A bicycle odometer (which measures distance traveled) is a k k )/rmrovk4/s35yde z *8irz)mcso0ttached near the wheel hub and is designed for 27-inch rmz/iym8s)) *ckek4rd o s/0 z ovr5k3wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloi n ohtse+t6z* l)rq;;n*m/ppcity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would th;snp*hlqe;t +ptro )6/*z mni e magnitude of either component of linear acceleration change?

Could a nonrigid body be describe ji / xj,59c7rx; 89mamgb ekuccso)q5d by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaij2o ut 1b6bv x94d+zgln.

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 wit)vp+h6lnnor jl tk73k2f io49 h your hands behind your head than when your arms )7 l9+n32h kk 4vn ofr6iojlptare stretched 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 (u7knkynv ,m mf/5gs/ three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear mknvfm/ k5 /,(yn7sgu 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 havegw8jy6l lh 63x 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 distriblw68 ylg6 hjx3ution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?q0 hhem 4a2z:lrv2c4x

If the net force on a svqz v48iq9 qzs 7*zy-pystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net force ze789 qsz*v -zvqp4yzqiro?

Two inclines have the same heighty5 f p7tmf9xrdq49:es but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the5syq9t 4dxf79 pmer:f bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down anwr9mx1. edubao; kjeb 8 1;n*lq9s* yb incline. One sphere has twice the radius and twice the mass of the other. Which reaches the qlu ;b*m1e. xk wa89e *jnso1dyb9r;bbottom 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 radiusfee:(iv8kw -p0 kuff2 whi5j gunp,i(xa:v), and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Whicv) :-0n:p upkeh eifvg85ixai f(jw, kwf,u (2h has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Ym w xiy4(ddvj-3l p;s*et most moving or rotating objects eventualld-s4jy ld; vmw*xip 3(y slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would iux9eo4)mxg n:8uw ;l this affect the length of th;m ui4 :gue9onw8l)x xe day?

Can the diver of Fig. 8–29 do a5xn 7ug;fv xzy3/d 8yr somersault without having any initial rotation when she leaves the b3n z8/ ;yygu7 df5xvrxoard?

The moment of inertia of a rotating solid disk about an axis through its cente (pwdabesanio3:3n9 /ai. c(tr of massp9a :na( i.i(tdc w/a3e3ns bo 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 tvjd83ijps- ,k8ns,glb1m v4 on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop43gpbm81i, sjt8-skdjvl n v, the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical a.pjxv2q0no svzih -jf3s)b 41 nd have the same mass. However, one is hollow and the other ivj. sxf- 3j 4n)1vhspqo02ibzs solid. Describe an experiment to determine which is which.

The angular velocity of a wheel el5 9)v8td btyl1 bwuyr/bc55 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 e w t5b55 d18yclble/r)vbu 9tyast, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge+yxywie fp(49s6,llf 6 7jm7g9weh pl of a large freely rotating turntable. What happens ixplgwplfel467,h yfy9i m+w 6j 9s e(7f you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he4ailfbdjq2/( 1 jp2q 2 y5vauy thro u4il2vy(b2y 21aq5dq/fjjp aws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservationf3o+l 6o r*m+yro dhe2 of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can ope l 6e2rryooo++hf3* mdrate to keep the helicopter stable.

  Express the following pl6jz44 l+e.nvzl c4 ,sq-eweangles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincik8 l9x wix(8(,wjpga hdence. Calculate, using the information inside the Front Cover, the awj9 g((k ih wx88apx,lngular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergebhe1r 6o 76dlvs at an d1 r76 6hovbleangle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderpx4s:vjca(k 8 rotate at a rate of 6500 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 blades slow d(x p4j av8:cksown?    $rad/s^2$

  A child rolls a ball on a level ,wvv, 6axa (fp5(nef ofloor 3.5 m to another child. If the ball makes 15.0 r6,( eavfof pnx 5w(a,vevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travenfa. n+78mdqw ls 8.0 km. How many revolutions do the wheel.m nf7d 8n+qwas make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.ukmr6exnlo; : )e+n n4m .fg7y qv2y3w Calculate its angular velocity 273e;6lxuy+ne:myq .f rmnw )vnk go4in $rad/s$ $\omega$ =    $rad/sec$
(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 r+rh:q o /ih /z7xu2cfu)ua(rmevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center?/uhur)r/hx ai+7qomc(zf2: u    $m/s$
(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 Sun bks9+e p:w tf.   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi o9kvwjg1 );z. l91yjpffe( ovnt
(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 fromv(j )-vjl ii, yoi4zw(5;dea.dxax3d the axis of rotation is to experience an acceleration ,;x av (i axj.i 45jlw3yiddo v)dz-e(of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cwy6frzsenmd dtot3;0, cq x2qu;*g14 enter from 130 rpm to 280 rpm in 4.0 s. Determi1udrf wy;o gdts6q*ze2c0 3m t4nqx,;ne
(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 radiujr)b3:g i*iu is $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themn 6htuhzi0 eei. qdv; :/n+q;nselves into a slow rotation to distribute q60d/n. e+huie ; in;n htzv:qthe 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 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 you8 2fat,( w8un/a1zjsl04v 4gkme m s. Through how many1g8sf, zkawt(uejy/v m4l4 u 8oam n02 revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ctlmalcvz4dw97d o 5 :;alculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlwq( gvdpi(z 4,ing “human centrifuge,” which takes qw,(iz4gdv (p 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uni+8d5 /9ihab90khtnc j m:ip zwformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thhza/ 0i5 mtncbk9w8hij :p9+ dis time?    m

  A cooling fan is turned off when it irr,l :yi8nygce8) s 1hi/ri9 ks running at 850rev/min It turns 1500 revolutions before it comes 1hik :yril i/yrse8 89,r)ncgto 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 spee2 ik*jv; mnn3id uniformly from 95km/h to 45n*;23i knj vimkm/h The 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

  The tires of a car make 65 revolutions as the c)tq6yt jk0. qit6s.taku(j :oar reduces its speed uniformly from 95km/h to 45km/h The tires hav6q6k tuyis.jk0atot ( ).qt:je 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 bike puts a-e,3mqp ja5wre5z.qyr 4 r i-oll her weight on each pedal when climbing a hill. The pedals rotate in a cirzpi.w qq,4yr em-rrj5aeo35-cle 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.abn+vxl) q-f of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forcen+ f -x.vbaq)l 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. Assume nu13ym:nx 3sz.z,zo xthat a friction torque.m3x1ynzn 3u: zzxs,o of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of macx- i/-nq 9 chzg2f9noss m, are attached 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. Calculate the magnitud9o- qx2-fn 9iz/g nchce and direction of the net torque on this system.

  The bolts on the cylinder head o58e)njh 1p upnf an engine require tightening to a torque of 38 8 uh)pep 1n5jn$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the a: ecel47jc cu1jqnb *0xis of rotation is thr0eenuq1j4 :lcj7c *bc ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia nxqwcz,56 c q4vz+y+jof a bicycle wheel 66.7 cm in diameter. The rim and tire have a xnc6 q5zvy4qj z+,+cw 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 j vj1: wox ywc8k8pms5)6t)ezin a horizontal circle of rad 5smye:xcj jt8)6wv)zko1p8 wius 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 f2f,p3fxl fof se 0e0j5* )9j-jfssy mpotter’s wheel rotating at constant angular speejsm p) l5xsy fj0e2o 3e,0fj*-fff f9sd (Fig. 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 1m+r6t bkfrf3a*j 2k rb5q tm:a.qs 8jarray of point objects shown in Fig. 8–43 aborf 1* +mtt.jq532frk kba: ba86 qrsmjut
(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 t,bm ecig as, 3f2e;90;xojzkootal 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,+wq;kcfrgx7;jc(6l /m d mih spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has,r cqhxm(fgi kc67;dj/ lmw;+ 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 ray0(foj88 jcbk6jynjzae 7 r74dius of 8.50 cm and a mass of 0.580 ky7 of0a zj7j6jby4n(8kj ecr8g. 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, acceleratir,0-5ywv3byqz6 t zp ong 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-dr/4u v s:14t,yrgak hpaiven merry-go-round and is able to accelerate it from resph4a va1,s:tgru /k y4t 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, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300rdwms )lfzj8(wm 9 :e( rpm is shut off and is eventually brought uniformly to rest by a frict rz lfs8(dwemw) m(j9:ional 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. 8hutx:)0kt h *v–45 accelerates a 3.6-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 solely by the action of ths*w4o:ql iqm1q l0:l re forearm, which rotates about the elbow joint under the action of the tqs1q:li40 ro:l lwqm*riceps 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 considereu kvwr22ugemg5wz,7 *iab( g9d a long thin rod, as shown in Fig. 8–46. 2kb2g we(mz a7,9*rui5u vgwg
(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/5mgh+vs-tpc of two 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 hamm:r9d etdz i0b tac 3gp8:sg.0per from rest within four full turns (revolutions) and releases it at a speed oa rzg0 pt ei8dg0:btpc:sd.93f 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 mf pos.e)f;pv69 49 tm hvhf;;gee/kanoment of 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 d*yjp wju cd9yil.2*n (evelops 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 and radius 9.0 cm rolls without *.ce8nd3l -ou 7 gx-(elxfs7sqbs)ubslipping down a lane at 3.33x8egx-) s-7s fusoq*ubel bdcn .l7( $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to)zrthol*i+du 6k eg++ the Sun as the sum of two term+6e) tkgl*z+o +duri hs,
(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 an -th4g ba*mopd 83ga-md a radius of 7.50 m. How much net work is requapg-m- d3g4aoh8 tm*bired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm / g7r hrvc,vzf1z ph:i7+b *teand mass 1.80 kg starts from rest and rolls without slipping dtf+vc7 i r7 zzhb/ep*grv1 h:,own 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 on its tip. It sta zrv3(wv3(hwxrts to fall and its lower end does not slip. What will be the speed o((whr z3vwx3vf 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 rotating on the end vyx*d tw7c5;lof a thin string in a circle of radius 1.10 m at an angular cd*xv 5tyw;7lspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindingx: *ued+-x3(zgnnb gmfs(sc 1qpw )e* wheel of radius 18 cm when rop( q*ee-sd3 )bnfsunwgczm (*: gxx+1 tating 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 roy 2sych8yo4.auhp35; a3ma cktating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation aa;8y4h33c apyhsykoumc5.2 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 in a dx y,ihcx,-q+us5 sq;6 kazhbjxt r.p0:8v0ertia by a factor of about 3.5 when ch ,8az;vsjq+rc0bq .ias5u-th xxhd:,xp y k06anging 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 rotation rate fro 5it /ss-gwkr-cz :2ykm an initial rate of 1.0 rev every 2.5ktk -scsg/ wy-2 zir: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 32yw t8 e;5mskfy- wvj b:-jx j8nki*aaround a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of m5kax* j-ys;2n -fwwvijtbj ke8y : 83mass 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 mo8l mf;uxenwx;5xavh0(9b:fm 2 o tnk(mentum 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 of a 66:l0,4 ei q 8cfmhvnjwuyt.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 da//kj5y,gebv5 go wz4 g,kh*q zu -fk(isk of moment of inertia I is dropped onto an identical disk rotating at angular speyeq5ku,( zah f b/4,*gwv/5jz -kgkgoed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnstxzgt cq;qo /7v2xqd2c) t(s+ at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cern0eif1a jq6o 3(7xaz nter of a rotating merry-go-round platform of raa0361qae i( xrjonz f7dius 3.0 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-rou4* xeoja4mlbb5p.sv +4n vn:p nd is rotating freely with an angular velocity of 0:54e4 v4bxbm+ns pj *.vpalno.8 $rad/s$ 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 collat9y7 x ,mjr;bypses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate 9j; mr7tyby,x 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 120+*;bblr3yd g xehi(1g $km/h$ 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 a- k0h(jny ,mo+y)lgh1 gtxp b.,pmq -$ 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, init: b9(j3t;sl s7 qbjsznially at rest, that can rotate freely without friction. The moment of inertias:jq7jb9tsnlb ;z s3 ( 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 etwk7t2sp lue8io*3r( dge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless i287lr 3t(* ekpwtsuo 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 of the rope 5f5 sqba-eu h.a7dz i0lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig uahb7sf50ei a5q- dz.. 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. Dete)ajywp wh :g*flf 6h56rmine the ratw)jy* 6h w6 :5halpffgio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest ataa6v5y7m7iy v/h n 8cl a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0ubsoiqjy6 r;n w+ +1l,.20 m acquires a rotational raqo,y+ 1 +js6bn wu;lrite of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks,a9p o6u, c;b ifwexl+3emf(j* each of mass 0.050 kg and diameter 0.075jm(ewf3,6 xf;eocuab*i lp9+ 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 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 irb 5*kchleo9e97g+gzs 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 bx 8;-(kw hpfoour 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 itkwo-b;f (ph 8xs mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it .mbxa-5tvt.+n( uy dato use energy stored in a heavy rotating flyb5.m ny axt+tv.(a -duwheel. 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 illustratey5et-m .v:o:ffejfn+ px.qr 8s 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) isq5p eolr,z2(g e rsg9krvodv47 6sm ;, rolling 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 length L can pivot freely (i.e., we ignorew0epw)h/7 dttauj0q 3 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) theejuha ptwd7w0 t03/ q) 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)d8zq bu5okh9 . 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 rests (Fig. 8–55). The skob8h 9 )qz5udtep has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tf-q+r q5zn7- 3v:(t dkpljof s z02ss,uk 6mfiurn with a radius The forces acting on the cyclist and+r-j2dffq z73qomtiz (n-s0:5uv6s s pkl,fk cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rocf4:2pv1r2dko div1dr y.qf 8ok into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, 21r q2yr8:dop1if.o dv4f vkd 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 dxvmxxv seq7f :c,85 tv:a/.psolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio ofx, p78vxm t e5v.v:/ds:f caqx the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, of / liwpjniot-b-cag h /y3.c,: mas.w-t-oipal/gync: hc b ,/ ij3s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra7vcav/z p:j 6vdius r rolls along the looped rough track of Fig. 8–58. What is the m /v7pzja c:v6vinimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a-y /84rocluyr d8bj t:ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as e gkp*zcrc ;k90 9bwcrb0,1 qsthe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wh 1e0c;s9,brw*0c cp z kkbgqr9at 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