A bicycle odometer (which9rss/k*fm ,9.zpzb(rwk f2w r measures distance traveled) is attached near the wheel hub ankf r ,.wfb2/*k (r9pzmwsrs9zd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on theq3.2 t;w+ eggka-9ip k+zghkm rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acce.; i+q3eg +hkg2w9t pzka gmk-leration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descri apxah9) p.p1xbed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater td+k4s7;nsq f:.z u( jqzi3ohcorque than a larger force? Explain.

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

Why is it more difficult to do a sit-up wipr fumg*2 nf26- bb w6t(i 0ma7fgoqb4th your hands behind your head than when your arms are strmui6 r4b*n-bf om(b fwtf07 pag2q2g6etched 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 three at h 0fxbfj63od( 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 3 f 0jofdb6(hxharder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fasntuc f*-)y q0lz/a.(g0f rry jt have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On th)(0ya u / z*fng.trrqc0-yfl je basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow be5aa(umb lfej68+ o+ pbam?

If the net force on a system is zero, is the net 0,karo :u.t 4u6ioztd torque also zero? If the net torque on a system is zero, is the net force zero?,o0t t k 6uua.iod4zr:

Two inclines have the same height but make different angles with the3a 6vlg + u2prvq,gy;q horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the u +g,rlvyq62pg; a3v qball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from r0 o4mqn 7pk)tcest) down an incline. One sphere has twice the ran 7tmck4qo p0)dius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mqd1a s nwa460wass. 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 bot 6qn4a dwsw0a1tom? Which has the greater rotational KE?

We claim that momentum and angular momentum a,h 3 5son-svy0iq pri; s(w5jzre conserved. Yet most moving or rotating objects eventually slow down3,y pv;sqs5zin5 -0(jro hw is and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would,ea81jaw2vpb fr1yd9 this affect the length of the day?2p8j bdyf rva,1e91wa

Can the diver of Fig. 8–29 do a somersault wrq)b g)5-hhy*,kuvwtyt *4m(evscn*ithout having any initial rotation when she *)yrnvmb*s-t* ,he5 tkhqw)vgyc(4 u leaves the board?

The moment of inertia of a rotating solid disk about an ax-crytnmd 4)nk7d;l ouz h.q10is through its center of rqc)ohd7 01nl-ytunmz d; .4kmass 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 rotasj7,q av, av9 (hbgapy)wqfu)s, sdh4q/e, k8ting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular veloeh8gd,ufbq ,qaqhv )9(s / svay) sw,,kjp47a city increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hob 678 ddpvaa*5n3rr vsllow and the other is s*ra3vdrb p5sd6 87v naolid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontif 1g npw-3aj5al 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, descrij5gn3i1f p aw-be 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 of a large freelyl7r1*d(. .3xp7wc0a 4c ptqokdw mtfk rotating turntable. What happens if you walk toward the center07kpwp fkodc t 4c(1 wd.mq t*x.3arl7?

A shortstop may leap into the air to catch a ball and throw12,(xhgw7gh br 5 fipm it quickly. As he throws the bawxgh2 g51,hfrm7ipb (ll, 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 conservation of angular momn(.nf3vpg(f- lbhj +u3n e4iaentum, discuss why a helicopter must have more than one rotor (or propeller). Di3 (nul-n3iep.gf a(fvhj+ n b4scuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following aned pln-92fal5 gles 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 cn.lwa tl4b2k*xv48dnoincidence. Calculate, using the information inside the Front Cover, the .bx2 4dv* awknt4l8ln 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 Mow4t9 3 sdnas l z8oso-n5dvs6*uj-m6ton, 380,000 km from Earth. The beam diverges as ss-dwt6jvn-s a4 odlu85oz*n9tm 63t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor .y5 ,6 nib-dd7h6xank ep 7pgbis turned off during operation, the blades slow to rest in 3.0 s. What is the angul -6b5b kgxpi.677dyn eandph, ar 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 ball makes y7ey24y/ b huwtkg6j0 15.0 revolutions, what is its diameter?gje0ytyh246w7yk / bu    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. H;tf+,goe laz(y5;e viow many revolutions do the whl5aevyz +t(;goie, f; eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 dj,)v28po5 ;wwi(m(sa w xj mprpm. Calculate its angular velocwjd2vpo;jxa m( ip5,)m w(w8 sity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 sm+tyut:(b u,em ez5w l8 btx9) (Fig. 8–38). (a) Whaw) um95l,m:bb t y+ utz8etxe(t 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)8buhow0e x4 0iv)(se bxc)f/f in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po t155gshkgo:q4 v7wpn int
(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.*dzh/ .06lsg hnp q-qx0 cm from the axis of rotation is to experience an accelera x-ghpq*zh.d0 sn/6lq tion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 28gkb;*r/z hp yih4ai v01m*2.hgx 9vj b0 rpm ivhm4*ia0.; *1xjk2pr h/b h9ziyv bggn 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 radiu7 uje-3g0ujl us $R_1$ 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 +;c itbc(5 ig)t /kknyndgcc::zr;r 3the 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 time interval. The spacecraft can be thought of as a cylinder withd;gzc:ry; /kcn nr )cc5(k3 gtiit b:+ 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 s. Throug8m3dl wqzqyn r)b8 7*l)tdg12fnq ml1 h how many revolutions did it turn in this t 83 n*r ylq72dqnbmfw)zt1mgq 8l1dl)ime?    $rev$

  An automobile engine slows9.ntwn9cop d/o yt1c, down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flyi o k0mn.6d -f1vikmqipxd2i4:ew s(e8ng highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through e.kd1 i6 ed0mq8 ( onm-42xispfi:kwv20 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 acceleratesnisj7myn9m :+ uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge +jisnmn 97:myof the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rhlw j g7mb:g94unning at 850rev/min It turns 1500 revolutions before it comes to a stophb4j g:9lwmg7.
(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 reducet+ii*l +qr *xs kybn 1,h,bwe-s its speed uniformly from 95qb+it*,lx n,*1ebr k+i w-ys hkm/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

  The tires of a car make 65fkrbsmv - ;,r. revolutions as the car reduces its speed uniformly frobf.m,;svkr-r m 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 ridin ngac p, r,o;4v7tibf,g a bike puts all her weight on each pedal when climbing a hill. The pedals ro;vg,ac,bp7 4t, i rnoftate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is th5otnx eo,ksbbg7 ;3h6e magnitude of the torque if the force is exernh sgo ; 37t56xb,bkoeted
(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 wh,3 pql)jo sqxd1rp 54i,xd0mceel shown in Fig. 8–39. Assume thpx0 ) q4qo5dsjixcrl ,,pdm31at a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a ma(zgart+ mq m v s1b6r2hxh5syf d7c21*ssless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal positio2amc 6 sqhxf7z1 t+(2myhg1r5ds *rbvn and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requirez:6huo55 +nx3h d e1*iebcyeu tightening to a torque of 38 c5x i: uhed*u zbe+eh3yo156n$m \cdot N$ 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 o:ld8z,+w) rocrlw h.p f a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cerc8:r h woz+wl,l).d pnter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheelejztoqj**f rf79 -a 8l 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignor-arfze*7lj8j q9t*o fed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin kenoo)+l2 bv9, light rod is rotated in a horizontal circle of radius 1ob lkno e+v9)2.2 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 pos(7* kv fv/d8yhx*pvb;vxj. h ;w1lmdtter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands av;k bjh1h fdyd.wv/7l (s;x8 *xv*mpvnd 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 ofdwk*j kec ,c x eei3qi6k0xp i/m,06)g point objects shown in Fig. 8–43 6kg*ji0mw,ikqce ) 3 ,6/de0iexcpxkabout
(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 con b0jxuct2 33(owe8 upbsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindricala: ebcb;d,vn9 satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fi vb,c;e9:band g. 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 wpor8xz5ldu 7 om ica577-fhb9zhbh vg-.y2k0ith a radius of 8.50 cm and a mass of 0.580 zb5g hlr7yapm b8- -dh2o9uh zf5x0v .7io 7kckg. 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 1p+b b.6zbukug91fblida t 0:bat, accelerating 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-drivegih2 o;(3wjx 3wxawkeo f1* 7ln 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 radilf7(owei*x3wg h31a2j;ow x kus 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 10,300 rpm is shut off and is even os z2jj3)4vcntually brought uniformly to rest by a frictionan)jo4jv 23zc sl 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 accelerj h(in;*dtnutg3n a bg5+4s/t ates 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 so j gu 3:/ex.6fbsqghkz (4nx*bpc7hk 7lely by the action of the forearm, which rotates about the elbow joint under the action of the triceps musgp hnxk(c6 h 4*uqxgk.7z3e/jfb7: bscle, 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 con ,4 glzu690dhmm;so duxb qv -u-l,/vnsidered a long thin rod, as shown in Fig. 8–4bz0ud-, q, d6lm -vomu nx9lhv ;4g/su6.
(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 li- , k3yngn ctxx;3vj/.n c)econsists 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 hammer :52j(pnudr 5*tu3xpm9ctx m y from rest within four full turns (revol2 xpy5*r3d ( mjt u5xmupn9:ctutions) and 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 hasp r0zysyj-ilbc 7b7+/yiq*z4l sf 6c; a moment 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 g5i 6w:cyl fdxljqag11x6l,)develops 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.yo;)zw j- a7i bfeo5j and radius 9.0 cm rolls without slipping down a lanw7fje.a ;jozb) o5i-ye at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as kdj,w*rnqz,y abocp557p7 fb17l6 d bthe sum of twoo17 yz d, 6wl5bpafpcndqr*b,7 k7j5b 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 muc(ukl6cl ,hhw6gkwy ;b - / cu1rjbku91z+*mkdh net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume itkm6/hk,y-j *uhr 1ucu bb+ lk;l91(dg6wczk w is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stac(.g jo69s(grok c5ikrts from rest and rolls without slkc9kc(rj(o .5o6s gi gipping 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 o vgi9)7ge7lej lmg8.on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conserval)8 l e7g79ig.ejgvom tion of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg 2i0vp ml- r/kmj0hut:o c: kn y9h:sao6cxk+ 5ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of kc: ktu5ahx+0lshy:o - pvoc2m/0 ir96nmk: j10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentt:e f)a.l z nzw5u ;yozxd11h6um of a 2.8-kg uniform cylindrical grinding wheel of radiu51 n t ewxa;6z oylz):u1fhzd.s 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hi; 6usk* o6zjuqa*g6iql s0npi : huh*.s side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation deca i*;o6iu.z 6*gh hnjqs*0pkqlus6u:reases 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:ijw*xd9:wf6dnh4b rv 2 n+ fp–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straix wf:629 nw fdv *npj:d+4irhbght 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 from u8j( exrl *1odv4boc7an initial rate of 1.0 o 7 l(dexb4uocj* v8r1rev every 2.0 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 rotatipoy:s/uyf+(iie:8 k i ng around a vertical axis through 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-o ik+(i s ::feyiy/p8ukg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figu f*je ivg* b6xalvk-avk87cfb8 9(zc 1re 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 3npr 6* +v(uyswxsu8fof 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 momen-bo;nbt0q 49 vvno ti2t of inertia I is dropped onto an identical disk rotating at angular speed 0vtvo bn-o ;2tb4n9iq $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 c9o+kdp tq9o ct7 3af5$rev/s$ 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 mera4 tvo0ep9tv4ry-go-round platform of radius 3.0 m and 9tov vpt 4e0a4moment 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 rotatinvc9 +qo-0hrkooy8 ( tih6ke ,cg freely with an angular velocity of 0.8v8o9 ( khiq6 , 0tyhoo+-cekrc $rad/s$ 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, losin8 rihlbh. 8/tsg about half its mass in the proc./h8l8shb tr iess, 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 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 2e)sp- s . pv0wj/qvjvexcess 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 .us5+kigj,mq , ;2hzfn6yc gi $ 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 rotate freely withoutqond.dpb3t:w 6 x7ys1 3yda,qk; ni*r friction. The moment of inertia of the person plus 3,3p .7:r1byw6 ki;dqaxno dts* qd ynthe 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 mer dfud6jt0+( ;tely m(lry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ulej+((mf6d ; d0ytt lof 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 .f(,l/6.upfmpdq qko the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s ce qdko(m .fpq,l/u.6pf nter of mass move?

  The Moon orbits the Earth such that the same side always faces thed9 ppx,a5k0ya1k m1 dngsu./e Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter casa 9n,1dexudgpkp 51k/0my.ase, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 jks5d n1/ cfnk3o/pp 3m/$s^2$ 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 m nywi8sic6 huo9 s vah/ g(t:d.wbt8. .acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivereg: c(sn i9iuo..yvwb hhdw8. /sta68td by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of masqg:onei3tbt6lf 2/n 0s 0.050 kg and diameter 0.075 m, joined by a (concentric0l6qtgonn/ f tbi:e2 3) 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 is the angular speed of the rear whm ;aog+qe7nw( eel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, ya 3uql:aj nqjg,.*ob3x.d4kwere 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? Ass*b4yga33 ,dqqjk .ol .u j:naxume 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 aru1 a,3m:z k:b.nn ytrhyx7b. utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 2u7 1mnb yak.:xrrhn: .t 3,zby40 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 illustrates anhscify b r+1/0h4az 9m $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at spe3s+gc+7fp pmo/ct8yi ed 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 leng-wl- s0ylbh pg/1u/a3;gksw ath L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rbk-0/3pw ;/au asgg yhwl-1slod. 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 vertic2. h r-z,udxvwally 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). u,rx.hdv w 2-zThe step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wtl 8:bw;e gad.ith a radius The forces acting on the cyclist and cycle are the normal force:.t; la8 bgwed $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 bcc;+ugz zdb0jk/ )3esegins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5. ;z3ck0zj+u e/scb) dg0 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 as th j:)0n8+mte 9 ks: wlnwn+cqz uw2vg)gin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretchedwg:t9swcn 2 0n+jgzq)lkuw8e+:m n) v arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, o3dlk;lg (3z+1ojdff - f*dmrr f ma1-flf;dgj(rfmk 3 d3o zdr*+lss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollkvnvadi (wue);05d d7s along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that tn 7v5dd vik (aue;d)0whe 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 assume nx: o;/,lvvn q$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unitle7s07zna l -formly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration o0-lt aesl7nz 7f 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