A bicycle odometer (which measures distance traveled) is attached nea4lq i/ or.6czt8j*qcpr:tm- mr the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with*ctimolr /r:- 6pmjtcqz84.q 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim y-z85c/t6mk;iu ypxf have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleratiot-m5u6yy/xzf k;p i 8cn? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desc6ty l9 rv: jn 36ccl5 mnctm9zo;w(xi+ribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explai m )l- i;no1bpyq,6aain.

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 ytef0a;;*1 vjvcpm e h4our hands behind your head than when your arms are stretched out in front of you? A diagram may help you to tcej af1;pme4 hvv*0;answer this.

A 21-speed bicycle has seven3xbb iq 7jsq +x xbcfxr-l;j7)xy6 m-1 sprockets at the rear 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 harde)1rml73q-j bxc;6b biqs7x y+xjxf x-r to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able 0fs9ex;sp p u-to run fast have slender lower legs with flesh and muscl;s-xp us0p 9efe concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narr, b xdl9h9sk(fow beam?

If the net force on a system is zero, is thg d,. 3f tsy )xh3jky9(nxrko;e net torque also zero? If the net torque on a system is zero,xkf;hgdky . 3,3rxs)(n ytoj 9 is the net force zero?

Two inclines have the same height but makxz/as1 -j0vd tw1 ljr3e different angles with the horizontal. The same steel ball is rolled down each incline. On which incliner 1t3jasd/ 1vjl 0xw-z will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incli tbro:7 bet6j4ne. One sphere has twice the radius and twice the mass of the other. Which reachtt6 bjbeo4:7 res 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 thnlz3)qat (y9 k +q8hnde same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has tqz h8k(9dn3+qnat yl )he greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are +rg)hv yei8-ycd 8f0m4 u3rncconserved. Yet most moving or rotating objects eventually slow down anyiccnr3v8 4du ef8h0mygr-) +d stop. Explain.

If there were a great migration of peop/6xudyi q z-n(le toward the Earth’s equator, how would this affect the length of tqui dxzy/-(n6he day?

Can the diver of Fig. 8–29 do a somersault without having any initial s 4ce7g )oktb6rotation when she lea67kg ebt)cs 4oves the board?

The moment of inertia of a rotating solid disk about an axis t9 ppw csgtn-kl:nhy: 5 5y,wi,2yau. chrough its center of maw5y2yi g9clshn.twp5-, n a:p kucy: ,ss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating s7z 09k9h(.hm -3jd vuugz3ujmerhxp8 tool holding a 2-kg mass in each outstretched hand. Ije -zmv.3zh k87 hmj9udp3(x90gu uhrf 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,8 *juk5a6/rr pcn lat4 one is hollow and the other is solid. Describe an experrc4 jup 5rt /6a*nal8kiment to determine which is which.

The angular velocity of a whe /:u7egw-o) (oufpvdq pws4*pel 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 lin 74 (o*)oqeupfgw p-pv :uw/sdear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a lao1u;91 st )qv z;h faenv93m uemks,7gql58earge freely rotating turntable. What happens if you walk toward the center?l mue,3 15 9)fm 1a 8e9sq hnaqeogkzvutv;s7;

A shortstop may leap into the air to catch a ball and throwfdzvps+ 4 )ct7d2xp; h it quickly. As he throws the ball, the upper tx zdpv)2+ 4sfh7pcd ;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 conservation of angu,eikk* yc0 eud c3fk h:;mf-s*lar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter sta-ki ;kcf0cyu m 3ed*eh *s:k,fble.

  Express the following angles in radians: (a):o8dl* b414zlatgd br 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 kw bte3)gq ;5oof an amazing coincidence. Calculate, using the information inside the Front Cover5;kgqb te3 wo), the angular 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, 380,00 yztzexn/;4k-6jh sgk ++q,t4 il, zau0 km from Earth. The beam diverges at an angle-/yj,x 4it,nhkgzau s zt l;ek6++4qz $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender ri ekrn pdf03,(,vce6 kotate at a rate of 6500 rpm. When the motor is turned off during o ev,(i,erd fk06pk3n cperation, the blades 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 anotheep +q*6:elvjoh5:cwwi/u el0 r child. If the ball makes 15.0 revolutions, what is its diaoi j/e+lv:h5p6e 0uw*l cw qe:meter?    m

  A bicycle with tires 68 cm in diameter travels (wz pf/ap -q2w8.0 km. How many revolutions do the (wp/zwfp 2qa-wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm2 7.gniu,yrvn w:rbn12 k9n jd. Calculate its angular veloci:knr r972g uwv1yndnj 2b,.inty 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 m7mzfr(dizt-f -51(kq + ox odhakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centertf+mqoh(di5 7k-rdfoz(z1 x- ?    $m/s$
(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 itse*xd k d+wlnraq6,)i* orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p3k;b efq;y.3,qy 2x b/ zvdvxzoint
(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 o- c)5i/t nxzfaug)g4a9 tt l4cm from the axis of rotation is to experience an g 4oz9 )a/ilcu 4g5-)ttn xftaacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center5jqp9v84gsu yo:vgi1 from 130 rpm to 280 rpm in 4.0 s. De 4g9opj1 s :5guyvi8qvtermine
(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 1f6d kjyx- og)$R_1$ 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 Apollojb6ag 8wu; 5rxtfl 3.kq i30qz 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. Thq3 3qu.5 wjrb;aixz0lg8kft6 e 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 u6o a-5i) , 28ljquz igy2wmdpaniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this ti2jolziyw,ua) 2i5 mpq6d-8 game?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. vsp .egz ,5y-lCalculate
(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 highs )v+6f-i ava o:3ektxkpeed jets in a whirling “humae- kai6+k f)oa:vx t3vn centrifuge,” which takes 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 uniformly from 240 rpm 9-e,bahh qwq+t/m j )3h 7)r(oyyynfpto 360 rpm in 6.5 s. yrb)qt7f - p /hj,(omyqhwnhaye39)+How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turnedbt/rh:qxwec5/ z-n k )atts19 off when it is running at 850rev/min It turns 1500 revolutionzk :rb5te1n qs9t -ax)/w/ch ts before 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 r xkv,.sdur 86ccuq7 j1qphq1+educes its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.q 1sk7hd.,pxc8+ qrc6 uvqj1u
(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 revo,0 /i 0o qfmuve2-wtwhlutions as the car reduces its speed uniformly from/-20 iof t0,wwuhv qem 95km/h to 45km/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 heryf r,45sln/ekjv7 od0 weight on each pedal when climbing a hill. The pedals rotate in /o5 dyfsrjn4k evl,07a 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 forkx-ze6o9y-k y ce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if thz-y9ek6o- x kye 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 mbx*8/ak+*fbgowyx +j4qv *iFig. 8–39. Assume that a friction torqu k8oij a/+byw *xv+*m*xf bq4ge 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 rodahopr9d r* q846 lvn7l+a :wxd which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude oahl v6r*dwa9r xdpn8 :+q4 l7and direction of the net torque on this system.

  The bolts on the cylinder head o ),vxfo*w-bu - y (luquolz6n;xd9i2lf an engine require tightening to a torque of 386f u29dyx*uwozo ln;xu )- ,vilqb- l( $m \cdot N$ 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 e,ba:k: d ak,dl 6x e5ni4.btfa 10.8-kg sphere of radius 0.648 m when the axis of rotationfxk4ia b,:6:dt,.nkbld5ee a is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.d5 kd)div ic677 cm in diameter. The rim and tirii )kdv6 7d5dce 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 horizbr+ -c cpx:st(ontal crsb+(-c cxp :tircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bv m;75t lerry:.scea8 owl on a potter’s wheel rotating at constant angular speed (Fig. 8 75mr8yslare. c v:e;t–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point object2hw+d7vzw1;6xvr orv s shown in Fig. 8–43 abouto xrrvvd6zv27w w1 h;+
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mas6(g2oy0 rx;sw+; ii5 maiahd es 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 .zm j3la8xt()0vsle+g sjphev 4kuc v.*p76 ksatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the saptl+ (s a .vjg6.p*3cjsxm8l4zkh7)eu 0v vektellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with q p;btwosc87. c(kycs8 5cy2)c vx8roa radius of 8.50 cm and a mass of 0.580 kg. Calculat;c8 tb)ps7rvo ks5oy2 qc8c( y. ccwx8e
(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, accelera) z (jlf0,ioraou;mf ,ting 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-drivengm u(jeut d,nm2i39 7l merry-go-round and is able to acceleratmj3 uil792et n ,u(dgme it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at oiw;v 0krujsv hgs1*7tbt*9st10-hd10,300 rpm is shut off and is eventually brought uniformly to rest by a frictio gv*bh0tso ;ku9*-hrstj 7wdi1 1v0tsnal 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–i8 61 k zdy/;9jogddmb45 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 isu:f(hw g*lj 0aiz3vh 6 thrown solely by the action of the forearm, which rotates about the elbow jv:hl 3zfu6w 0(ghj i*aoint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a longwn p0afc1tn .f bq i93jo*,5ij thin rod, as shown in Fig. 8–46.9n5bf0pfj,qaj1nt .w oi *ic 3
(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 rvzz we7g7c oetf*5 q(8 /pzs:masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within ffhb(pc8 dzp; (.w;kxgour full turns (revolutions) and releases it at a speeddp;xzcwg( k;8ph (bf. 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 momenindrprs +o,/+ s3vxl0k ;*gj ut 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 develops a torque of d jgg/ c(zb8-:bn gij7280 $m \cdot N$ 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 cq.ngkt alfnx1 -xo9 io(1p9-4p2swgkyz9l( vm rolls without slipping down a lan 9yp gxn21ll itq4k(zx(kn9sow-v op9 -gf.1ae at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en /n+ qfbuhx-z,ergy of the Earth with respect to the Sun as the sum of twoqb, zuh f/-+nx terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How muchg0r/pgzgav 5i) e:o-9acc.si2jvqv9 net work is required to accelerate it from rest to a rotation rate of 1.00 revolutior/vgaqcov02ap5v9 g:9z s)ji .g-e c in 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 and rolls )o;k+t9p * (g;e8ro xpmel p6fl;gsxkwithout slip)s r6fp gt+l*ol ppe;xekx;98g(o;m kping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip.4c-w n0 8cg1ulmhr*so It starts to fall and its lower end does not slip. What will be the speed of oc4nrs8*mhu- 1w c0lgthe upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum ofggu: z-z aqrh p1+,zr5 a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular1ra-h :p g5+grz ,zuzq speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of sd,lvrm4 /584rjoiyaa 2.8-kg uniform cylindrical grinding wheel of radius 18 caj/ol,rvd i45sm 4yr8m 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 aur6z o6 e3*tkvt his side, on a platform that is rotating at a rate of 1.3rev/svuzo6 6r *t3ke If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can renlu00xtn 7* oaduce her moment of inertia by a factor 7t lx0o*u0 annof 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 spiwvv 7suz1m56 ,as zw g .rhes)aq4o,r3n rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate hs.1 ,6 aruq5v 7zgs4)w,vswm roz3a eof 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at yrt+)+cpx 3 4zlf e*kia 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 of radius 8.0 cm, onto the center of the rotating wheele)+4ykzt*l+p r f3xi c. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3 le *j6.vxxn5zud..vz.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 momentu oca y);0avuy;m 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 idb8azijd0*z:0q-t a hgw 0h*d d. fcj7kentical disk rotating k zdad c8h.h0d*jj wa q0:iz0t7b-g*fat angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns9c)c l ;tdhac. 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 tht kw1 -3d9yxthe center of a rotating merry-go-round platform of radius 3.0 m a k3wx9td1 y-thnd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0a)pol 2m rv/k;zpih r b*k,n83.8poh;ank r*)m2vl b3z/r, kp8i $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf,dp6owb:ztax a j/g ;,3t8wde* losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass adpx63e jb o*,8gt; w/awzdt: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 involm40bv k7jgpfh q05pj /8iwp7ive winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass vzo0t +1 2zqsi$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate5 ahq 6brabi/ 6rdt m)s-ybi;689svbg freely without friction. The moment of inertia of the person b)mg aa9qvbi8 sd-b/ yhs5t66i6b;rr plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merrm7g0hk9tg :wnfr5m9i c/s9n a y-go-round turntable that is mounted on frictionless bearings anw 9rntg0mhf: c9iam/gs79nk5 d 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 ogvr/bu s*b7o cz1 15tjggwkat/ ( y;4wf 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.54v/cw *gzgbw srtoy 11k/; (tjg7bua 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 svajjq *v84yt j;zv( wps/*w8kuch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to itw8p z8ja*ty k;vq( 4wjj /s*vvs orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rn5x2,(pu d 2uaop4vppx5 zbz.est at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 (dr3 h7door okm2. jp1m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the mo13(pk2d7odrh o oj.mr tor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg 3ymjz+ dc frdk f3 v48s5*5mrsand diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter km* r3zssf j3r+85f 45dyvcdm0.010 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 t;zk 62y/5apq7.x6uu 7ynpyzn l :w ldzhe angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wt h ge87asb j4ta9d)s*ith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that tj94sa tet b8*)ha7g dshe lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for iod a. oif797pt:dwzd)x5n ;kr t to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 a79dwf5k:)ipxor 7o tz;d. dnkg, 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 illustrateksjb1(4l n (ors 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 rolz0 a62x/w pefhling 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 (im5-g4 x g/lcet f+(qmo.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release,xg5(gqf/4o-ml e +mtc determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on tzec73*lku a8sjtr-d -he floor, and we want to exert a horizontal force F at its axle so tha zcakstr-ul8e 3j* -d7t 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 flat road is making a turn with avi5msy2aa,3p 1s6eibekhb/ s d1f, 2a radius The forces acting on the cyclist and c he3a kbi2ss p5 ea/f,6d12,vysambi1ycle 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 0:jb7qx+uk9 q rts-lx9 egem4into a sling of length 1.5 m and begins whirling the rock in 4u bm9s0e j t97gxeq q+:rx-lka 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 where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms 3,*bokq7f, gmkku yp:as thin rods, making reasonable estimates for the dimensions. Then calculap bkkgyqfmo, u:*37, kte 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 assemblyqp 2pqdbr1:0v grn5. qnp( 5n *iq;igwfu9br3 which consists of two cylindrical plates, of b 1irg 3fr *i; pnwdn5 npv.q:2pg9b0ur5qq(qmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 along3l )deererzv+e8zsm0cu- rrr umu ,k7*;.oq* the looped rough track of Fig. 8–58. What is the minimum value of the vertical heig r*. sv8u,mzque-k)r3 rl +dzre7e*mu;roce0 ht 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 an4b66.zs:leopplq. d j-h /dh(nu( abssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spyj55 l r)4feq.i c uqhz7a.z7oeed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accela5j4. 7lf)7zziquhecoq yr. 5 eration 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