A bicycle odometer (which measures distance traveled) is attached near the whe180n1ni nnz8kbmijpp 9il e48 el hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-ini1pnk89b1in0 e pl4 8nzimj8nch wheels?

Suppose a disk rotategnb,2m-zzuk(f22c oor3 c0fejxxz2 1 s at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have rad20 n,g2e(boo3 x12j xfc fmkuc-z2rz zial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desgo uv g15 f895o 7yr3mnswotm c-,i;o fj-lm3jcribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gref2 wn mm+g-( 4bgy;csu 0xn/hhater 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 ht3t*. l2c0bd- pdnqveal3w2w ands behind your head than when your arms are s0at* 2-v n d.3dcwet3bpw lq2ltretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and threedim u+8hherusx)o0;d p30mli 6w(9dw:tywj 5 at the pedal cranks. In which gear is it harder to pedal, a sm ulm9 6dujdmd+0 (:t3)ep5h rs8 ww;iiwx 0hoyall 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 fasag/0 (9ytx qzg9n /bnet have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advang/y ex(0 99gn/tqnaz btageous.

Why do tightrope walkers (Fig. 8–35) b/ +.( udf7vj3nzz ln zo7kyd2lbi 9n;carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If thmb.vqyk ,aml:6.txl )e net torque on a system is zero, yb6tv:l)m q,.mx .al kis the net force zero?

Two inclines have the same height but make diby xpw f6,piu26+w6eifferent angles with the horizontal. The same steel ball is rolled down each incline. On which ini ieupb66,2w6 f+ pwyxcline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simqu+yy* fq yw-7ultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches wyqy+ 7-fqyu* 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 startk(oplsp92)lb x m)3bi from rest at the top of an incline. Which reaches the bottom fk l i3xbsb)op)2p9 l(mirst? 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 anguug;vw,-co)yo 2 uj6xmp1fawt x5y 0o0.dj ux:lar momentum are conserved. Yet most moving or rotating objects eventually slow down and stopwm cu0 oj-,yo.dpj5x u2v6: xwu oyxgt)0 ;1af. Explain.

If there were a great migration of people toward the Earth’s equator, z4y 9t3s4lxfj how would this affect the 9f4ls4z ty x3jlength of the day?

Can the diver of Fig. 8–29 do a so*yu2a/eh:/ j1jyik)(gcu c ivmersault without having any initial rotation when shj(cg)2 ya:1i/ vhcu/iyj*ueke leaves the board?

The moment of inertia of a rotating solid disk av: /nzzfq/56avg d9bd bout an axis through its center of ma/f5/vn : 6zzqbd9g davss 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/bcb y2onlnh7j s692j holding a 2-kg mass in each outstretched hand. If you suddenn j9ch7nyjbb2o2 l6 s/ly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sa p9q0 nkj/f gp -budc7gz3;/iyme mass. However, one is hollow and the other is solid. Describe an expe pg /pc yjq9fzkg/ -d3bn;i07uriment to determine which is which.

The angular velocity of a whezvxhp ikiww4.+ ze+ 3o z9:0i)c+gyutel 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 acceleration points east, describe the tangential lineav+ 09uz kwo.+ c zpt)3x:iei+ygwihz 4r 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 turntable. What o0ec6 h.jhy6n happens if you walk toward th o0jeyc.h6n6he center?

A shortstop may leap su 6j2 b.q spl+fq myz36 muey1r1/m4rinto 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 legq3yb+p2rlj rq 1esmmu u6m y1s64 .zf/s rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law x 68cwg -gyec(yk )zq.of conservation of angular momentum, discuss why a helicxyc e 6)zq8g( ygwc-k.opter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rad 34bjyy/rmsy )whl kyk+,6ft3ians: (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. Calculrdw;eq1/hp. n ate, using the information inside the Front Cover, the angular diameters (in radians) of the Su/.rq neh;1pw dn and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mou wyr, /b+a1gd*n0dkq wfnx2cw*ba//on, 380,000 km from Earth. The beam diverges at an ang y+ka/ gda2*rnc,/u0nxd wf b*w /1qwble $\theta$ (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 motor i1ta uv g3sr0d5/p h/yls turned off during operation, the blades slow to rest in 3.0 s. What is the angula vrpl5s dg3h/t0 ya1u/r acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flooo+vh9 m-xmh)ho *md1cr 3.5 m to another child. If the ball makes 15.0 revolutions, what is i ho v+mo1mmxdch9*h)-ts diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many h9i(n v1p;7 xah 3ew(mh-uewqrevolutions do thep uh7(x-hmiwe(9 3a1;nq hevw wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate iardv1s6ct,c :2rjvw . ts angular velocity1vs w ,tr2dr:vacj6.c 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 compleb4t, g2pgwi k.te revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thepibw, kt g42.g 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 ar6(m)e 0 bb k (tm:xwbu3jbg,pbound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o*:5a0wxx tg udf 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.pz6tgfy/n /zo-k( w9;vlz(c yz 6v ni40 cm from the axis of rotation is to expef nn6/z/(vzzyic 6k4z gyl9(p-otw;v rience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2v+hniq0-b y,a) pg tz;fq (6lz80 rpm in 4.0 s. Deq f ;ygza,bv06) (t+i z-pqhlntermine
(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 radiuscel n--sojh3 + $R_1$ 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 Moof97zc gu +uq1qn, astronauts 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 rotation to 1.0 revolutif7u q+9zug qc1on 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 15te g02o5-h-pj6 zbxxu ,000 rpm in 220 s. Through how many rx-et 52zu-g0x hp6 objevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm xqjva4ghb xzsy:;39+ 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 folbsqe lcu2dp 2r0(1s3 fu21egr the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complc3 res(f 1sg u0qbll22uepd21ete 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 unilp 99oxt6fqy0 . pz9zzformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of thef9.ty qz9px9l0 zo6zp wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It//81hzyoz3nzz l d6 vu turns 1500 revolutions bef6 y1n8zzl //vouz hdz3ore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revx4en23moy jen.s6mv7g62sc rolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0ss.xjvnm632 y7g 4o nrec e6m2.80 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 ra.jhq p p3y4v0n)t9czevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.a p9cpj.y)03q4hv znt
(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 wfx- awymf(d x-uhnd.b dp+ *1-eight on each pedal when climbing a hill. The pedals rotate in a circyx .n(-dp-1 ufwm f*+hdb-adxle 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. Wh9hwjaa .x m-auu:p:;prery/m fx.d0ee4 q(( dat is the mag; xhwd4x aqem./ uaaej:(9 dr0rp-:f.u(py me nitude 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 9+srqo;ucpbbs 02z0q of the wheel shown in Fig. 8–39. Assume that ; uz0s2qbrcs0b+p9 o qa friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless r*itxx+cke eah6 +bnh 9(a:b7god which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magn+a:xx6nkth+age* (bc 7i9ebh itude and direction of the net torque on this system.

  The bolts on the cylinder head of1p8zvc m3,(mvvopgl8t671wl t 8 2ghrsgbpy* an engine require tightening to a torque of 38 smplgr7pv ( g2*ot6mlyzv,1 88pbv 3ghct8 w1$m \cdot N$ 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 o-(2-syh 0swq ms dp:naf radius 0.648 m when the axis of rotation is through it- s 0s(s :h2npdy-waqms center.    $kg \cdot m^2$

  Calculate the momentl-5z0 9r+vm0,vvzbsk o0m rerd w*x,o of inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combinzw0 r0zeb9 v* +ovkro5rm,xvsdl-0m ,ed 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, light rod is rotated in a horiz3 ce h 8susnl7s.r:er9ontal circle of radius 7reu9 :lns8s3cshr. e1.2 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 rotating at constant angularned+/ qdwlvat3g02a7 speed (Fig. 8ln 03/+ataq gwded72v–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 i.l9 eqv6fht1c .u8murb 7t ls-nertia of the array of point objects shown in Fig. 8–43 about b 7q fe6.utcsr1lv-t9m. 8uhl
(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 twor3 .z,xx3 8 pmov v)ma85sffxp oxygen 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 at the correc65;7y lrvhzbw9x *na3v*waju t rate, engineers fire fou nr6a vwyax9 b7w;jvlu5h3 **zr tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.ir78ivi gl8 kq5m(gc) 580 kg. Calculate i)irc 5v l8i(kgq7m8 g
(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, acceleratinyikjt)li3 gswx8 /lw; ox +-l,g 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-roundpc0/z(xt7 zzn and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglectinz (7pzz 0/txncg 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 evrpdb,34 3bxmsp0 y ubdea-j )6entually brought uniformly to rest by a frictional torque ofeb4pxu0y,psmr d3- 3ja6db)b 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-k0my+sb)/gru/4osay h g 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 foq i:* 4a)h/ga3 zztyobrearm, which rotates about the elbow joint under the action of the triceps muscle, Fig.zza/tb3*4aoy:iq g h) 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 loq9xbg no. )wk3,ntbp60nuf (wng thin rod, as shown in Fig. 8–40wkn3 ) 9.(qpx wuon tb,6fgnb6.
(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 massesd8j .w4p, kobin0(nfh, $m_1$ 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 acceleran 02(nil wqc,l;sdk/x2y8wxq tes the hammer from rest within four full turns (revolutions) and releases it at a sxsyi2 l8;dw ,/wcqq n 0xkn2l(peed 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 7h g tyt.z; n-sss3: ;fre :a;ptpswp6of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develo,wc/9a1 r-cd .z7cxj/d ynqdzps a 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 and13kqfxj vak p5k,lc,q9d0f+ a radius 9.0 cm rolls without slipping down a lane at 3a0 p,9 ljak15ffc,kx3d +kqqv .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en a x2g -tfx5;gyjjof.1ergy of the Earth with respect to the Sun as the sum of two terms,jt-. 2ayg xf1;gojfx 5
(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 16 n* 9 +gav1jl)hicte/a40 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution pe/19 c)*+iljgaah te vnr 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls ws*yl/+grxuht9k9*aqh+ e3 t nithout slipping down a 30.0 x3u9*+rany*heq+kth l s g/t 9$^{\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 balancehh4.prl 3 dot* u1mw2ed vertically 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 ground? [Hintmh .ue3phro*d l2 4wt1: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin s+/gd(cs qb wrh:t8;ggtring in a circle of ;tc:g+8d( rbhwgg s/ qradius 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- -jq fqlc 5rf4n)marlj /(3-rb;lh;ankg uniform cylindrical grinding wheel of rhc54;naf- r(rlj/m lfqa)nj;rlb-3 qadius 18 cm when rotating 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 9:zu)ejak x-w at a rate of 1.3rev/s If he raises his arms to a h-z xa9jewk: )uorizontal 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 inert (j8k5*ri td+lzjz+ssia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2rjz+j+ 8k5ssltz*i( d.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ratx,q0mkxei7 f(;h w)rd e from an initial rate of 1.0 rev every 2.0 s tox)fde;k hwr(0 q ,m7xi 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 ikb/rft oi7*5)xb w zn/ts center at a frequenn/wk* txo)7 z5ribfb/ cy 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 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 spinnk( g )mwuo/ q/ mudla+io0 s-v4;ybg)qing 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 momeft k1w5njslr1 q4xv 5lh4i1 m.q *mw9ontum 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 dxcs,b 2a :nc:hropped onto an identical dna,:hcx2s: cb isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4e j+v4i4 x0rxl $rev/s$ 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-roulqb( 0v gkf we,e. dt*v6pf,w7nd platform of radius 3.0 eb70tvqdw*ep,(wg. lf f, 6kvm 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 velocw kum 5.nnozygvv er.,68qz+pk(w:m( ity of 0.8 yqkn 6ur ,vg+vzzk( we5.:wmm(. 8npo$rad/s$ 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 nrjh c 63ake98into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radiurjh98eca6 3nk s. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 15dal6 t)3 d kflwt9wv420 $km/h$ 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 pw().jkgia 5i(d( s4j nxlo/ j$ 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 platfs8bd(6xdf 8)6vfgqg qorm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platxdd8bqg6s (f6)q vf8gform 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 fv7 wa ()aapy56.5-m diameter merry-go-round turntable that is mounted on frictionless beariaay) wva(7f5 pngs 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 ys4k nzw0jt ;4bj63qw on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end jtzk0 4b6 jw sq3w;y4nof 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 center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Det :hkuo1mbj e64ermine 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 o:m1 b oh4u6jekrbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resoazwgjr ou-*/ :ip9ue. : v9mct 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 cylindehvyc,z*kd9ynf/ x, r fmb 5m:;r of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tod,z;5m:* nm9y y/vxk,bfc rhfrque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of i*a,jey th-u0jh lm-9mass 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 theuje0 my*h- hl j,t-9ai 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 isov b ;bgjq1vt7-6xc7y1y 0obh 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 Su*( ma twu:ihpzd7ct3-n, 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 itst- (aiudth3:c7m * zpw 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 vj,*.wy5qtxq 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 be able to travel 350 km without needing a flyw*jqqywxv,5. t heel “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 illustrateshgphr;g4u)9c7k w1nzn k xk-9 an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at s52.ud cegz8piis2 6 xb2rftm+peed 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 lengtprd/ ne1eu8:fmre i 80abo 54zh 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 releaseparrnu8ze0 me5i: /o4 1df8e b, 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 standing verticallyazqsx9 u 8qj/tx4vf+: on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has8ujsq9faxz t/q+4:vx height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn witd/ );sy5wb7nl e+q)fj5 evnkf. - dgivh a radius The forces acting on the cyclist and cycle are the n 5s )j5kqw-;byfdl+ eefv vdgi.n)7/ 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 a 2l *ab+5q0 pmk+ighuhnd begins whirling the rock in a nearly horizontal circle above his head, accelerating it fr lk 5 bau0iq+mhhg+*2pom 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 cn8a+w43rlg l +- fztpiylinder and her arms as thin rods, making reasonable estimates for the+8 tl3razfw-l 4ig+n p 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 cat+i- p3xac. zlutch assembly which consists of two cylindrical plates, of i+t-xacaz3p. mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58a+r h j2y9zcu-59ll ap4:tqk r. What is the minimum value of the vertical height h that the marblt9 ulyk2lr4aa9h+ z r:jq -c5pe 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 ass.wcjn1/ooe - tume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces i-ue:r0.gcji+4o7tzd e:;n osetum;s; wc 9duts speed uniformly from 90km/h to 60km/h The tires have a dite oo;g7u e4ejw.z u-i09+td ;cnc;:rs:dsmu ameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s