A bicycle odometer (which measures distance tr,hgb3njk v7y.;mf vh3aveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use ib, hvh kj;f3v3 n.gm7yt on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point tm/t (n5akdz:tlf l8fg, vq1* on the rim have radial and/or tangential acceleration? If the disk’s angular velocit5*tgd:/ 8f1vkml(z, ftatn lq y 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 descril*uv ob0 jir+y+ot* 8ubed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu +rw hyrcdv-/f-:)wu le 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 handn5pz-r)d2 ewvs behind your head than when your arms are stretched out in front of you? A diagram may d2-e5r )nzp vwhelp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at 5); unzgg qme4b q+08v1bnbtg the pedal cranks.ggbq+0m15 gzn)uetvb4 q8bn ; 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 fast have slender wu .*z4 +nev-:pnnixk lower legs with flesh and muscle concentrated high, close to the b4 pein :*- +.znuknvxwody (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, narrow dw9de7ji9 h*yhgy ) v,beam?

If the net force on a system is zero, is the net torque also zero? If the net tr pxf:x(9mx0n.x9+nxvmc 5s83q dvjb orque on a system is zero, is ( xqvb 53fxnmxd n jcsp+vr8:xmx9.90the net force zero?

Two inclines have the same height but make different angles x m2huh old(t.x67zdd; a8cg9with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom.o(;7dl hu dg6c8xtad x2m9hz be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down 22iwm c.frvn3nves i x-*mz63an incline. One spherifs vnvrm3x*3. wnm6e -2c2iz e 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 total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They smlaw50 01yk mhtart from rest at the top of an incline. Which reachesl 0ka15mhyw0m the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ar30;khyn jfc*hs,a: dse conserved. Yet most moving or rotating objects eventually slow do, *h 0 fh;snydsajck3:wn and stop. Explain.

If there were a great migration of people toward thot d yjdkx /5.( v+2nfojy7+ 9kjguun4e Earth’s equator, how would this affect the lengj+uf2o tn(okvn+5j kud/7dy jgx94 y. th of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial ro0pjk2zgg5 w b0)5l.oj,tyzoe1 v rr5btation when she lea1p0jy rr5)og5z2 ,vwtbl .z jko 50egbves the board?

The moment of inertia of a rotating solid disk d* ly3;.zfrsd,w j y,nabout an axis through its center of ma, f,dywl3 s. rjnz*yd;ss 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 rotatin,4vqn7rt p(trg stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the7p v4rtt, (qrn same? Explain.

Two spheres look identical and have the same mass. However, onez t0lzkc)jh *8 is hollow and the other is solid. Describe an experiment to determik0lzcz8 h)*tj ne which is which.

The angular velocity of a wheel/ug pu1 +z/wy1t x+jpp rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acc1+tppuw//+ ujyzp gx1 eleration 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 thedvwh *ad*1 lbe2*v y3wa+y u5c edge of a large freely rotating turntable. What happens if you walk towardv adcu 5yww3v*2h*la+* ed by1 the center?

A shortstop may leap into the air to catch a 8v.wneux4c c*cy7 b.bqh0ac 4ball 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. 8–3x 7 n0y*cvbc uwecb.4. 4qah8c6). Explain.

On the basis of the law of conservation of angular mombze6z 7l0l-azentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller cazb ll 70eaz6-zn operate to keep the helicopter stable.

  Express the following angles in radians:cceb0 lnd5-pe82ogje :kt9. tr-js .n (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 amazina+zf:fvuj 5qe,eswm2(bs , s0l tu* e+g coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of tlb:0s5 ,mee+ e u*av2,fjtz w+u qfss(he Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mcrqlgb8ua0cb 19o, p 4oon, 380,000 km from Earth. The beam diverges at an angle bcp9 84uogb 1q0rl ,ca$\theta$ (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 9ihc..qgi1w 1w; 1bswel l,lqrpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is th1,w.hci1lw eb s. lqlgqw 9i1;e angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a levele*r w+ sz x4f)s6tet5(d ,cmot floor 3.5 m to another child. If the ball makes 15est5m6,4c t*t ( wozf)+ dersx.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k9p y- rbw/3dk o7inz2km. How many revolutions do the-wk 73pn 2d/ori 9ybkz wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250f 9dx6 ,tuzdn,0 rpm. Calculate its angular veln 9du,t6,fxzd ocity 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 revolu hsh1e icg+t66tion in 4.0 s (Fig. 8–38). (a) What is the li+6i6h1 thgecs near 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 Earth (a) in its orbit around the Su-z9j 1dbs4 ocup9ye7b n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sn* st/2u se1xepeed 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 )o3 ki1wczdcpdk 5.2m the axis of rotation is to experience an acceleration of 100,000)5p.w3dc comkd1 k 2iz $g’s$?    $rpm$

  A 70-cm-diameter wheel acl cvuu//m.e z-celerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Du. lc-v u/mz/eetermine
(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-m bnugug85y 5 $R_1$ 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, astrj y w,1+iyyhkt:90g.d 1nkevg onauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotawye91dvn .gj,:0ky g 1ythi+ktion to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly fro uopd2a*2 s-b* ane,tgm rest to 15,000 rpm in 220 s. Through how many revolution tsapbe 2n2 ,-ugdoa**s did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s- ;4ehd7ceu cj. 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 stre-1g/4 evee9mk9dbnoksses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revole /bke1gvkd n9 9-oem4utions 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 diameterfn3je;7vt 5xt*pz 1 ds accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How tzx5dvj 7eptn;f 31 s*far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mt *qhyuv)4-4j/tswv uin It turns 1500 revolutions before it comes to a stoyw*4t- u/utjsv4qh)vp.
(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 0hosgfsmg8 .3y +mz 8xmvk4 u865 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a88gv0 m4 hgfz.ys m+8okux3s m 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 unifoaq .qh)he 0z2wrmly from 95km/h to 45km/h The tires have a dia2.e)hzq q0 awhmeter 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 q t9sm80dvy,w a bike puts all her weight on each pedal when climbing a hill. The ,89mqdwv t0y spedals 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 40 t )hipm2lm,jpoya 5cm wide. What is the magnitude of the torque if the force isymim4jh ap) 2o5 lpt0, 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 therc2oxnb7 ac8* 5/t*tzrmdh:e axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 2r / 5cdc*z*rtbxth8onaem: 7 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at)yy s1 ynhif/1tached to 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. Calculy /nyy1fs) 1ihate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an: *m 3qyd /ejpk5xf/ta engine require tightening to a torque of 38 mt:q 3 djp*fxey/5a/k$m \cdot N$ 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 inerti pzalj6s0-y vt 5:qun-+x9xyk a of a 10.8-kg sphere of radius 0.648 m when the axis of rotatiou-5v njp xxa:y-z+ysq0k6l 9tn is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclek srwmd:l/z3a +uj d0( wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whkj swa3:0zddm+l ru(/y?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod isake5,)h4tgp 9xk m5m.wm1tma rotated in a horizontal k ax4m1m,5 9 5mt.)apgekmtwh circle 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 o9.r7 itc,+z-6esy6q lp bdw kpotter’s wheel rotating at constant angular speed (Figi9plco-k byt ,+de 7rzsq6 w6.. 8–42). 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 objectsi:4lik yn sy,5 shown in Fig. 8–43 abousk4niy: i ,l5yt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass is vwbfon4c0cko riz , ,*.e v5-l$5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, bb8i3jv9 v crjj1nf3 0engineers fire four tangential rockets as shown9 bv 1n3crfbvjj8ji3 0 in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a r y0 delh4(lye6adius of 8.50 cm and a mass of 0.580 k lyh0e64ly( deg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerati(cmy*+7 v lcl73k i0dp2v x fvt;*mlitng 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 a small hand-driven merry-go-rf oku+i.oef3/ound 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 tokf+ .uiofe/3wo 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 rotac3cv 3eouhbaxt :u*7 7ting at 10,300 rpm is shut off and is eventually brought h7x7vccu:3o*eba t3uuniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ry7j(l2yb0 bc7e s kqnh/jq98 ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forvin9b4zxnfq zdu(7; 5 earm, which rotates about the elbow joint under the acti ;q4n( id bzz95xuvnf7on 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 Fi kx*mo4 vdhs3+g. 8–4 ms*3hvdk4o+x6.
(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 tw cc g+vckp62ycl9s(.o zz7vw)o masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest4;8fspk qkt*pcr c1e0 within four full turns (revolutio*scpr;c 0p4 ef8q1t kkns) 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 ofijjzb-4.l ;my y0xq5w inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tor(gyod3n zwtb:w 1u46aque 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 slipping down a0o s dk vsej7 nd3)fj9njj/)s+ lane jnje9fdosvs/dnj+0)7 sj)k 3 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of g65 hy rkh9v0nw6ay 2qthe Earth with respect to the Sun as the sum of twok hnavyr 9wgh62y5q60 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 muchlr; jngk 0xbb7:yikgl 6jt495 net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a ; bj9n7xr0tybkg5g6 j:4ll ik solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts -u,zjf an8tl rwc91z( from rest and rolls without sl9ja t1z,l8cwru zfn(- ipping 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 vertically/ h*sx20+ . krnstzvq;thy ml: on its tip. It starts to fall and its lower end does not slip. What will be the + thtx;2zms nrvh0*q :l/ys .kspeed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatxv3vg w6d qh1,ije-g y2y10oaing on the end of a thin string in,iv10ehw 1xo62yg-gj qv day3 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 of a 2.8-kg uniform cylindrical grin pkb:05o2rz hkding wheel of radius 18 cm when rotating at 15op52 0r zkh:kb00 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 ipw d43( ldq9hon8 kg)ts rotating at a rate of 1.3rev/s If he raises his arms to a hor8dndo(k9thq g l3)pw4izontal 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 r wx(g 9vn,sd y*2.7 mkzze.7k*cnxvoyeduce 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 in the tuck position, what is nksc y.*vng.*d,zm7vow e9 k(z x72yx her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increjs) e;co-z1a*uubd4 hase her spin rotation rate from an initial rate of 1.0 rev every *;zsudb4) -eo ujc h1a2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a fre+;u9ko 49wccyhjq yd4w :bx,equency of 1.5rev/s The wheel can be considh kw+d:yy 9c449 ,jobcuwe;qxered 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 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 mom3du 9lyz+yf 0zentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angulaq) m-f* nq(hrhwx0kxjjy 4e 51r momentum 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 dropped onto an id :* bern*7/*mwnzk.tzbi17ga/i v6krudk/qx entical disk rotb7 .w*v t er:g6m*uakz 7nik/r1n/kb*i qzdx/ating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 x(fy:o3y vl.p $rev/s$ 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 cent so1l 26n.d3i*.zkzce8isw *n kfwmz8er of a rotating merry-go-round platform of radius 3s em k.*s2.ln6w3z*i 81fozin zd8w ck.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocis)zxjp4;ld4ll7)0 9g nxkqdm ty of 0.8qsjx4l; 4dd pl kzx7))9gnm0l $rad/s$ 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 ithq9 kc8ggxd8m:nz6 z4s mas9zx4: h z68dcg8qn kgms in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round 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 cu;/r htbj7q 2winds 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 szcy z,9 -lxa9$ 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 ro ynmw0 ty9xs4hdi u37,tate freely without friction. The momentu4xntdsyi,y 0 379mhw 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 diamgu4m9 i:ktg1qeter merry-go-round turntable that is moun1tmu i4qg9gk:ted 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 ov . yyvcdx*9tvh3f80 ef the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s c3y0c9fhyxd 8e v*vvt.enter of mass move?

  The Moon orbits the Earth such that the same sideirh c9 kq91qo+ always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abou9o+ hk9 1crqiqt 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 frow yc;6qk hdo(8m 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 uniformf ,ej *c0gff.w cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelerecf*wg,j.0 f fation. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and di2+io*irxz o6 /n7v,fiw elhs; ameter 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 2 o*;6ohi n/wx,zvf ler+is7ithe linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel (ceyev1z lh7+7. s evskv8fk4*$\omega_R$) 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 o c1cfmjq0. zdsh3e6 pfm20t ,g(gr nk6ur Sun, but with mass 8.0 times as great, 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 11 km, losing three-quarters of its mass in the process, what would iqd(pf0j0zc.m m gfec 6s16h2r ,ng3ktts 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 stored f4j4)rmo4f d 1kqjck( in afckkd4r1ofjm4) (j4q 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 ) wjijm*3lpnn(u0eo1hy 4n.v((r qy ian $H_2O$ 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 rolscwo-wi* d*m; ling on a horizontal 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 len/ho2gt4x ru-ppu vn2f 2/vtc 1gth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume tpgvt2r n/u4 vcfu/ht2-op 1x2hat the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl2p:y x i6:, z4,iyogv kab3ono v9d*caoor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55z ioi:* d:3 , 2yanb4cyk9pov,vx6goa). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turin 50ibw n(rt p6xua4;n with a radius The forces acting on the cyclist and cycle are the nor6 an;xp(wri04 utn b5imal 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 ikvjxrr+ nq/mgb w-65( nto 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 qr+ v5(gk/6bj-nxrw mof 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 and heo:oh5j8y55kb(g n ftor 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 tightly an5 h:tg(oo 55yfj8kbogainst her body.   

  You are designing a clutch assec7jmmf /5x2q aluqd :af 3.)dhmbly which consists of two cylindrical plates, of masq7 /.fa 5qmfddc 2jl:ux) ha3ms $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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–od6 h hp;)lf 3ttn;.uxoj6/5r rvc8a u58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest po5odat l/83;jv)f6ntuhu hr xo r6.cp;int of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not *j9tcz/4 gg pyassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforml38wlvb)1tufdax el 6 0y from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each ti0b )w3d 6le8t1f axluvre? $\approx$    $rad/s^2$(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