A bicycle odometer (which measures diswqc p(ur:h2(ftance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheelsf2u (c(h wpq:r?

Suppose a disk rotates at constant angular velocity. D4-n+z8j pn1 xk-dcwtd n8 lh0yoes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accelerationn 0 th8nklncdzyx -dp+j4w8 1-? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described bd0 +bk2blcu :*weh. ewy a single value of the angular velocity $\omega$ Explain.

Can a small force eve 9-jpe(vrmj ebyb9+t2 r exert a greater torque 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 with 9qpo l fyril twoorph02ru-n++ xzu3, /0fh,9your hands behind your head than when your arms are stretched out in froo ,uw0+ 20rfhr+lu9pip lq9, /t3yofz-xnohrnt of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheexs20 g*qbb h;ca; zgoms,0eb ;l 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 c;b ;x2 zbem,sb0g*ohagsq; 0which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on2zv6pydr +u x9 ee6g8g being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). Onp96z2g rye 6gd+vx8eu the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?i(isvy;wm-2 a

If the net force on a system is zero,ic f+x;5mf,y n is the net torque also zero? If the net torque omfyn icf+x,5 ;n a system is zero, is the net force zero?

Two inclines have the same height but make difrtwk dlfgx*)s,98 . fdferent angles with the horizontal. *f xgdf9,s8wt.)rkd lThe same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollir-( v afyp.9qtj; zi2h kk4(pdng (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bo; ar fykv-(qpt. 2 di(pk9zjh4ttom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylindpba;x pr0f1 z9er have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the g1;a0z ppxbf r9reater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. cd vumsin9uk a20(*;tYet most moving or rotating objects eventually slow down and c a;d nmk*vt0u2u(i9sstop. Explain.

If there were a great migration of people towash ls u(i3i*jpm):a. jrd the Earth’s equator, how would this affect the length of the dayij l*ju.3is: mh s(ap)?

Can the diver of Fig. 8–29 do a somersault without having any ini4 vusrgj+g8: mtial rotation when she leaves the board? 8rmu4svj +gg:

The moment of inertia of a rotating solid disk about an (rqz p)fz*m7 faxis through its center of massfmp )f7* qz(zr 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 rotatdihbwg2 l5c+(ing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, hg5d+bl wi(2c decrease, or stay the same? Explain.

Two spheres look identical and have the same mashxb-ubp- k52sif)z wgp eq0 .(s. However, one is hollow and the other is sop)2fp-b5-g .e qwhz ukbs i(0xlid. Describe an experiment to determine which is which.

The angular velocity of a whhpoq0i* snc5( eel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wsic5*0 (np qhoheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. l8h*:t.d3 fp; npfr*yxv k5t hWhat happens if you n htxvh :;p rkf5.3lf*8pdty *walk toward the center?

A shortstop may leap into the air to catch a ball and throw it x 8k(sb-etsxs2b*m,s quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that 2tsms bbx , ks8e*s-x(his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momg;1hb fcf()(bo1h i +bbq j*ko.9lvrdentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller canjflq+1hrvbbg ibbkof9hc1o;(. (*d) operate to keep the helicopter stable.

  Express the following angles in radiync1swar;,;i; f qu1b7 fw2 ratc9c g1ans: (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 anrawh*ay ,r5 i*o)g uf6 amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters fyoig6rw 5a,h*u)r* a(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. T ds2y+ll-7 w07 2 +eaq2cbzmysfs/t pdhe beam diverges at ddly la/+tsy7 fesws222+- 70 qpmcbz 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. Whentzp5ym 4c0gtap va4- tg0tc)w un,/.c the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladesn,ga-g.p0at t5tz4ccp 0 tuw/v) cm4y slow down?    $rad/s^2$

  A child rolls a ball on a level*:x*is ld mk-;xp-uyz( pc0eo floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is(i-dy*lc*:s0zpo meu p - xk;x its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manyf +x)tyar++ x i)kqyf7 revolutions do the wheels maxfk riy)y+) + 7ftq+axke?    $rev$

  (a) A grinding wheel 0.35 m in da(yd 8(vnb(y diameter rotates at 2500 rpm. Calculate its angulavdb((8(d yanyr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-roun q(elt4t dlm40d makes one complete revolution in 4.0 s (Fig. 8–38). (a) Whq t0d44 lmtel(at 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) in its orbit around the Sunhp n hoc.3 8y0;iefla;fqnd //    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 0 -7azq ac mzpe0 qk)x5;eonxk/d65kupoint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the dxf;g1( j69*ayr(z;o bi lnb kz- -sphaxis of rotation is to experience an acceleration of 100,0rxon z*syhb;6g-(abj;p(- 19zlk id f00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fro ry n8*eqegw1-ly x-n7k8ipz.m 130 rpm to 280 rpm in 4.0 s. Dk8e le.-* -q7rxzwyn1ng 8pyi etermine
(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 radiusk7* 8u/ogpj4wtp 2ps d $R_1$ 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, as/n.ryube) d0h3xc)v htronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute duri.vdu b 3e0nh x)rcyh/)ng 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 i (oamy8e3ba ci0 ,fs-vklv93un 220 s. Through how many revolutions did it turn in this o s vekfyic3m(3-al ,v a80ub9time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200160gi8ym/qj c6kggniu5 m (hb 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 fuy2d/ag0g w-tg ,3c peor the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its u3-agt,g d02 ygpcw/efinal 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 +pcx3*e4eee4uw5 s n jrpm in 6.5 s. How far will a point on the edge of the wheel have travelnxpc44e*eus+ew5 je3ed in this time?    m

  A cooling fan is turned off when it is running at 850rev ;z o77ix. mseqz3:va4ccv1o g/min It turns 1500 revolo azo7 cx 4ie1v:v.73;mgz csqutions 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 rukhpd,pa7ven l*1k;67byw bi :7 zqt:evolutions as the car reduces its speed uniformly from 95km/h to 45 puqa te*;n76p7 bvb1w7kzli :yd,h:k km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduc9jfb nx8e.9i5po6ggoqjd q /: es its speed uniformly from 95km/h to 4o8pjn i ed69g:q9bjx /5q. gof5km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedalxe+em)t(r z+ g when climbing a hill. The pedals rotate in a circle of radius 17 x( +ezmer)gt +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 f:lj k927u)kgk;szwk door 74 cm wide. What is the magnitude of the torque if the force is ek7)u2kglf9 kw:kj s; zxerted
(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 torquea0yg. qlclbd ;v5q 29 8utt-iy about the axle of the wheel shown in Fig. 8–39. Assume that a friction to-ttq; q0culb2a y8 9.idgvyl5rque 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 rooz(hw g1ogyvf8,s4x +/s wg,q0x i* aoiwf3o:d which pivots as shown in Fig. 8–40. Initiaix ga+ :wfw o,qsw13zs*yi(,hgov4 fx g8/oo 0lly the rod is held 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 require tightening to a t 7.gq1ddm8p5zhzqr 1rc. 6stlq0vrg7 orque of 38 v65c17lddprztrh q0 gz8g7m s .q1 rq.$m \cdot N$ 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 radi85.vxopd br:iev fkkn 25i7lo20hy u8us 0.648 m when the axis of rotation is through its center:hfv2uk87r5. o bn 8yedliip20 oxv5 k.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy qka(j2 k1ks:hcle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (h 2k ak(1qjk:swhy?).    $kg \cdot m^2$

  A small 650-gram ball on t;zc xg49dey*phe end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. g 9pzd*4;yexcCalculate
(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 angulgq3 gs2b ljko+wnjr3 )p/i2 eg b2uh47ar speed (Fig. 8–42). The friction force between her hands and the qeg+7 p j 4wslb2ug/3bg ni3hj2)or2kclay 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 inerttav q/t4c0km )r.ym vk,pf1v8n4ap5nia of the array of point objects shown in Fig. 8–43 r45q1fv apynvc40vn kam pkmt.t/8 ,)about
(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 twohdzq3s -g)y5i 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 cylindrical satellite spi pijz6w h uoo2ilvxbw**3/(; unning 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 satellite b*u 3upx2( l v/*ow;zoijh6iwis 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 cmn*)tq6tksp ( w and a mass of 0.580( t6*wpts)qkn kg. 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 from rest to 3lui 0lfyd a5.2 $rev/s$ 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 ol0a qfp9r: ,pj0glud6 ftq*;0v dqmi9n a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2d,g m0qadj 69qfvlifpl;0 :tr0p*u 9q.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 v0cu0)-vm* yoaqss2a is eventually brought uniformly to rest v2ou s*0 c0ya)amsv q-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. 8–45 accel 8mzo. 2/ynnhohm6hj4erates 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 soieq703 2 ki8lqsm9z pylely by the action of the forearm, which rotates about the elbow joint under the action of the triceps musci 2yskmqz9e8pl i03q7le, 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 considern 29f mg1/xhmqnm4 wr6ed a long thin rod, as shown in Fig. 8–46. 2fmh6 w/xnm14nrqgm9
(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 t coba9ri t tu48c*q*s-wo 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 (eyq n(*pn7vfthe hammer from rest within four full turns (revolutions) and releases it at a speed ofp* e qv(nn7yf( 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 momen6y54cdspgd5tz6 uw 8vt 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 ofaw8oo 5s-p-0tcrgzoyqz96uo2 b.d1c 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 eatxa.)vwjr9 k;,ks -and radius 9.0 cm rolls without slipping down a lane ar ;va9-),tke x .swakjt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth w.cpo-vim ;q6,/dkwl zith respect to the Sun as the sum of two tero,-/d6m pil.qz ;wvk cms,
(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 tttnc4 fq *)k;a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? ;fkct q*tt4) nAssume it is a solid cylinder.    J

  A sphere of radius 20.0 drs7ca1sas9 4 cm and mass 1.80 kg starts from rest and rolls without slipping 1aadsrcs 97 s4down 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 on6j ca 5jvho0z9b))x8j;bhxur,1rl u n its tip. It starts to fall and its lower end does not slip. What will be the zrj;6o,xahj9ulh uc8v5bnbx) j 1 0r)speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the endy4z:z4+aq nqah 8 4l5ctq.nfm of a thin string in a circle of radius 1.10 4 lhn4 4fqqaq:.z8znat5c+m ym at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifora u65vt5z/3e4nop7h/ ogm xz qm cylindrical grinding wheel of radius 18 cm when ro5znq6m /hzxtop u3gv ao75 4e/tating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, .6sbohp( mld b.l2 puo/ ix4y)on a platform that is rotating at a rate of 1.3rev/s If he raises his arm o(so64xm/iyp .bl .blu hp2)ds 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 redu zfoo. s/6m3n6acopcg7 2+ug gce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what zcfs3 2/agog7 6npc om+. 6ougis her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ratj)s u/ b1exjer.c,7k5hfhq-ve from an initial rate of 1.0 rev every 2.0 s to a final rate hev.er jj5b, -1fkh)7squ x/cof 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)6/pv pr4 hi4x2 cisyn 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-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cyh 4x4iinv6 p2r cp)s/m, 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 spinni w1utg.77+ka lmhj0 tds8j4znng 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 momen)ha3 *q(f;xut.u suagtum 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 igtxlq 9ujgw-mm6 2 4v7dentical disk rotatin q467 j92mml-ggwt vuxg at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 h+orvv9 bvspdj7x7 2z26w-n n $rev/s$ 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 kgwv(qu/: lsv e+rt*4izku:*b uuhe:p 5 stands at the center of a rotating merry-go-round platform ohs v:e4rul uvq :( *t*iu /up+bkewz:5f radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity o9o r 8-bpa:+h r7cgmdd*5nfojf 0.8 8o jd9r5h*+7 brn cfp:gdmao -$rad/s$ 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 abq. /ar93qk-v8wjkwbmbe-i t 5out half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries jbai.bmqt 8k9-r/qek -w5vw 3 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 evl)i ul.o )2qixcess 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 x vji,wcu-;z9p6+ 010trwa w6uvhdif$ 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 res nscgp a+999 wuuz.2j(jyc.tgt, that can rotate freely without friction. The moment of inj+awtzn j9 .9y(9 gp g2ucsuc.ertia 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/nu pn83orl/d er76l a stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170/8plero/ln6au7 n d3 r0 $kg \cdot m^2$ 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 kg+r3o4wh5jde rq89h with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance Lj 5+ dr94q ke8hogh3wr, 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 side always faces the Earth. Decobk3-11jjlw j5brp n:* -kedtermine the ratio of the Moon’s spin angular momentum (about its own :k1j -3l*crwejn do-5 1bkpbjaxis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rern9g 3i ox**i6de;n5*jqx 3hppz5k yjst 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 ;l5uw f8 l vp59m0lvnh0.20 m acquires a rotational rate of h5mll lupw8 ;v0f9nv 5 from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical h9yt v*+yhzo+disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long9+yho y+hzvt * 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 rey.o4g f mra1ff i8+iulu6 +a1f )ncil/ar 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 times as great, were rot)l,y: v to 9xum4ki+ilating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 kmxy:o ,lmtvi+ i9 k)l4u, 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 the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for itk)1bn9ald q /9:ibknz;vi6 ka to use energy stored in a heavy rotating flywheel. Suppkvi1a) bkb; i:9zqk/nl9ad6nose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an4,hu.3cyyhnfz -l(x7d pkrc. $H_2O$ 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 .geu*m:ufx vx-) :)u*mfx p fsrolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictioyw 5)/runvp*x4qay n* n) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontallrnnqaw*xp*)yuy5/4v y 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. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vervf:z22sr*1 xrp pa snyk5 ou-:tically on the floor, and we want to exert a horizontal force F at its axla: 2ys5 r1rzupvfsknxop*:2 -e so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn pe1tb u:)4r0. -zvqiwrgq 3ilwith a radiz1iq)4l gtiev0ru.r: p3b wq- us The 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-kg rock into a .a)r9skwg u*kyi5b)i sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleratingsk)yg w*5ui ib)a 9kr. 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 as t ,.vvloo,p md2/90 vnih* osyqhin rods, making reasonable estimates for the dimensions. Then calculate thpv. 0yv9/nv2,o liqsm hdo,*oe ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch8m6yg)f ev/ -yure* xq assembly which consists of two cylindrical plates, of em6qyxey )-8gu* v/frmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 vr)qfnbt.946 llz.(q j,uq1 vd zim0pthe looped rough track of Fig. 8–58. What is the minimum value of 1p rm.(ul vbnz qd4v, .)zj6ft0qlqi9the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do no;b*usuvqexo f ke-v9)sc, ) ,ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 sgrjk,ft s;:+revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. kt;r+f s s,jg:(a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s