A bicycle odometer (which mea(4hf*0 r m0w9ig/1d+wxf o)kun zxywc sures distance traveled) is attached near the wheel hub and is designed for 27-inch hkcr+n*w910 m)0zx 4o/ wd y(gwxffi uwheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates n02 5y;jdlaunat constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? yja 2un5;nl0dIf the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the an9t0jv1 zlcphk+ 4cec; gv)xh+gular velocity $\omega$ Explain.

Can a small force ever exert a great,c ,wn+en7mym p y,cb1er 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 ym,dn7x xs9s ae7b2 7ftour hands behind your head than when yourdm stxe2s7,a7x 9f 7nb arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear i 6;torjm9qm+ wheel and three at the pedal cranks. In which gear is it hart+jmo; r6m q9ider to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend ox+e +4bpgo 8p;5 tzksfn being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the bg+t5kp p84e+s fzxo;basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lon tm:h i..a(eoig, narrow beam?

If the net force on a system is zero, is the net torque also zero? If thebcq0id-cy p ky/25d2h net torque on a system -2bky c 5 i2p/hycqd0dis zero, is the net force zero?

Two inclines have the same height but make different angles wiwyqqo r.8 q(-m(k vh6sth the horizontal. The same steel ball is rolled down each incl ohqr(qq.8kw-(m6vys ine. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolkygsht5z c57* ol*i 6rling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? W5t6zyih go7l** ck5sr hich has the greater total kinetic energy at the bottom?

A sphere and a cylinder havetg/4d p5p:c. tut )z0i*tl urgmy9g+s 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 t g0tu r.u*igpy: sm/) +9cdtzt4g5tplhe bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most 6q 8)9udkppk kbs,:ei moving or rotating objects eventualls,:9ip8k 6 bup kdekq)y slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how9b4m7 c 5 + xa.f4 kzh1qw-nj,uqy zuaf6iwx;h would this affect4;cm 74,.1ua qz baiw- whyk5qjuh69zf + xfxn the length of the day?

Can the diver of Fig. 8–29 do a somersault without h3vrd i7f5;s,mups7tzaving any initial rotation when she leaves the bopis,md zf5rsu v3t77; ard?

The moment of inertia of a rotatinh mj.l* 5zn4cgky-(3dud r 4lng solid disk about an axis through its center of mdgz- n.rkm 4d 54u l3lyjh(nc*ass 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 3ab8ca// mynvrcwg; .on a rotating stool holding a 2-kg mass in each outstretched hanyacr3w.;cm nb/v 8 a/gd. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look idenpn)jidtn sn+ i5-)* zq7e;a k ;vqmzn-tical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whqv s*iknt5;-+d)zq;eazn 7pi n-j)mn ich.

The angular velocity of a wheel rotating on a horizontal axle points west. Ih6rhlut -gsbno: ,5 ng.g1ii8c :0k ovn what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linea g6.,ng5hi1k: iulohv0sbrn o8gt-: cr 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 freely rotating tukxb:rq bc 1/p:rntable. What happens if you walk toward therqc 1:bb:xp/ k center?

A shortstop may leap into the air to catch a bavu29 cbzhb5cfwip. 1u jryw( l.:f9 5bll and throw it quickly. As he throws the ball, the uppe l59y bu :bf 2j5rhw1vwzu9fib (cp.c.r 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 angularx/v nb1y0two1 uc7 3pr momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or mo/rp01y3 u vnxc71twb ore ways the second propeller can operate to keep the helicopter stable.

  Express the following angle4:f2ckp5 pso 49mj*l:op joyq s 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 coincidence. Calculate, using fzf7 la ikpif6,;8e2wb;) wpothe information inside the Front Cover, thea;iflpz;bio)e,p7k 8 w62 fwf angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divehw )f, nvf(9anrges at an )whv(naf ,fn9 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 atvtgt 5(-,gjfhdz4 7vo a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelgt(vzo4 vgjd,th5 f7-eration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level,9 v mli/0ocvj f(tz+ 88soj0ippdu o42tdi*o floor 3.5 m to another child. If the ball makes 15.0 t8 diom2j0p9/ool,t0z 8u+d voi4fj c*v (ipsrevolutions, what is its diameter?    m

  A bicycle with tires 6b8z 8)faejp *u8 cm in diameter travels 8.0 km. How many revolutions do the wheeubjz)e*f ap8 8ls make?    $rev$

  (a) A grinding wheel 0.35 mloix(;r n g;b i+mizmqb 81(gu5ut o;+fe/z3 g in diameter rotates at 2500 rpm. Calculate its angular veloi8r(;i 5 mxib qum+ goe ;(l+ot g3gn1;u/fzzbcity 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 s (Fig. 8–8y n i4. l5 yplr)xj:6b;xsrk /g3mbfo38). (a) What is the linear speed of a child seate.l6xypbli/) kogxn8;jry m5b34srf : d 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 thvza1 n;;jy g z72a9nlse Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofm): esm8sqp+v30 hpi r 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 ir5l9yk+/lqgj- qjp s *f a particle 7.0 cm from the axis of rotation is t r+/j9 sgk5j- lyplqq*o experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accele(kyq+dccxzt (h(x02 i+ 1fygb rates uniformly about its center from 130 rpm to 280 rpm in 4.0f2bdyzct( xy qhg+( i+(1x0kc 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 qka::yx-osy .$R_1$ 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, b14nwn m f4k/op lk9m;w c+0ugastronauts 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 kcubg4w+lnfo4 mn 1; 0k wp/m9rotation 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 2*xsi;r3 4fap c20 s. Through how m i3 *sx4a;cpfrany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200*zf-6b c2 urto 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 jeapi7twn)gr+eigs uf(+ lw s/d.j*/ b:ts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speef +grd./ijupn+g) s*w/(wb7t a l:s ied.
(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 kf .+vys o1.srwf7ohi-i,t;v rpm to 360 rpm in 6.5 s. How far will htfwis1,rsyvv.ifo. o - 7;+ka point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1 f;sb2e k1g8t0ass f3o3wut9r 500 revolutions before it sfwe gf 80 3;tu3ksobar1t92scomes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its jz/vnml.2t 1dr. gj4d speed uniformly from 95km/h to 45km/h The tires have jj2 d.g 1ldrtnv4m/z.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 reduces its speed u.byj qmv 1t62bniformly from 95km/h to 45km yvb12b6j m.tq/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cl7 hsvu,,jou3iz5 i)m vimbing a hill. The pedals rotate in a circle of radij5)vo 7iv3umi zuh ,,sus 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 is7yn;m4-:b ;q uwxryb t the magnitude of the torque if yt47 ;bq-r mwbn;uy:xthe force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in ruw0:6 valx (nqrbhx/ p7 3y6aFig. 8–39. Assume that a friction torq3pnq76v(6bha rwaxxlry u :/0ue 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: 2apbsgp8a:y070vlq, qz tsm rod which pivots as shown in Fig 0yabv 0g salm2pq,p:q78: szt. 8–40. Initially 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 req uwe8eogas k90 ln2d62uire tightening to a torque of 382es eg 0 n6oaklu98d2w $m \cdot N$ 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 *p r:cfg,h-uwradius 0.648 m when the axis of rotation is through its center. fpg,c-r*uh:w   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel drp 5hu wwh54;f5s 8jhar :kklf92la :66.7 cm in diameter. The rim and t5 a a wd8k5u k :2ls:l9h5rffpwhj4rh;ire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the-kcqy4b ium:o-.u m3l2ihu7x end of a thin, light rod is rotated in a horizontal circle of lxmmuu:7k-q3 u- by.h24ic io radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating9i-1h5bj(b7lvf*qs,0kc irmh kcg x ,bys,n9 at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is sgbrc1hkh,0k,( bil cy,smfn*5qi9 jv 7 9b-x1.5 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 arrxt x63),1izq0xl.zmhnj ;k euay of point objects shown in Fig. 8–43 about )0qnu j3zx61lekhx;mz itx , .
(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 oxyg:t8gfw jcp (/xen 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 sat iy24z5b, avcioyjo6/ellite spinning at the correct rate, engineers fire four tangen 6yozb ivayj,4 5o/ic2tial 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 is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinde4hg:sb mi6c rz +4f8pi+b/ qacr with a radius of 8.50 cm and a mass of 0.580 kg. Cac/+i zaq +4rc 4hmb8bspgf i:6lculate
(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 ba hc.+ nv2lov:p,d1azd d7v x1 m5j2vbtt, 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 0pbcm+l1 i2 rea 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 ofep2+1mr 0blc i 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 rotavo3a73ci7u -i+stm *iq 9h ih4um96(znwqdae ting at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torqu ie(h7htzid3 q7+ac6nmua 3iiu4*-sv9 o9qmw e 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 accelerates a 3.6-kg ballv-9kpjl 8kg; n at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, wb-rlvm3m9: :khneep 8yr,)v shich rotates about the elbow joint under the action of the trimnhp,k9-:m83s:vrl eeb y vr)ceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can b3+n7)bwbm xs:fo-cdqu d7q ,le considered a long thin rod, as shown in Fig. 8–46. w:7ul7 dm sncdo+q xqb f-b3),
(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 of18bhvwuwxf .a,*6zgly 6zuz8 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 within four fuxom3; tur9a x. 4xcxc;ll turns (revolutions.xxau3tcx9 ;; mrxoc4) 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 has a moment ozpr v4is, n*n/f 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 torqum6ya4zzne2 +w04-wl0 49nznqkfh n r3lba6jfe 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 rad g z06la,tf0yaius 9.0 cm rolls without slipping down a lane at a,t06laf zg0 y3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of to 7 e; o(v0cl/tq:nqhche Earth with respect to the Sun as the sum of twoc/o0vh7qcte; lon:q( 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 radiusv a4/9cy*7kropev ,jh of 7.50 m. How much net work is required to accelerate it frjvh 47p*vo9,r/ky ace om 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 starts from rest ajvovu- .5vft 5( hezub2i :0qvnd rolls without slipping u:bv(v -0eov5u vf5qhi. jtz2down 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*on*mw r.c6v m vertically 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.cnm o*wrm*v6 hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end6,p8zg q ekl1t of a thin string in a circle of radius 1.10 m at an angular speed of 1lzg,q k6 1et8p0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uni ( w )yj9nvj/i 94urdrx1b2qorform cylindrical grinding wheel of radius 18 cm rj2rury (9j )b / ovn49qw1ixdwhen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating atxhy,8yw9 kl x: a rate of 1.3rev/s If he raises his arms to a horiz:yx, w8hl9y xkontal 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. -i7 ym.hnfhi3 u7o7f anpjr:/8–29) can reduce her moment of inertia by a factor of about 3.5 whemp.hhi3:in/u 7a -o7nfjyrf7 n changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1tirc io:3/rymxy16y c z3q t(gg-h6/ m.0 rev every 2.ihmto1363-6cqymtgzci(x/yr :r /yg0 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 rotating aroun jq 2b-0k4zaakd 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-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, ontb4 q0a2 k-kjazo 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 3t ltxf*hbv: x zxn0a5*fn //o4.5 $rev/s$ 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 n +mpuz-w)0kfmomentum 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 in;yx8c xqr f;8o;l4f6c2:px lchbe yv2ertia I is dropped onto an identical disk rotatxry6q8;l8xcv2fcxlc 2o4;bh:ey;p fing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnstq8fg p 5hi3oeb (a4d q2*:iud at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at thet 2p/ jw-j,yyk center of a rotating merry-go-round platform of radiusjw kj-/y2y t,p 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 rota tb:zn 0 *a3rtv+/ziwcting freely with an angular velocity of 0.8 3+:antzz0wctbri / v* $rad/s$ 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 mh)it sd 3 :jex3.(d4 dlz2nzwpass in the process, 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 n(3 e)ljz4z2:p3wsdt .hidxd 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 wik4mesg3qr7 v7 nds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass n kc t wsf :ck4sr7*r,tg3-;-pk(c eqz$ 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 rota 1(ny1wbbcj.ihz8r h- te freely without fricti1h1 (jbiwy b.c -h8nzron. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person standsywe-,ag04yt zx ; nx0v at the edge of a 6.5-m diameter merry-go-round turntable vx-yy0ztw;xge 4n0 ,a that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on thw2xaqqiune)9 )gqr )0 e 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 fqxwgq0n9 i2e)) r )quaar does the spool’s center of mass move?

  The Moon orbits the E*z oo(m.we,ntarth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular mom, ew*no oz(mt.entum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from r old/2u )dfgyg*l1u4a est at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone fujce-, s8js8oe3;m i in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torqu3 ojeu fj8-8si,;me sce delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eac):+ ex(q wleiegf9,op6umb lb q5a 7)ch of mass 0.050 kg and diameter 0.075 m, joined b+bq7 :q(la6b i )ewf 9l5c,egoupe)xmy 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-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 ang /k69q xg-ksux8s+zid ular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with :51pzgez+,t9lb2efunfh *dumass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If t1gdfb2ule 5efzp:n9, * zu h+it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored i owh+2of.r9e6td4ffbxs+ q tjr x6i.5n 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 240 kg, and should be able to travel 350 km without needing a flywheel “ot hsbt 4+.9 fr6d5eo.2f+ iwxrq6jxfspinup.”
(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 illustrar9fuqfrt gk* tv-*5) ap6za8z tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling *mju2:d e r:nxon 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 :zioi:+iui2/8 scwq tand length 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 rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]/ i2 +iioucwsz8::qti

A wheel of mass M has radius R. It is standing vertically on the floor, and-4y* qldb48oz+ p wqus 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). The ste 84ulsp-oq dqzb *y+4wp has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn witarsm puk*wc pfbwep)(z(a99s 0 7 jo r/1ib1r*h a radwa1/ppbzi c (r au 9b9)7wsofrem10r k**s j(pius 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 sling)3 +l9m lhqypw of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating hy)pw9q 3ml+l 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 thin rods, makr:/b rok1ck( )1uha9 finvi h*ing reasonable estikio(i1 rhh*ukvra) /b9f1 :cn mates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindr l1p/a- sq, qt37jhrdrical plates, of 1sqq rh,d7-/alj3r tp mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 hapbu(t8 fp1o:gm:p nr- pi4)z*gxg+ rolls along the looped rough track of Fig. 8–58. What is the minimum 8(: -un g 4ftg1gpb pr mz:xoiapp)+*hvalue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but v2cfbrb yi:a5 5s(ex(do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniuhz3qj -yr2jqk8 z/;hformly from 90km/h to 60km/h The tires have a diametery2; qz3 uh-8 rkzh/jqj of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s