A bicycle odometer (which measures distance traveled) is attached near the whq2 46no6zx;c f1wlzkpeel hub and is designed for 27-inch wheels. 4 6nzo1c2zqpw kxl6f; What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Do6le 4dwicwzgiw m(-p(9 pf34p4dm*dies a point on the rim have radial and/or tangential a4p (giiflp d9 cze* -wm dwm4(4i3p6dwcceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describedqay:5 p ,rh:1)t yrilfmm3yf4 by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcey6j*dls t 40wz? Explain.

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

Why is it more difficult to do a sit-up with your hands behind g 0i*orkw-8 yiinbr*o ikd4ng,/)oc,l 8gz9v your head than when your ar)g ,gok98ocvyiigd* 04-wo r /*n8bnri i k,zlms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear7:i2hlkd*f.lty wl(r wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Wh r(: hy7dllwi 2flk.t*y?

Mammals that depend on being able to run fast have slender lower tbc/64/e boe fw16 ynqt94xy ia8ep0jlegs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distrx4 /tc e6by 8wa/914bpnoq6y jtfee0i ibution of mass is advantageous.

Why do tightrope wals; rj2 :gz/stfkers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torquef*4k:j-vo .qt3w t fke8/ja df also zero? If the net torque on a system 3-a .tfwf*f/:okkjt4 8jdevqis zero, is the net force zero?

Two inclines have the same height but make diffet b2o2 gxh)* t1uxrcf1rent angles with the horizontal. The same steel batbrc2f1) o2tx hgux*1ll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously stag5al1 c*rk i/krt rolling (from rest) down an incline. One sphere has twice the radius and twice thecar51kk */li g 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 st:zo r( lt6okch7yb+76 m ghygtj6d;m 2art from rest a6+m 6 hlzjoc7yo h(rkt;gy:gtmd2b 76 t the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentumop/ /fmgn,cz s-k6st* are conserved. Yet most moving or rotating objects eg f,o*6cmp skts z/n-/ventually slow down and stop. Explain.

If there were a great migration of people toweyzx75fg jgkwrp 6 fs* 7-)4heard the Earth’s equator, how would thih ergf6wgj7y5k px7zs *4e f-)s affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having w y/+b2nmpt h6w aq/a9n)7gum any initial rotation when she leaves /9b+t7qu2a)6 gnwn/yp ahmmw the board?

The moment of inertia of a rotating solid disk about an axis throjy2/a9chr 1 uyugh its center of mas9j huya2 yc1/rs 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 mass in each oi4 er*byc;id0 utstretched hand. If you suddenly drop the masses, wil4i0e bi*rcyd ;l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. ird e0)m,h)kdHowever, one is hollow and the other is solid. Describdm d),0ek)r hie an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wes0ym 9n- syse1xb6a1oet. In what direction is the linear velo s1b1m- 6eyoxny9s0a ecity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing onib a9q; y82a-dfaw/r8/ vmjve the edge of a large freely rotating turntable. What happens if you walk j/abm9varadw/;q v-fi88ye 2 toward the center?

A shortstop may leap into the air to catch a ball and throw itccj99 z ompt2c6u,z* sfydg/ pg*d3,l quickly. As he throws the ball, the upper part of his body rotates. If you look *ctjms/o6 cp,u9dgfd9gzc2pl z*y,3 quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular;*ivesr o 0y82s znfu1giv30wg 7 y;ia momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more waywizf21yoei;;gr78 s30i0n svvy* agus the second propeller can operate to keep the helicopter stable.

  Express the following angles in ra3n87/yi hb0oa)k gtscdians: (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, usinzn*d 3nn:k jd/wq48 mzg the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as se:z n8k3mqn* /zwndj d4en on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon,v+v+5 segn1 bts 3v bdiy;z+5x 380,000 km from Earth. The beam diverges a3+1b5vx; g svzsvi bn5te+d y+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 rate of 650o(u(n k cfg;a(ri*dh+ 0 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blfhu g(*io(c;(kn+ rad ades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If tpx u ywop h*ab8ufdm(o5feg.(:2 2 q0uhe ball makes 15.0 revolutions, *8expb5 20uhqg2u .afm uyfwod(o (: pwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 0+e :trised0h8.0 km. How many revolutions do the whieh t r:ds0+e0eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcul64hd fvm)/ qvmdo2 a71f/nullate its angular velocity in l4 md 1/hl/f u6od2q)nfmav7v$rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38).t/z *973fu j xkf2jpdp (a)* f39zd7pt/pujk j x2f What is the linear 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 th0p;z0e miuv 1ne Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeowlw6-gb3; r,lddmk 2d 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 centrifuggao s,bfr;so*c7;;) mhy r60 cikiq.c e rotate if a particle 7.0 cm from the axis of rotation is to experience 0c;*i7,; 6 obkraccfqsgrsh)oy.im ; an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rp l d6jvdh6)vf, a+2t3lgg lnz7m to 280 rpm v)j6 z23+vhdd,l7 tll6fnga gin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius 6im2l qs1f-mdozn- -n $R_1$ 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 Apollok8,g pmx/hz ;ll:b-b6xi-szi h.7n kf0/gl bi 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 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.g6f smz- lkix/i h.7 ;blnbp,h:-bi k8z l0x/g5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throjm.lq rdp7 3xeborav2 4/n. 3ough how many revol /rna4.edp3.vqr ob o2ml3xj 7utions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpmskb(z0(d5 eo3v ;uggs,w l,wq 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(l/6ti ylv* s0l9wzym flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rl6z l y /(v9mwt0y*lsieaching 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 fzayhje;c3-g 75ked w5rom 240 rpm to 360 rpm in 6.5 s. How far wile h ;y7awcg5- k3z5dejl 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/min It turnsc :6eals ep38e5y (bmc h(t49pg zq.c9*qvmwq 1500 revolutions before it4zc.scp8 hmqe9y3 ewacmlbq*69 gvp:e q((5t comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unios 1k/zdu :iuh5o d19tformly from 95km/h to 45km/h The tireszuio :k9 ddtus/ 15h1o have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 rev-)r.ne 3bqny )wkfyc,oswb**olutions as the car reduces its speed uniformly from 95km/h to 45km/h The.)-3bwb* yrk e*yw sc,nf qo)n tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a a7akl* 2g1ikx bike puts all her weight on each pedal when climbing a hill. The pedals r 2k1a*7lkixga otate 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 p r1vf,*dbmzut0f +.z7 f/bfe 3bzgx0 cm wide. What is the magnit/b1*zzvurf ff30b.fe+ gt, dz0p bmx 7ude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. mkby+wpl6 n xl9(lh+p ll050pAssume thal0xlby +pnh5wk(pm 0p ll+96 lt a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whk zr s0f5:pyiip1j t)+ich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Cr0y p+)5 :kfpis1z itjalculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteqe.t wi9s)ys+ ning to a torque of 38 se+sw)9. yi tq$m \cdot N$ 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) a3ayt a3prt7 a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its ce7tp a33y at)arnter.    $kg \cdot m^2$

  Calculate the moment 4z,0 blqax gnf/63ne;m9r cw zof inertia of a bicycle wheel 66.7 cm in diameter. The rim and ,wgaqxnmr 03fzc 9e /6;4bznltire 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, light rod is rotated in a horizom , zn*, n0nirqsdnh:3ntal circle of radius 1.2 m. Ca0d3,rzhnn* ni, mnsq:lculate
(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 rota fw 8sqopq;7jrwr (p1p(4qf4t ting at constant angular speed (Fig. 8–42). The friction force between her hands and thrfqr;qq w1 87tp j4p(s4( fwpoe 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 iner9bkk rn u e8f1j,unc 1l9tc43b1s0lictia of the array of point objects shown in Fig. 8–43 48f1 nbbkictcl931n01cr9lu,e suj kabout
(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 co/3mu kbs *xq4 s59ttppnsists of two 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 spinni +5ly uvgkh/ a072xfqfng at the correct rate, engineers fire four tangential rockvfylq g0a+k 7/25fxhuets 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 andaui- ch.7zm 2a0zki)rwk3 q1elxq ((4v d*kj k a mass of 0.580 k* kk7hi(cjd2uixl. (zq r w14k v3a)e0mzq-akg. 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, accelerating it frafu.jk+ 6eb irq m(t 8ch8e.h*om 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 s qbqzv,vm l30(mall hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite e0q3( lbq mvvz,ach 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 rota + h5s 8ope*f6wxtoh1tting at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of fh+o8tow1se5 xt*6ph 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--gv8.h:dcoy n kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown su ncifj5a:ep ok 46dq-p)n7 )uolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45.n):k pp)o754jfcnei a-udq6 u 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 ibyy,cb rguy)7t l4b4-t6 -ymin Fig. 8–46 ycbtil,4mbu y6y4)g7b tr--y.
(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 mas pd j9b8ck8q 50tglp :bbwob(5ses, $m_1$ 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 rest within four ful ni:d ryac/.w-l turns (revolutions) and releases it a-nra dwy /i.:ct 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 inergd/u,.jdrig(8q.3tcea(ue b tia 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 torquehao, 9hhk3gj 7 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 ke f+h onk4n: 87 /xxqstx s+unoznk;,*g and radius 9.0 cm rolls without slipping down a lane at 7n/nxqfe t*s sxnxk8h+kno; 4z+ ,uo:3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sumbnjbky:m( 8zc+ tp(3;xikq ;p of t q(:xynb ik +cp(;tpjzk b8m;3wo 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.j)bm5oyu.g jti :2 67xrctqw2 How much 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 solid q 72b:moxwi2jry 5cttujg.)6 cylinder.    J

  A sphere of radius 20.0 cm and mas5eyg4(naxh8 mwdxm2yka+* +cs 1.80 kg starts from rest and rolls without sl5da mc 4ym 8y *kw++g2exxna(hipping 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 balanceves7z*5g:sd.- lk3()gu sb /toqhr jr d vertically on its tip. It starts to fall and its lower end does not slip. What will bsgr h vg j/)qbses7tzo(: .k*5rdu3l- e the speed 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(k/3qls y q2dx/oa6mk ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed o3(adqk ls/kx/mq6o2 yf 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindri u 00u( d4wuxmfi*f)h+ vuzxxv8+ cp)xcal grinding wheel of radius 18 cm when rotu +xu+f0fhux xc4pv80 wi)u)d(* zvmxating 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 hisso s3wo5h4 t;g side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8w3g sot 5s;4ho–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 reducj djt;t519e5wlmf. mwe her moment of inertia by a5;1d5ej tjmw m t9.wlf 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 her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotat8k.tl*i5eab9o* ,n nn bl7xcfion rate from an initial rate of 1.0 rev every 2.0 s. n7nk t9cblfane,ixl58 *ob* 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 verticohmy+ mljq7l,,lj2 f; al axis through its center at a frequency of 1.5rev/s The wheel can,,hj llo;7fqml2y mj+ 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 momentusux -jc2t 05i rfynw,*m 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 angular momentum o9tv4 9coib23cw6 yil if 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 iap/-zj e,d x3gdentical disk rotatgd3pax jz-,e/ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at pv tkwmq8 ;e-92.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg sta ti:ldb*lrs5,q60;izknt vy4 nds at the center of a rotating merry-go-round platform of radius 3.0 mt bz; lvsl6 ,rt ynkq*4:5d0ii 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 wicqrn+zh gg ;vgl8tfvxgr7 /is2ug*g1c /)t 67th an angular velocity of 0.8 * )fvsq g/7rg76ggihrz /gc n1u2;g8lxtt+cv$rad/s$ 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 whitel/o3j(r 2nlib ygh*2j dwarf, losing about half its mass in the process, a y jhjl/2 ro*3b(ginl2nd 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 winds in excess of 3u7r ga d.9cnxfcb.7y6 q7.ghu g,xtp 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 jh vp69),j:zfxn6k96 soybrd $ 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 g8dod6n / aeh0can rotate freely without friction. The moment of inertia of8ge0od/n da6h 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 standj5 ;+djr:,hwr unv bjw,w3o*ks at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of + *vnkwj:r hwd ,5,b wjjuor3; 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 wit3n/6zq.al qks c;n.f jh 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 behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of kan.6lsnf3q ;j z.c /qmass move?

  The Moon orbits the Earth such that the sdz jh k 1+b00lb*zaxb:ame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momen kbz+0b jd1ah0 l:*zbxtum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest0ckh a -*lmk2yttf(mga,a 1b- 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 ,846tah34+izsrgmhyy3 gjhkthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. r zamhks+gh3ty g44jhi38y6, Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two sovzqupo745ahdzbw/p:rqcb iu o;bp9 ()f( u+2lid cylindrical disks, each 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 the vq;wicoa:zr)u4fq 95u + 2puo((pdb7 hbpzb/ 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 angulr s.k2 q-8lp42ctf7- m umrtlrar 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, wer*p- ihw;j0 q,6 /mhombhvq bn*y7otm4e rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravhi 7y h m-*;t v,pm0ow*bnqo/ m6bjh4qitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-s-v *czl/lq6 bnxz 1p4pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 24/qlvb-* sn pxz14cz6l0 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 illustratkz4dpfp9m,f.e a( j/d es 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 hof:bdwx w,a.*lztw66 trizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., k0m 6nxtq9 z61bavh;qwe ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then releasedxvkh6b6naq m019 q;zt . At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R55k r+c6mra0r+k iz*ac ug0te. It is standing vertically 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 zcur6+a5*errk +5 0ik 0cagt mrests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn ng+t s oz0(r0 m*ber4fwith a radius The forces acting on the cyclist and cfrtb* +4z0sge0 rm(on ycle 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 slyd1h2br vx+/ qy le) c9-0d 2felwuz1uing of length 1.5 m and begins whirling the rock in a nearly horizontal circle a2h)cy0+ lxdu ybe2/ wr1d-eq vuz9l1fbove his head, 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 solidahondx;hkrv a;.1q zs 0op.yg )vo* ,tg;0vu3 cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning ; vt3agxd;o0pv0*zny ;hho . )kvsr.oa,guq1skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,(w yv(acg 0g3 fe mvmfh;;ey9vyk 77-q of ma -egm;ym( ;w7vq(v9fe c73 gaykhfyv0ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rat(jjh,w f; 4ob0*dscp dius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is0,t; d obhjjfs* pc(4w 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, bu)to2jfqf: kxxoo 1x+ .t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make pr.q8kfv 0x0np3jfx57 ,y nas 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angf xysq0pn8n ,0rpv f5ka37xj.ular 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