoi4i a. eventually slow down and stop. Explain.

If there were a great migration of people toward the Eart q(w6i)oyl0sh.gq nt6x* n/jnmp .+pyh’s equator, how would this affect the length of the/ qy0tn 6. )oxywin.pp*6+ qlghnmsj( day?

Can the diver of Fig. 8–29 do a somersault without having any initiat-j4p5 fc1vxwl rotation when she leavesc 1wtjv4p 5f-x the board?

The moment of inertia of a ro.fy,5.:o q8dure1yvew ym ur(tating solid disk about an axis through its center of fo.er1.ye y:( rumwd q5,uyv8 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg m62urjebk *p:nds c4.nass in each outstretched hand. If you suddenly drop the masses, will your angula:*j 4 srb62cen p.ukdnr velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevv y :zoxavn9g,hi9p:ak.+xh 8er, one is hollow and the other is solid. Describe an experiment :vixv.hx:8h z ,pk y a9n+ga9oto determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poinobt*r 9. - jc1hskcv3qgtt 3ea75)nq o7sf.nz ts west. In what direction is the linear velocity of a point on the top of the wheel? If the angular o-3v7ft7g.9qq n c1csjezt*srn3ok hb. t5a) acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating tu+6k w/nd kwa,urntable. What happens if you walk toward the cudaw 6+k,/nkwenter?

A shortstop may leap inp7 q4 .l aorg7ytn) +x7/fsfu1zx mu8tto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explat glzs8uyo1fqt7pu7/m4xna. r+) xf7in.

On the basis of the law of conservation of angular momentum, .fm8epxfl;fn8 l)e) (dnyj3d discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller c8 d)mf8fjfl3 n)d xypee.(nl; an operate to keep the helicopter stable.

  Express the following angles i. n(bjny1d9 s1+op ycpn 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. Calculate, tl:p895nm udousing the information inside the Front Cover, the angular 5:muldn98 tpo 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 beah f-q0fjq3mlxr mdk84)r b.j:a oq*w i6bh/1am diverges a68 k*fxm3qj4o/fr.mrqd0hb1 a :lb)ihjqa w-t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a r;;gx b i5,wqfiate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in iw igxfqb 5;,;3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a balld 0t rs:5rgk:iu *bkg (4fy7io on a level floor 3.5 m to another child. If the ball makes 15:(ors57 g kt* gibk4udiryf:0.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hon f f.z 8qdu7sn) v6yk-r/akl,w many revolutions do the wheels m-n y,s6/f.a dl8)k qfvnr7uzkake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatctw1z7))i nz/kj.y *pbtwqy ;e its angular velocit1z7 pi t))wwy.;kzn*/yct jb qy 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 ueq6v-lv* v u 5oir c)w+co5;tmakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chil vo+)*eq56wvl ;5viuu -rtccod seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit ukywuw8;n lb 7h4( q:ib twtz2izm68;around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of abl m 24b3f;amx 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 centrifp85zz /lw 9t eu7rjrw,uge rotate if a particle 7.0 cm from the axis of rotation is to experience an ,wru5zwr/7 l9j pe8tz acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm -yh-liswl1w ,to 280 rpm in 4.0 s. Diw-ysllh ,- 1wetermine
(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 e-u +gnt 93t/s8elztv$R_1$ 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 themselve fzwk0frw48j8s 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 ej4 0w8kfr8wfzvery 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 acceleratew 7;jey5 5x(ib9u+qvgu4j **mjoip4ex1i dgqs uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thisjuxqu4wgqip+5b71e5 j; m(ig*d*4yo ji9x ev time?    $rev$

  An automobile engine9 cqgfhdzcd)fe2 ( e/+ slows down from 4500 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 highspeed je 8fogyh( 0ap*ax;c -o +k-upn5by5emjts in a whirling “human centrifuge,” which takes 1.0 mp*aufy ;y0eocbpmojax(h-5g8 + k n5- in to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fromw pw ( f.bjl)*ply:u q-tzx*s7 240 rpm to 360 rpm in 6.5 s. Hqz.p jfp*l lx(stw7- u):by*wow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off wpx (x awx..+h0ax3hso hen it is running at 850rev/min It turns 1500 revolutions bef xo.a.0hxsw(xx 3h+apore 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 65s+70i r9 azh3ucylqc8 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8039c zlch78a ry+i0 qsu 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 f(bhq m9z0 +iup5ss8ke5 k5l acar reduces its speed uniformly from 95km/h to 45km/h The tires have a 9s 5 z+u05hq5smbkle8ki( afpdiameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each ped5il*shk( br 1zal when climbing a hill. Thl*zk 5is1( brhe 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 of 55 N on the end of a door 74 cm wide. What is th::)bxp ghtf1p e magnitude of the torque if thfb:g1)pp h:tx e 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 Fi 1;6y 92sncgynbwaxr9g. 8–39. Assume that a friction torque of 0.4 byns ;r6ywx 9n2a9gc1 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to th.sai x ef(za r0c;*o vy)75fhle ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rele*0si )5c v.hyezoxfa laf7r; (ased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an en8wo1iiti28j hq55ovbgine require tightening to a torque of 38 288v1iojwtq5bhi5 o i$m \cdot N$ 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 radius 0.648 m wh*b q+fj e5:a+ hucbk6ben the axibbefhjaq*cb5 u +:k+6s of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia obtaxz-1r m1t: 7-+nnbijd/t v f a bicycle wheel 66.7 cm in diameter. The rim and tirt+-7nbbtdr1j:va -mt zxn1 /ie have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin(6* ilauo smu3, light rod is rotated in a horizontal circle of radius 1.2 m. Calculaum3(s*i 6alo ute
(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 rotating4jz8:1 b 1j vrhnp *zl7jsnx9n at constant angular speed (Fig. 8–42). The friction force between her ha p:h1nv9brz7 *jxlnsz8 jj1 4nnds 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 +jar ;( wffhry.gt5/o of the array of point objects shown in Fig. 8–43 aht/5yjwrr+gf af o; (.bout
(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 atomuft+w :5mx/as6i3e2o5owr jd r4n n3 ss 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, unifo wrxp a8p9zzfe322 bc0rm cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg a8p cw x9e zrfz02pb23and 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 a uniform cylinderk tbi87i)sg x) with a radius of 8.50 cm and a mass of 0.580 kg. Calculatetbs g)i7xk) i8
(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 frol1h.jz9arbo(n*,d,ezi q p g 2m 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 tangn*iy5hsl p*wj va8tgt 6-44jpentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency5atwts * jnj i pl4gh-48*vy6p 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 10,300 rpm is shut off and is ev(sd4chy8 e:fhk 84zs wentually brought uniformly to wyed4sh4 h8cfz(8 :sk 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-y-qh1 koifr c ,3js;8rkg 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 rb4o +b -le5 kwz45ifswl+k1p ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the a 5p41eb+w4kz-ll kof s+irb 5wction of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in m n).)my2npa os)jtot2.me 2fFig. 8–46.j. t)aso2mm ep)2yto.nnf2 )m
(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 masse cv+v:qr9wwqu75v/5 x dk1naps, $m_1$ 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 2ay7ft v fzu2*xjc s5,bhc2.v hammer from rest within four full turns (revotua2z ,c xfc*jyf.7vbs52hv2 lutions) 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 (h9 5loysj1t rinertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2804 ok;2i)cb jx:/doy q/:krop x $m \cdot N$ 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 ca1 7iefyw :z ihjyqu*,/ane8v2u .ea 7m rolls without slipping down a lane at ah1 qa/vua8f,7we:*ey7 y.2jzienui 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the S+m8rkbd7n;w6lclc -s .5 rhl +t sj9j2dna;mgun as the sum of two tn h2ga 6s8+ -stdr+cdbj79m.mwl rlkc;l j;n5erms,
(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. Howa5*5 wj x)dx3hx akai) (/wlmj)8 bpzz much net work is required to accelerate it from rest to a rotat i *3z)ax5(5 al8pmwk)hzxd )/jwxjabion 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 reshywkl8 a6j/8dr6fi 6e-ykt3,) wx ptet and rolls without slipping down a)h/d6w6t6r,aek3ti- ex wkjy8plyf8 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 verti c8ez8:g0yo-z7q u yvically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground8zq0y og:c7 ev-ui 8zy? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of p z 2s bux/d:(dn7tc+h3s9xhca thin string inz/u37pn(d :s h9t+ 2cdxsbxhc a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum k9rq1pbha+ 7gzx2ua09;d 50ta px pagof a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatk02agpx59q7grt z+ a;a hp1p xa90buding at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating neh+qa0 kzonbax6030 at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed o+a k a36q0ohz x00enbnf 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.s r v)lomgvcn8q7* l)34+eorq 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when 48oo*+r3q em v)) rcll gsv7nqin the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase hemf8xeu hk4 3c.r spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate 3u8.hcfmxk4 e 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 t izz4c:r ;xree.b ukr:zh8d:(mj g;*d hrough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass*ehr: e(z u8 rd:c:rx;dzjbk.;g m 4zi 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinnin kqjtmn+b9x)9nm93cl g 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 thi0s( a2ng /xzf;gax) ve 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 iner-4x4oip6.j pbr+kj mxtia I is dropped onto an identical disk rotating at angulajoi p4r+kj p.6 -xx4bmr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4i9jxcu,g; 2c ex:3a7 tay /du v,w,afo $rev/s$ 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-round platfo/xam8 kwq(a7/nxi8fnd2 ri4 2rw(h;eyijxr,rm of radius 3.0 m andxy2awdiq nie(/k8hw 482xa ( , rrn mx7r;j/if moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocitwb+6 i ;yj4fag+api 8wtba95i hwj,. oy of 0.i9f g p6owb yt ,.iw+i;a4hjab+ w5aj88 $rad/s$ 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 collazj-suwk/hl) z+r o34t pses into 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 angular momentum, zut- h/ 4)r3+zo kwjslwhat 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 excesshm / rk(blw1a*9oa sql/y36sm 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 nr(nw)vrwt 0-/to6qf.as;ns $ 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 cantr8s2se zp/eut(b9 a 0i88vkz0gr 4qz7t vs3v rotate freely without friction. The moment of inertis8ai kb802rq7 tprg z3ves(/su0 84etv t9zz va 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 diameter s41tnb1ox m/f4i4 ayrmerry-go-round turntn/ma4foi s xt44by11r able that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope -4i1 mwte*//d7s bwfyz4pef erolls on the ground with the end 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 bed t p -e/z ewi4s4e*fmfwby/17hind 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 face :kx4v ad3m ifbr4)i 6xe46iurs the Earth. Determine the ratio of the Moon’s spin angular momentr36xkaif6 4:m re4d ixu)vib4 um (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 lol m*.))o+)hnb aqg1s/h; n, c1esge.bc mlarest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 :a9 smp udxq) ofw/c 6dpq/u5vz:tb()m acquires a rotaawudp9)6c: v/b(z) /d mqt5:fs xqouptional 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, eawukrht t 0 64j2:qp+slch 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. Use conservation of energy to calculate :4t6+ qsuphw0kjl2trthe 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, hoog36j9k zv7l o(o0ijui )/rv yw is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great,tl r 9um*9i3url n48gw were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 1 gmlr*urnw83t9i 9 4lu1 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 energy stored5 ike rehafo m.m)k6-/ 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 be able to travel 350 km without needa-/ifo6e.hrmmkek 5) ing 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 adzuvnk3p7+o dty l9s0ia sc,)dn(a;* 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 horizontal sc:1,x z yy,t ljs.uzd7urface 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 frictiol,1 /y- so894jn52usifjeqhs adl2s s n) 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 /-9ou51j ns2 ea8llq2jihsd, y fsss4at 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 verticacge p w9 jt1hcm utef8i;)370blly on the floor, and we want to exert a horizontal force F at its axle so that it will cl89t h;wigbu1 0 pcetfmec73j)imb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road vqc: k3c1zxm wef6ngs+ k ,l;/is making a turn with a radius The forces acti:; m6k fxgz,cl e3+/1knsqwvc ng 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 into a sling of length 1.5 m and begin pcch6f.w b m6og1n-s*s whirling the rock in a nearly horizontal circle above his hcwps6n. cmf-h g1*b 6oead, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder b fdz6izy;fw ;fgu89) and her arms as thin rods, making reasonable estimates for the dimension)w9zd 8uf ;yfbf6g;zi s. 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 asse03s8e5x57f zs leak a 28pm*zzta4aa qmbly which consists of two cylindrical plates, of ma8s5melz4zq3x a a 0ftz8ka 57*aes2ap ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tuzp0lym2m 3)g jqn61k rack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highesznj6upql y3)m k201 gmt point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but f9 /6/ khs6f2ehnyp; xb rwp8cdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed u t yjk6/vxg*,lw s+.tgq.exj.niformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angula/ j tgg+j.s .xx,eqtl*.ywv6kr 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