A bicycle odometer (which measures distance traveled) is attached near thqh2 2 xbn/zy4ge wheel hub and is designedq2 x/ 42hgbnzy for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularz nxq4hb*uy.m-r6 s l2w*eoc; 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 radial and/or tangential acceleration? For which cases would the magnitude of either component of linear ahl;c we6**4smronu2-. yzxqbcceleration change?

Could a nonrigid body be qfqtdi1;p:8 sm;x/zg described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg*mr)wx2 :asqc. tkvq6 er 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 hands behind yo8.zd;pzhajk 0ur head than when your arms are stretched out in front of you? A diagram may help yojdha0pz;8 k z.u to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedall6 v8q,* 8;cvwkvgjjb cranks. In which gear is it harder to pedagjbkjw,8 c vq *6vvl;8l, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower+h5ws)gg pr 2n legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explai)gh rnpsw25+g n why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) 1b rrt6t;5jwq* kx am7carry a long, narrow beam?

If the net force on a system is-(qs(m7tx6 o 8z.s sfhlr4 3-do hzetd zero, is the net torque also zero? If the net torqu.7e-mdz fts o(4q s 8 tz-sx6lhorh(3de on a system is zero, is the net force zero?

Two inclines have the same height but make differenokm r9ks7q/qmkd dv3xge)2vo7 - ,q-lt angles with the horizontal. The same steel ball is rolled down each ioqkls9)x-d/k eq kgd -om2v3qrv7,m 7ncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulv ex k zs:en6:5.0cjjdtaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has th 0e65.:ejvxk jn:sdc ze 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 ma+6e+l 2jezq zv (uaf1uss. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed f(a6uq2uez+ j l1e+zvat the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum arez53e-xe vr0jc conserved. Yet most moving or rotating obje3ejce-r5x0z v cts eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s ehybpu n+:ana:m5a0uk (zw id jt..*tfa+. rc0quator, how would this affect the l kftd*a:jai0uu5. aycp++b0ha n:. nr.(tmw zength of the day?

Can the diver of Fig. 8–29 do a somerxz4j.1s ; r.;fg/bhsttqs/ zzsault without having any initial rotation when she leaves the 1z;g/ ; fszqtr/ hjx sb.zt.4sboard?

The moment of inertia ofr6 ;u:knhuf*j a rotating solid disk about an axis through its center of m*;uukf6 n: hjrass 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 s7 l--tyxbo jo(3 b0dqptool holding a 2-kg mass in each outstretche7 -qby3bl d pxjo(t0-od hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollowpb2* umz)*5 o:7k c;c apkiczx and the other is solid. Descri*:km)k2px pic c5z;cob7* auz be an experiment to determine which is which.

The angular velocity of aeczyt3,0 g f l);dawa) wheel 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 linear acceleration of this point at t 3,gdwa))f0; zacleythe top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standi,0puc vao)jz y (+xk iz*51dhhng on the edge of a large freely rotating turntable. What happens i5u j (ap,kh0c )+h vyi1*dozzxf you walk toward the center?

A shortstop may leap into the air to catch a ball and throw itj+t pehoji/ 6w:mt49f quickly. As he throws the ball, the upper part m9:fjjh+ tipo 64 /tweof 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 conservation wm y uts-7bba fp-l9sd-2h5i5 of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operds7bm5 phu5i-9l t y-2a-bws fate to keep the helicopter stable.

  Express the followin5j g)jcc dr.;fi6l ;+*zhpdr dg angles 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 Ear5qm(fg+6 u d:q j) vpf7(jfiotth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diametjj+t(fdomu g:f ( 6 qqifvp)57ers (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direnjwaooo56l7 4 i.bx7 jcted at the Moon, 380,000 km from Earth. The beam diverges at an angle j.n567bxai wj7oolo4 $\theta$ (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 t6xuv0 m+klu w3gelzp 4 l+8:zphe motor is turned off during operation, the blad 4zlpkuzl v8 3we mx+lp+0gu6:es slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the dk /xyy.tzpd;g2 ndz4sx3) 2h ball makes 15.0 revolg2zdx/2 pxt k;y4h.dzyn ) 3sdutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.+/dz/w x (k0fpm/hom g3w tv*p0 km. How many revolutions do the wheels mw/h+ xow/mzv0/ tpdkm*(3f gpake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Caewt11p.pg;dy) s)crd*,tzzr a5 rag1 lculate its angula p1 z,e .51tw)arz*dd) 1cpg rs;rygatr velocity 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 ma sr4,. ql;mzz/*hcc rakes one complete revolution in 4.0 s (Fig. 8–38). (a) Wha/ r*4 q,r hlz.a;ccmzst 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) t: 1lb yj j)69rg;kfsgin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinvrqa0f/65 cliu( 0z0 e5q pktaw1 ,wsft
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from thnjb)eqewh8as2b 02d+ a 7se: ke axis of rotation is to experience an accelerae qbdn22j:abs 0s 8e 7ah)+wketion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abou v 4djllwk nx7*q+c.n+l-xb/zt its center from 130 rpm to 280 rpm+ 4nl v z-7*lcdxxl b.qwjk+/n in 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 zh8 m)kyrn6;8i/ob ur $R_1$ 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, astr, bdpa78q7e p;hqnow a;x2fh*onauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min tfp;7qh7,np8ahq* ;xwd b 2 aoeime 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 220lc951f7exx qisg-7 dk s. Through how many revolutions did it turn in thisxilx1c5qg s7ek97fd- time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculat o-qjdk eeq)9*x .m9kte
(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 uw6e cfd ea;r/58zxdkb 7i-4*bon 2dpof flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutioni6/57 d2zd4k f 8-eb ecnprwa;dob*xus before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm iimsbz8j ;g9,f-bftq 9u: eur8n 6.5 s. How far will a point on the edge of the wh8frs8;bf9 miet gu u- jb:q,9zeel have traveled in this time?    m

  A cooling fan is turned off when it0t jdn 9fy (1poaxv:8f is running at 850rev/min It turns 1500 revolutions beforevj(x1 08ny:pdaf 9tf o 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 revolutions as the car reducey)h.:zxt:t/dif + 2dxl ,wadg s its speed uniformly from 95km/h to 45km/h The tires have a diamete2+ya l:h g xd:,f)dz/dw.xi ttr of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed k t4mv4 7ujy4ulx1a j:uniformly from 95km/h to 47vjjlyt 4k4u:ax1 u m45km/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

  A 55-kg person riding a bike puts all her weight on each pedal wl) r 3df ,9hv8k4wobykhen climbing a hill. The pedals rotate in a circle of radius 17 cmfw48y9kr) 3blok,dvh.
(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 wi0hci .+ovp cshynjh5.f,xv, eh( tx. 8de. What is the magnitude of the to.nh.yxxvj(5f+8 hh copc ev, ,.0itshrque 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 torquqk/kv bb 6:lv2 /vt/rhe about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0t v/2bq/:/k 6khvvbl r.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the 23alfurw .n m0ends 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 th nf0um.lwa32re magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteniu8z*kmphh ;)n ng to a torque of 38hp hznu)k*m ;8 $m \cdot N$ 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 10tmn8i m/8os+9uygx6j a:-b id.8-kg sphere of radius 0.648 m when the axis of rotattyb8 uo +gm-mn :i69xjda/si8 ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 * 8t qhsno4(aw.vmpmgg c*wahlgb7-b1 i-,+xcm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the huai(*-go1pn s bat lbxvqwhh+w48-.m * 7ggc m,b can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thintbgh, dfv1 ti3z07pi/, light rod is rotated in a horizontal circle of radius 1.2 ib 7g/dz1h 3tpfitv,0m. 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 rotatin40);urhuzj yk8cxv0. a)i o; rz9nf ttg at constant angular speed (Fig. 8–42). The f4r0.)n9 xchtu;tk8 r o0y z;if )ajvuzriction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig.ph5btk ,bz2zr/uma- 3 8–43 abouk b upb/ zr,3mh5z2ta-t
(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 totaonu))yh47ueh yzr7m n i2:b;17 kndetl 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 thp2 j; hmuu674bs4s7 mn3piuhve correct rate, engineers fire four tangential rocket;i6us m4 jnvhp74 p2uu smhb73s 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 t kn,ai-c; d,umass of 0.580 kg. Calculk- a ;udti,nc,ate
(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 from rest to c0ar8cby*9vx*qttw qxpsx0,7j5(g x 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 smad )rpwudky9+ +d 2eo5wll 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 opposi25p dyd+w r) k9ow+uedte 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 /a )qad or/ahl31c x(:dbt2uw10,300 rpm is shut off and is eventually brought uniformly to rest by a friction/)x2u:rh/adtol 3b(1ca wd qaal torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accj8e 8wc;u ox 3ul/,rayelerates 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 thy 8vdmn) js.72w x2 5xsitk(bye forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10xkt 7ndm8)2sy52sijby(v.x w $m/s$ 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 cakgvq) ,ln) sn+ elji27n be considered a long thin rod, as shown in Fig. 8–46.enkl 2)i lgjv7) n,+sq
(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 s*pno5 :.khrk .tc sm55nrv7er6; xzitwo 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 gyj wf.) mfiyit/))h-e9; adt hammer from rest within four full turns (revolutions) andyfj ).iaf)t/)it gdme;y9w -h releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia zmt wd1;r5) :r((*wepgjp e giof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develoq4n3byr0bf g9 h)hq2n ps 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 witn4;bpz x,46b* ,e(zb)z pavdiyw6uz qhout slipping down a lanewqaizexb;uz, bb,p d(6z) 6yp *z4vn4 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of theb( l9y1s;i yy2w7xccq Earth with respect to the Sun as the sum of t cxyisyq7wb9y1(;2lc wo 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 ma yefjxkg+5+snr 2,-4y h+fjkgss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00gyh+g+e ,jnf2rkk5 j+yf4-sx revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest-n t*6eb cc1ee and rolls without slipping e1tn e -c*6cebdown 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 star.nq(3awqs+drn kl;(g.yt k-ots to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before itdaltk.rw(qkg(yn-onq; s. +3 hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kt jw/d6qu0+ 7gwf45ict sm7 dxg ball rotating on the end of a thin string in a t4swfuq0x/7wj di c7m+td6 g5circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mo2mwdj c (5* taezl9gz d()q8nhmentum of a 2.8-kg uniform cylindrical grinding wheel of radius 1m(al) 2n85 dgehzc(9*j tw qdz8 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 at a :mp, +e,z d8 3sbwsjaorate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreaseszeojm:ba 3d, wsp8 +s, 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 inertia hr;/zavv4d o c/5ke 8vby a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed e8 4 /ka5vcdrh/ vz;ov($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin /8t* gm k9jtw6l-a9 ac;chpl urotation rate from an initial rate of 1.0 rev every 2.0 s to lt6cgp ah 8j9m/ u lta9c*kw-;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 tm;hx70pb tr7ni9wzs9hrough its center at a frequency 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 oxb 7srt;9 p0m 9wh7znif 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)tbo;an z );sm 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 momea5we oz4+jh c:2 y9mzontum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identf( xqvvg zd p;m*yhe8bzjgk6y :,39t1ical ,:gv6z3dkfxhyyg tbqv9zm p*1;j( e 8 disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a7cn0x tvi/ +2ijy+ hx4uegb 1;nxs 8fxt 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 center of a rotating merry-go-round plrfs*/nsypg 48atformsn8p4 */yrs gf of radius 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-round is rotating freely with an an bfpc hs0189rplc5 q la,.p3pngular velocity of 0.a5qlp1 8bl ,c.p90cnfs p3ph r8 $rad/s$ 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 eventualz)g;. 4zr0ibhlrk ic( ly collapses 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 carrkigz b(;.4crihlr0 z)ies 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 ez 65agebf53 av14 nifwxcess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass awhqt m4j kyo**i/(d7la-/s w$ 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 ai)/ 4vsyv 7i7o ocnp0i platform, initially at rest, that can rotate freely without frictio0snvvyii)i p4coo/77 n. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6gmx: : uefz (g)20wtwu.5-m diameter merry-go-round turt 2xg :fw):ugu0emz( wntable that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the eryn6oesfa(r7p nc52 ; nd of the rope lying on the top edge of the spool. nr;( f7s5a6rn 2cpo yeA 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 mass move?

  The Moon orbits the Earth such that h,mwx +y1zi;i.t bq.q5,. r)lh yp xbqthe same side always faces the Earth. Determine 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 orbiting the Earth.)btbrq)xy q..mh,.5; 1h+pzqw iyil,x    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/ vz,qscpli92r. qu xbg:k.6n.$s^2$ 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 )c2 ,tq+f9+,mudmbufhtn* x gin the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval +ftu+2u*qntmdb, mx g, hc)f9at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg a*o.alr +w,tsand diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of ma ta.,sl*ao+rw ss 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 is the iakrd 7 p dvv2 /jtz.x9gaa254angular 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 mgnatyy:gdsjt)fyt( :276 *r x ass 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: ynar)y gt:sjg yd6*2fx( tt7 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 aik5-.k(3e- gz7mx lq v0pqzl dutomobile 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 cylin7kl35xz0 zmivl q. p(-kqd-egdrical 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 illustradh-/r3dmzjm (tes 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 horiz - ijtdy()os,:s5nw mgontal 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 ignore fr-716smr;nlyzkv jx e9 iction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleraty ;vsz9rxl1 nmejk6-7 ion 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 vertical pm 65hq6bsjvkpr6 ;pk,x;l1fly on the floor, and we want to exert a horizontal fxp 5hmf lbk 6kvr;pp16,jq;s6orce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fla 1drjgvza 4c5/:+ruqtt road is making a turn with a radius The forces acting on the cyclist anzqta:1cg r/dv4 u5j+rd cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and beginsu9 8 :x17l5ggspypjyx whirling the rock in a nearly horizontal circle above 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 wpu7xl9:8j5pg y1x ygshere does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her au*3tkzkk; 7sggldlg/sj6h.y j3b02t ,xrf4 a rms as thin rods, making reasonable estimat 2jzgt4xjkgysu/7s tk3.g*;l , lh a0k6bdr3f es for the 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 clutch assembly which c7ub)a7k* (soyd2xk rgqalu pc- t, 4g9onsists of two cylindrical plates, of masssu7p *a kx4ryk o),c7tga2udlgqb(9- $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 trac vrsakq65 bh /j0uv3d8vgc 19ck of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Asgsvqv0/dj95u6rc8h3 1b k vcasume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not h* zaa,0yyst 0assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly f:c vvac resm.y20b6 ua50 f,ldrom 90km/h to 60km/h The tires have a diameter ofb60acm5ru2. dea:0 yv slf,v c 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