A bicycle odometer (which measures distance traveled) is attached near tha(uu9q70j au:* zh.wijtmi 2 fe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycljauqi.*uaju2i f z9 m70h(tw:e with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point onj: ydhe:w6mw ) the rim have radial and/or tangential acceleration? If the disk’s a y:):mdw jweh6ngular 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 descrin /s:bfpvc4ch4m0ve7 bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forceh1*q1 y: tdri+4q mhof? 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 your hands behind your tntvfg) z5k -j+f*n +ihead than when your arms are stv fi ++5-) ttnfgzn*jkretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal **w3wl pjc-ukcranks. In which gear is it harder 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?*-c3ju lww* kp Why?

Mammals that depend on being able to run fast have slender lower l-j; t3+ntgz; 3rgqnsbegs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationa-g3 g ;n srtb;+n3tqzjl dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrya7oqh90nm 4ps a long, narrow beam?

If the net force on a system is zero, is the net torque also zew h4s;nq( bjj5 +navi;; hw/dftpki/*ro? If the net torque on a system is zero, 5(t;+;najkj4wfn h hi i/v*sbp;qw/dis the net force zero?

Two inclines have the same (k hco+ia*vn ;height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater?ca(;hi+von k* Explain.

Two solid spheres simultaneously start rolling (from rest) 9- exx9of y,g zqxdf-*down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bx*z xo 9f-y9xeqf,gd -ottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder h+s on(x,fw s)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 greater total kinetic energy at the bottom? Which has the grea+n o w,hsf)x(ster rotational KE?

We claim that momentum and angular momentum are conserviq .wt7u-.qcu ed. Yet most moving or rotating objects eventually .- qti wu.ucq7slow down and stop. Explain.

If there were a great migration h-lss2-6avsnto 5 , god)n* xtof people toward the Earth’s equator, how would this affect the len s)o* 5ts-g,ons2anvd-6xtlhgth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initialbo, j12 5ofem(3d u jyqhd**kd rotation when she leaves the boajd1obeu,j ym*odh2q 5*df( k3 rd?

The moment of inertia of a rotating solid disk about an axis through0t2-b8j rska)l3mi gn its center of mass is) jtl-mribgnsa 8 k203 $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool- qsskf)n o39r holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, defs-q39n)rsk ocrease, or stay the same? Explain.

Two spheres look identical and have e;vv2r.o-qkzq(w1lf wt.v, e6jsdj9x6qb j9 the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whicrb2jvv-qd96tewx9sk f6 oz.1jqe;vq( l ,.w jh is which.

The angular velocity of a wheel rotating on a horizontalvr(j*.av hl6 j 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 wheel. Is the angular speed* (.jhval 6jrv increasing or decreasing?

Suppose you are standing on tyem6egf1 jq/5 rl *+yw3ni 4mthe edge of a large freely rotating turntable. What happens if q eefwt5+m6 rjy3 *1m /iyl4gnyou walk toward the center?

A shortstop may leap into the air to catch a ball and+0 ze;vx hlfn2 throw it quickly. As he throws 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; vl hz02f+nex (Fig. 8–36). Explain.

On the basis of the law of com95hi6bzj1mq w d ed2)nservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or m1m wd 6hmqde5) i2b9jzore ways the second propeller can operate to keep the helicopter stable.

  Express the followinjca-x t l+h:9npymx0 *jjg u:+g angles 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 t; m- -axwgr;uthe information inside the Frm-xt-; r;ug awont Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from 8e68w if ep1 xatfpcjt3(v:n1Earth. The beam diverges at aawv tf6nf:tp88(3j1 e 1pi xecn 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 oy1 uo;fox2 :qpelk6v ,f 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as th o:yq fo;xpk2eu1 6lv,e blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mak4+gvmfs psj6 b6..2 g8p+crev (ylllfes 15.0 revolutions, what is its diametec8fj2+s(.m 6 6lp b4 p+fsllvrgy.vger?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do u0zqii /1)tjyk/ ws ,sthe wheels 1 /swz),0/jkiyq ut simake?    $rev$

  (a) A grinding wheel 0.35 m in d:1:rw y f .z(dp y07isp*3vizfu)mtv uiameter rotates at 2500 rpm. Calculate its angular yr:m .wpi 3f:0z(dvuiup *1fv7syt)zvelocity 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 r)3h sbd) icl/*mh6./o u 4mej:cygwu bevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cenbh)u /wodu3c/m6bme igsyclj:h 4* ). ter?    $m/s$
(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 tm lssotsk(m1f39965bu j l/hhhe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp-x - -4x/pyuorce1nw yeed 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 centrifuge5 kpctsa-d 8d7i 5na2n rotate if a particle 7.0 cm from the axis of rotaps d5n2cktin5a8 d-a7tion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abo23 vg awhs0z c,h*:gjbut its center from 130 rpm to 280 rpm in zws*2 3h0 bg:jg,vach4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius 0stms5hv24 dy 8r y3mc$R_1$ 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 aboardjff(u(e w(*jryy zitq9d3 6 ((4obtw m 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 during a 12-min time interval. The spacecraft can be thougmojfy( * wfjiu y69((w(rztqb4(dt3eht 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. Throughcy j/(geeh m2u-yfuc49- wa1r how many revolutions did it tur-c/e(u4e92 - wmgj yyuc1fharn in this time?    $rev$

  An automobile engine slows dowlwa/5nc+ i0;tepjx /sz 0l mr*n from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying hiy:kkdr0 y2g 9xghspeed jets in a whirling “human centrifuge,” which takd0y :9y2rxkgkes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformlyes) 0yjyat*(ra4ciar **l j(4ezhuy : from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel havjy: y ez*ruay*t (4h4l0)(*icrjaae se traveled in this time?    m

  A cooling fan is turned off when it g*nqpcw: u:77jduz4 s is running at 850rev/min It turns 1500 revolutions before it comes to a stopg:c s nujz7q4pu*7:wd.
(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 speed unifyqz lt0.dh.4i5n9un c x+4 pzbormly from 95km/h to 45km/h The tires havb94t 5c4+py n.qx0 .nz uihldze 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 t06j 5h lz+ujtzhe car reduces its speed uniformly from 95km/h to j6j zl+uzh0t545km/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 eachl 4,e4:ffowe s pedal when climbing a hill. The pedals rotate in a circle of radius 17eoef4f wls, :4 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force 4y6td9twebv. n (dfh*of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exerte(byw hvn tf49d. 6*edtd
(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 i lp4 tgkknr8 0;gc4,qyn Fig. 8–39. Assume that a friction tonq,4t4yr kp;lcg 0gk8rque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachel:w y7htfpa py,*1:, dtw(lnz d to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque ly:,awyw7,tp: p dfln1t *(zh on this system.

  The bolts on the cylinde,c 8 5y:d qf9swisbm6xr head of an engine require tightening to a torque of 38 siby9q6 xs5wc:8mfd ,$m \cdot N$ 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 ine jp(*3g2q y 5,t2wmmw;cjxu bt*yf.e mrtia of a 10.8-kg sphere of radius 0.648 m when the axb;*p2g,3y5ey2cm.m*t qt(wfjx mj w uis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy cy lbwkd(lzys8.h c*fs))0b5 cle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Thedlz fl0)58 ycswbhb*sy ).c(k mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated w0ikv2c/f: c+hcq6/b ayk5 nrin a horizontal circle of radius 1.ny+5 k60q/kc2 wbi /fc:hrcva2 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 potulfm(t.g7,/ 9umu(v )c esnhyter’s wheel rotating at constant angular speeu7(l/y9m.fcu, t h vmseug)n(d (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array g9bbl,xe-dz .13x/bvge 9 iyk . /cm q9hs8kkwof point objects shown in Fig. 8–43 xyb3g 8 hg/emwkxd/9k, iv919-kqebb c..zsl 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 consis+cb 8qryzh c+b8:zz) e0etyt:ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satell4i863 hax yj7jlmvgr,:;ve n qite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of l3 ,;x4invj hy6aqge87j vrm: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)nf).1h dwy udf *l*kz with a radius of 8.50 cm and a mass of 0.580 kg.nud dlyf**k)fzhw1 ) . 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 jd,m g2yjf-4eaf .jz+ a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hf) 552d qhwnlilmmf -378yh qeand-driven merry-go-round and is able to accelerate it from rest to a frequency o 8i755ehf)-qd lhymfqwm 3ln2f 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpmeywpt.b1uu 2* is shut off and is eventually brought uniformly to rest by a frictional torque of 1.1t*uuepb .w2y2 $m \cdot N$ 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 ball atr a y.y ell-v9l2g:0a t(cse(w 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 thdylj/ i xpqvl6,o(5; fe action of the forearm, which rotates about the elbow joint under the action of the triceps m5 jqxld;lfyi,o (/v6p uscle, 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 ;)dd tzl5wq .1bopwh4 can be considered a long thin rod, as shown in Fig. 8–4614)hq.lz b wdt5wdo;p.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two m iw4.hmv 05rfqaq +h na1a/u4b+topb 5sh;1glasses, $m_1$ 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 accen0 wtfjps4,rkv; j7:z ue+) aolerates the hammer from rest within four full turns (revolutions) f vw,js 0pa+n4 o z:)kje7;rtuand 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 ofc2d ,vzgh;x g en37s*wkt*sr8b+* qsm 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 azt xme6vu,32dd:r02i f xxtr( 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 slippgm+,g5*y;: d2x )zbwh fz odjpnz96ouing down a lane *xzhdzfy d5 umgno:og+62 jz,9;p )bw at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth w:06q u 6djmzp0is q4snith respect to the Sun as the sum of two terms, m: 0 nusisj6p4qzq0d 6
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50dfc; i.wn pogjv41q3- m. How much net work is required to accelerate it from rest tgnf qj3d4;ov- .p1iwco 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 frozr2f*.jh x7bvb)5tqj ( 6aiajm rest and rolls without slipping down bxjq 7vi(f5 a )jr6hjba*2.tz 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 vertica/pb w- 7dtdex1lly on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use coxp -dw7/1tdbe nservation of energy.]    $m/s$

  What is the angular momento; 2w8 iig1zkn(* woawum of a 0.210-kg ball rotating on the end of a thin string in a circle ;2oz81na*iwoiww (k gof radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrica3b3sf+bu5 pf wo4wf l-l grinding wheel of radius 18 cm when o3pbfbwu-+s54f3 l fw 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 platftn*wzm4o,ofm3ux -pq wq19b.orm that is rotating at a rate of 1.3rev/s If he raises his aru9q qo.f 4 zw -wtm,*nom3bxp1ms 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 reduce her mo1(d4ei/qx dt lment of inertia by a factor of about 3.5 when changing from the s4 1t /e(lxqdditraight 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 inm-mg)g,d b:uqcrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final dm- ,bm qg):ugrate 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 /bk:unz 6sez)5wqx264 dezk(. c qny xaround a vertical axis through its center at a frequency zq kxck)e6(w6z s.4 dnx/2:q zneybu 5of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mome* m5ljv7qha mzurk yqk0x .pz0 ,l313entum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eandu2et0 .jvt pp7( l nu9*ag)jrth
(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 mo+ 8nm i ,rna;dgnab/ pq;nk9b9qz39 dument of inertia I is dropped onto an identical disk rotating at angular spn nabb;n;qr q99p,8u 3a9zdkin/d m g+eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2xg9j*tv z,le we+:ci 1.4 $rev/s$ 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 the center of a rotating merry-go-roohw c8qx bb o31v88w(x0jjac*und platform of radxva wo* (8co x1c8qj08bbhw3jius 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-rouzcx v )l)07niwnd is rotating freely with an angular velocity of 0)7 wcnizx 0lv).8 $rad/s$ 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 f0h3uwe o0y 5l 2:ob0/l 2zrj,psbuuivhf0 2e dwarf, losing about half its mass 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 new rotation rate be?(round to the nearest in3f0s2pyubru/of0 b :e,0 h0 o 2whlilv2eujz5teger)$\approx$    $rad/s$ (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 pi yryoafkz* 97ic 8e b69 iq:*4u.frjexcess 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 o oayb0+m4r)vg8 ze8n$ 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 plztubj k9li2/l *a/wzwi h6 nl -(-/kexatform, initially at rest, that can rotate freely without fricti2 -e/azlj/ i9x bz6ilwkk t (hw*unl-/on. 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 stands at the edge of a 6. ( ,c2bxrl ja8iz.xk0x(6gz eb5-m diameter merry-go-round turntable that is mounted on frictionless bearings andxi.lb(xe2 jrxg6c8zbak 0z,( 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 edxbh 4tcoy dk3c9hb, 73q(p2)vrgq2end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwindsb(qhhbyvx 3td7ck)r 32 oq 4e,gd2p9c from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such b0f4m+ava yo3that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its o ay3f 0vm4oab+wn 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 rx (4awxu* y,jzc+s/+jviw;w6-f rtjz 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 wet-oin2, .ob 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. Cal w, et2oonb.i-culate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical 1( alx*o6 qint,lw0atvg15zydisks, 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 conservaq1a vw tt*zg,(nal lxiy5 0o16tion 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 angu y9qb,9.vlpo tgmx 9w;lar 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 4p0+m wixlmi 5mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mam4pw i+5 mlix0ss carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ma+ oeunghu4x7o (fmr -6h+7ycg.gq7 energy stored in a heavy rotating flywheel. Su x+h fu ormy6ec4+o7. qn7g (um-gahg7ppose 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 anfbyu9l i*)1m( y a *t9fn2icjx $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at spe)f0hn 1m2x xe ymv+9weed 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 mass0:e6:v;(lf3zbs queefy i i:uxei + 0i M and 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 accvfz3;(:iie x i :e0:l0ebi uqefsu6+yeleration 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 radiunhc +g,m4r z(ws R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climbr +zw,gcmn4 (h 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 makpd rc txdqtbp7i.t*x .1*m/ v4ing a turn with a radius The forces acting on the cyt1tq t47v m/*xp. *cidxdr.pbclist 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 i l;0 uca* 25/cdbzcifwnto 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,c /w fdbl*iuz20 ;cca5 and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinde3)cilxvsj gj.j:+x)y r and her arms as thin rods, making reasonable estimates for the dimensions. Then cag+: c xjyvs j)li.)jx3lculate 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 cluen1d(( jr igofw*vv lf94 z3p2tch assembly which consists of two cylindrical plates, of maf(pe3nvjdw4( z1 rfilg9o*v2ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rady-x:o7jrh ml ;ius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of hm:7-ylxro j;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 do not assumqkalsv9 (84u he $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces i+hlk.;y zj ek u3isu-k /kq31yts speed uniformly from 90kmk.jshkul/31ie-q+kyykz u ; 3 /h to 60km/h The tires have a diameter 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