A bicycle odometer (ix,ixj5tnfru. xk* )7which measures distance traveled) is attached near the wheel hub and is designed for 27n )i xt.k,5ixr*fj 7xu-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant/vqu *- mq9 z10bcex ;pbt-vsg angular velocity. Does a point on the rim have radial and/or tangential acceleration? If 9u g/sxm -; 1e zp-vtqqbvc*0bthe disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angularaw 0r o*038 sw6tkzjmq velocity $\omega$ Explain.

Can a small force ever exert a greater torque thal:y0nyq8he6d 3pxf/ e 9 4/m; ytsfohzn 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 8p:5nvk i8ku oxe,iy8sit-up with your hands behind your head than when your arms are str5e8xi8 kyk v,:p8uinoetched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sev+ kx:8tkjs (hoen 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 gj hkk:+(ts ox8ear 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 legs w;1h 4g zvrhf bmu7v0mk51r9pqith flesh and muscle concentrated high, close to the body z p ;v1kmvurm9 hgb qr5f7h014(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 lwqzj7fl e(zyt t+y2l )an)z:dq,/hc0 mh :1xlong, narrow beam?

If the net force on a system is zero, ( r* 5wqjir8zxis the net torque also zero? If the net torque on a sysr*z 8(q 5xrwijtem is zero, is the net force zero?

Two inclines have the same height but make different angles with theiaph d 0 .bv-g6v)*ebac qk7f1 horizontal. The same steel ball is rolled downgaf .qb- v d6ieh07p)va c1kb* each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down auf/ na3zgyrw(r,ga.n wm 5: e5n incline. One sphere has twice the radius 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 ba/: 5,a.fmurzgn3yw (ewgn5r ottom?

A sphere and a cylinder have the same radius and the same mass. They s98o 9o2hrmh+f;qxpswtart 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 botrwp9 mh ofo28qhs9+;xtom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet m/che+9h5fsawj c k fbl4n/k 35ost moving or rotating k3 fwh/nfh+ l5 a/5jcbce49ksobjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator,-q 718oj cg p;ea/hipg*e/bpnazr +6u how would this affect the le qgor*b; /u ci1g8hep7eaap6+zj/pn -ngth of the day?

Can the diver of Fig. 8–29 do a somec/w+m p:htr4*n fcoky:c. /scrsault without having any initial rotation when she leak: srw.c*/tof/cm:yc+ pnch4ves the board?

The moment of inertia of a rotating solid disk about an9qwu .jmvo )8jf8:fk e axis through its center of mass 8.9kvej: u)o mj8fwqfis $\frac{1}{2}WR^2$ (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 sitti6 oge1 0vwh1sqng on a rotating stool holding a 2-kg mass in each outstretched hand. If yeqw011h6 gosv ou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look ident7 wcjm1 q:;aztical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to detecm;:a q1jwtz7 rmine which is which.

The angular velocity of a wheel rotating on0.hj y+w5a(fz tc)zk,uf*wt1d ;u trw a horizontal axle points west. In what direction is the linear velorh510j,zk *w d tc+wwzt ff.tu ;)uy(acity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge o57tsqmyrr6)u f a large freely rotating turntable. What happens if you walk toward the center?rtry )6su 7qm5

A shortstop may leap into the air to catch a ball andf/y6 otm5bro/ throw it quickly. As he throfo r6o5tym/ b/ws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation we3yo, 6upw4v*sw 6bd of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one ov6d p4,e*6 ybw3wsuwor more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30c,zpre+a /)g3ov6ss9ua :m 3hx sju l6 $^{\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 oph6s 34m sqxlb+u0wi: f an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (slh 0 qsw6pibxmu4+: 3in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from E2kr6 h;c2eq zzg4.n 59n np)zwgzmd :xarth. The beam diverges at ad4zgnkrxznq)z9wgn; 2 :cm2z 5p. 6eh n 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 mo4 6uuzxwt60l snkzf2 /tor is turned off during operation, the blades slow to resk6xuu z w26s0ft/lz n4t in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a levjk5 ; )nlv/cbcf8y kmws-3t9zto )c-vel floor 3.5 m to another child. If the ball makes 15.0 rvc 938 yc fb5vljm-)t- zn/ )k;swtckoevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 xy w76:bb7mvo :y l3,v.dj xim gepm85km. How many revolutions do the whedjgwv:myb omp:v8lxi 3b .7 6mx ye,57els make?    $rev$

  (a) A grinding wheel 0.35 m in di. p td:)kqwohoct91y/ameter rotates at 2500 rpm. Calculate its angular velocity in )k/1qwpd9tc .oyh:to $rad/s$ $\omega$ =    $rad/sec$
(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-rocr :c4d r3f9ogund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child d 4g3c rfcor9: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),6a;3c:rsgtmt+fh pi4m r :cc in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed -eyglue+zd2 /of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a parti(wn; y.z+yg4hhy-* o3 ru :h qwfujla,cle 7.0 cm from the axis of rotation is to experience g ,wyoyn4(l* w: 3fuuy;+h.aqh z-rhjan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abougm fnxl2u-n)eam. f9 )t its center from 130 rpm to 280 rpm in 4.0 )g-enmf.)anu92 mxlf 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 uw/kh4lrsh9f :+ ifz2$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themsel z0j0v5uple3)-xfrlw ves into a zlex0 uwvp3 r j0lf-5)slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 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. Ttfm, bf t)3e)dkcnhn)x p l*31hrough how many revolutionsptlf* m, n)cenhf))xtb3d k 13 did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm :p-z9-fs)+1: r n n fou1azj8qm wxgmito 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 flying highspeed jets in a whirling “huzpnlck.way-ar +us 4se vp.190 8 kp;pman centrifuge,” which takes 1.0 min to 4asew 8kv-cr+p 9 zy.0pp;up1n.aksl 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 to 360 rp /n.lgnk.g)n do3.t0 hwweu 3tm in 6.5 s. How far will a point on the edge of the wheel have traveled in this ti.3 n3 ntn klh0wtugow/eg.) .dme?    m

  A cooling fan is turned off when it is running at 850rev/mivbbhd:: ntg.vy4rpe q*1dd7q lw3e52 n It turns 1500 revol27 tdwyp d.re:gv1q:b4hl*3q5dvb ne utions 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 re btez, q2( u,ia.awm9s k;didj15y7 a-rtxx*kvolutions as the car reduces its speed uniformly from 95km/h to 45km/h The k;x7.at12 siba q-r,yz aewdxm5i k ju9(,*tdtires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 re02muzelnr4t.y*3(v t az zqo3volutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter on 2tu4 o3(vy .ztamzrle *3q0zf 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 bqjjwxsb4y,ym/z 0oo,m n -j/; ike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiu4ms0,; jyy-xj b z/oj,wonmq/ s 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 t7b*n q /cz9wwjhe magnitude of the torque if the forceqc*nzj /ww79b 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 tax(s ,jndehfe3 )7: q/fiuu+ yhe axle of the wheel shown in Fig. 8–39. Assume that a friction torque oiq (ye nfhxa,f/d:3 s+uje u)7f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to th v5*hmlmf0 7ztiox6 ,f,3fefce ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is hei,f3cf zto emxvf7*f ,m0hl56 ld in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine r) sw7tmxvvo(32oys0require tightening to a torque of 38vostm0y 2rxs w(3o)7v $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.tohsaopb. : ;ss* hjckn2 zbae)b/80n9 /dwr-648 m when the axis of renk9s-o;tcpabs ah)srd8/b:z jbw . n0/2o*h otation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Tbu rkm.fn *wsyy/-e+) he rim and tire have a combined mass of 1.25 kg. The mass of *u nm -.yr)ybekw/ s+fthe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated kh/ e4a 0lmf9f. ein3din a horizontal circle of radius 1.2 m. Calculate9/effdi4 nk3 m.l0h ae
(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 rotating at constant angular spccy 5as i; :5cwy2jnp4 bzrp43eed (Fig. 8–42). The friction force b5snr2z4ajc ypbp y3 4;c:5wci etween 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 awtl 3)hrjn v3ae4pinjp*.ic jca4156tz65e wrray of point objects shown in Fig. 8–43 about )5ejt h 6pnji5tlaaw63.pv1r4z3 jcwn*e i 4c
(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 mass i g3u a:pz3iwr5s $5.3 \times10^{ -26}$ 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 spga 8u cux7i0pluaa(n)5 4h 1lvinning at the correct rate, engineers fire four tangential rockets1g)uxc7naal(auv lu hp5i 804 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 mass of 0.mq ;r.in ;ut..giysp 8580 kg ni. ygi;;..mst8p ruq. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it frojudhiz4 .qgerw v 6ke8br6* u:h ie6/7m 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-driven merrygns 7jc (ct:oyl/0z n0lb sb74+vso 0x-go-round and is able to accelerate it from rest to a frequensv g/(+sn 7n :os c0cyj0b74oztlbl0 xcy 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 a pfn3 /ggvx:qg60hoeo. gp 6w7t 10,300 rpm is shut off and is eventually brought unifge/3xhw g0qg.:no v pgo6p7f6 ormly to rest by a frictional 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. 8–45zxa:+a7u)m n iqnu1(c accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the acti7 f xv1u5fhvzyn c(l62on of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball1v7uyfnxlv (62cfzh5 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 consiuz u45 v4vy(ddq l02yrdered a long thin rod, as shown in Fig. 8–46. drquy4l24 zuvy( 0vd 5
(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 r1a4g4xc/unug 9j0 pyconsists 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 from rest o02*,cr ol yabf6aqu ,within four full turns (revolutions) and re la2,abr,coyf60 *u oqleases 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 momlxz s.4ogaw61 ent 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(i39aqn bgx(z torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without s; efvq0wg 8(td8hj;dzvlo. e*lipping down a lane ah8d*g .(tevoevj8;f lwd0 q;z t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect t/ qxti:.y)-l 0h6h.y9m,hbgjq fkf ato the Sun as the sum of txfgf)kl- 0 hij q.qty:yba.6hh ,/t9 mwo 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 mqmat g lzmfz-y/tyh 62n6u6:;ass of 1640 kg and a radius of 7.50 m. How much net work i;y:n6h m mg-zzf2 yl6/qa6u tts required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stacjaje 1 lnh4 +f+3f)uarts from rest and rolls without slipping down fha cf+j43+a j1u eln)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 starts to fq * qm,xlvq/g3 9wwqiyl2v0w1all 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 conservat qmq/lqgw1, 3q9ivw*v0 xl2wyion of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin dzw9fwb.(t :y05i zjkstring in a circle of radius 1.10 m at an an9:t zw b(k0dyz5ji.fwgular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gm3y pqu2d,a5 nm+2op3 iw8 z75vw izuurinding wheel of radius 18 cm when rotating 33mv uq,ui npi8w52wya ouz2 +7mz p5dat 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 his6t/qmk9 eve7m 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 rotatitek9qmv76/em on decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onek7 kh:tvq rct.pba i9wq/((3f shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straigqif9kq3tpw(7c: (bar.vth/kht 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 an initial rate qg28nb)) k6bx)mmi mwn f-g96 xjssi1of 1.0 rev every 2.0 s to a final rate ofb1 sx9jw6 si) )i-kbqx6gm)mmgnnf 28 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 rotak3si0(- g9m 1;qqo d aru 6d4asflg(rdan+qv 0ting 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 the+qsmd04;o 6rn01saa(uqaqr93 d- ivg kd(f gln throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3. s; .esg6 cj-6ah l8afhbv2xaf p9(yh*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 stqo.9xu- ;d; d opl2gmomentum 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 oci+0/qsuvq gy 3;ya* af inertia I is dropped onto an identical disk rotating at angular speasqa yg/+uq c3 v*0iy;ed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 jm eon, tci +r-(oi8vs;q7uf1 $rev/s$ 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 otduk4xzwj2trlal:z4 d5yp7+ f0dw )( f a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920trj4(:u 4z+5d7w)p2 ydf wd xlazkl0t $kg \cdot m^2$ 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 t+sheoq -rkam::3q ge.b ag6*velocity of 0.8m:6qer hog +ab:3-et *.gakqs $rad/s$ 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, losing about half its 7oa arzk86 zrt5htad eu;ek5i 3)at4 -a65xhe mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost massx3d)r4ehezat;a5taaza6kt r8 ohuk7 -6e5i 5 carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 12 .kqyj*gu* q5gvpx+u0 j/t4kv0 $km/h$ 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 u *)28zhndlvap+bh) a $ 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 p7qmcud0e l3 -i3brbt0 r87yxulatform, initially at rest, that can rotate freely without friction. Tedq ruu7i0b338 tybc7mr -lx0he 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 6.5-m diameter merry-go-r+grash5a(jg :kl qd;3+0nl afound turntable that is mounted ;g g(j a:r ++0fqkhasan l5d3lon 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 b:;zx,espxk)w v. /hjground with the end of the rope lying on the top edge of the spool..e kxs;:bvz/j)hp wx, A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same qhoa 8-2t hp6lside always faces the Earth. Determine the ratio oa -phl6t2qo 8hf the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rau*nc az9y9m6zte 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 zgu5x4cz3f2g2zk hf6 5yb0yf radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tor ygxfzk h2f55zcyz62b34 g0 fuque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid x*q wfy)tv ;.*gjme/eg fh0:gcylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0hyvfeg; .q: *m* ggxjtfe w)/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 o 34wpaige-(xsf the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 tinujjs3- k v68v/,tezf mes 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-kunt z8 f/ev, js3jv6 all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollut6t bct2m,nyitt(f*u ;ion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a unntic *um(6ytbft, 2t; iform 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 illustrates an af av2 ,;0km v4;xakaeh9chk 5l ;evg7$H_2O$ 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 surjc gfh..)1 ci2yyn d;g((ddh kface 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 xbpw9gcu 6:d: zrb12 ipivot 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 rod. Assume that the force of gravity acts at the center of mass of the rod, as shown p:1b:w gu2 i9rcbdx6z. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticallqh .o3+aoks g /z7izta /+f alv6c9:lvy on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step aafz.v o6csvl++ /qa: l3h7i9/gtz ko 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 withwyk*e zr ,x;pjsjc*7) a radius Tscj p); k*jyx zwr7,*ehe forces acting on the cyclist and 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-khc1uq)0jazb ft6j2e -e e /zt7g rock into a sling of length 1.5 m and begins 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 where does the t zze e2j1j eca-0 q/h67futbt)orque come from?    $m \cdot N$

  Model a figure skater’s boh0709,s m3:lofb-c h ibwbh wfdy as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensionso0c7 ib0,wb:hf93 hmbf-lwh s . 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 consists of two cylindrical plates, oq h;y0x)lokc :f mas:)q;x c oyh0kls $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 sl90 gtw1voj+ e 9pc9trough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must 0gl esc9twp+1 o 9t9jvdrop 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 doa9vrqh4: ax 2g(5 mxfe not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 ru.mk e8z imk-5ee1c 4ch+5, g3vkcqvuevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular ac-i5mgeukk1 u qveh kc3e+v54z.8cm ,cceleration 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