A bicycle odometer (whichx wg3oe advv,./d+ ((sc/ jttf measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicgf tv3/(d,d ctoj.a/wv +x(esycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does v- :212zso 5gemylg2 r4dcxeoa 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 acceleration? For w xso2-v:g4ezg25moe2y cl1r d hich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by vx y(7 tbocifr.;z)4mqmz*,/t dzt7 t a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greatn 5supn:*zy8 ger 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 your handp ,2inwxjly 7v;:r3hf:av; lss behind your head than when your arms are stretched out in front of you? A diagram mrxy;i av23f;vwh lnps7j:l,:ay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whee-)pyr tp8t5ty l and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprockp-y ytpt8r) 5tet? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run 64pix9 y/m12dkmc oeta: w 9tifast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, exp 9: ym4mitwkatcx 26/9i epod1lain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bef ;4e9 usy og5xan).rqam?

If the net force on a system is zero, is the net torque also g +snk:ez- e2i2yb8- t rndq(izero? If the net torque on a syst+r2itnsyq g(2 zi d-n:ebk 8e-em is zero, is the net force zero?

Two inclines have the sa; y;js-h3y5colmk*gmr)jk ctyl v295me height but make different angles with the horizontal. The same steel ball is rolled down each incline. ; - sgky3cl5;c yhmo) k*2jmrlty95 jvOn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frc /nn(r* z.ki(ittwgl4-z: cl om rest) down an incline. One sphere has twice the radius and twice the mass w n4 *:.irkc -tclzlz(n/igt(of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the samee5.fjvnzbi7pzd+*eiuj06 ;7tk ysk 8 mass. They start from rest at the top of an incline. Which reaches the bottynv j 7i+ bi;j 5 p6eef0d.uzk*tk7sz8om first? Which has the greater speed at the 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 m/hlv/jg.m-:0 vrns 3v) krmu noving or rotating okmrvn// 0)vg j.mvs3u - rl:hnbjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator :,bvf8o zlcr;j+yx) r (afqd., how would this )r+y( rxv .qobjda8,fcz: l;f affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without h2a4pceq6 8ax xaving any initial rotation when she leaves the q4pe 8x26acxa board?

The moment of inertia of a roocl6 .)xumzg .tating solid disk about an axis through its center of mass ..6 cg)mozxl uis $\frac{1}{2}WR^2$ (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 holding a 2-kg mass inrtww ftbj*7p v x5-)i2 each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrif-vt5tjwb) wr*x p7 2ease, or stay the same? Explain.

Two spheres look identic;a+joepm+q. kal and have the same mass. However, one is hollow and the other is solid. Describe an experimen+. kj; mo+apqet to determine which is which.

The angular velocity of a wheel rotating on a horipxf3ah u: 8sdmq5 -q93xv66wxtj ) py/cjz;zrzontal 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 p)w8xjqa u tr;z q3y9 s6 m 6vjxh35d/xp-fzc:of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large frdko bskrc9q y, gz1x0859iz 8zeely rotating turntable. What happens if you walo9x,z y90zqb8r k8sizk5 dc1gk toward the center?

A shortstop may leap 19lxls(npy6 dz :c *zzinto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his z 9zd :(c xnl1y*zpsl6hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentt+,pt ueo /uo)um, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the ,e)tp/o+ o utuhelicopter stable.

  Express the following angles x4vi w5z/.i(hzulu: 5uc;cje 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. Calcu0dnjwjn jzu.ts 7eu6f 5 tkt)v : *q63+oyng,qlate, using the information inside the Front Cover, kjo6yfn qwv)3n6 tj5u0 ut +,qnsz *7j e.tdg:the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direcv+a6yyshh:l;ce qqxd *j cts./ ..4stted at the Moon, 380,000 km from Earth. The beam diverges at an angle cj/.qsv .s: dl cths *xqey.64ay ;ht+$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blen (i+pw7dedve0c4 dy f2der rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s.(vfp2i4dd+y ecd7w e0 What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makehoxf,7p0rny (u wa)x-s 15.0 rev0- oyh w)a rxpf,(7xnuolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travelsqd-jcrn/ b uj /z6+oi0 8.0 km. How many revolutions do the wheels make? q+ n06zjd/r/job-iuc    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calrcaj/5 *ak3tkculate its angul* k3/tka a5crjar 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-round makes one complete revolution in 4.0 s (Fig. 8–38). d rhplm24 zysvj2- 05-iylqo, (a) What is the linear speed of a child seated 1.2 m froy2vpd2- 4-sl 05 ihlo ,qrzymjm 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 opqh+2 ndv i4b1rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 8 /aluli0ze4k 4cu7plpoint
(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 axis ogar2m qa- 9q2cf rotc- q 22mqagar9ation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml y3zj a ieb9z5y1r0ua;y about its center from 130 rpm to 280 rpm in 4.0 s. Determ 390;euazar1 bz iyyj5ine
(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 radiusorn v28j xf-t;u/h4gu $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the A yetvsr*7(e rc- yc,lgn(w9w /pollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy eveng ew (y9wn- eyr7t/(cc,r*l vsly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelera9p3 l4e motci:3b bh/xtes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this timeb px93b cot:4lh/3em i?    $rev$

  An automobile engine slows down from 4500 rpynmtg (w4mkps96)g i3 4p.kbum 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:7r75 2fyqcrgf83 b vxqg,wcy stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionsbqxc,f 2rg8qcwyf : r73gv5y7 before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly*xq/pl* hvl 7y from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tp* hly7/lq* vxraveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1om ,. ( t/;lv.7obe amdbqd.1(txh, e b;tzovv500 revolutions before it cde vq7htz .mm;l.o(1(x;to,.ov / tbe,bdb a vomes 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 reducessap14jhti+av sezla 46 f /f22 its speed uniformly from 95k+plzht4a ea 6 js2i14f/vfa2s m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revo s,ouvs76ng 7ps+i6sfppyh77 /b 0vgzlutions as the car reduces its speed uniformly from 95km/h to 470iv 7 g z 6bsg,ypnssp7so/ +v7hufp65km/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)i*,; cl + ix3elnasht her weight on each pedal when climbing a hill. The pecl sii* th,nxe+;l)3a dals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forc/(, esyl ,vzdwr 2/kiye of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the fz/rlse 2y, i( /vd,ywkorce 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 Fig. 8–3gx f g lo8cy,dy;4-.:mluy(wg9. Assume that y - f8mgg(,4d .ulolcy:x g;ywa friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to thexi8ke.(t,wjl s.ml .z ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is helxis k.j8let.l,z.mw ( d in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder herh dbo8 )6(khvad of an engine require tightening to a torque of 38 odk)hhb r 86(v$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0 m0m ;7a :dnoab.l8ihv.648 m when the axiso8m:lv7 bd.a ni;0 mah of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia ) hm)egt1+)w g;oz,pb2j d cef7r2ysh of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub c2b h+jeg) rm7ps,h2gdct)wy 1ez)o;fan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a fb dd)okz r6h4yk hxtfpms5)9 ;r c+uov/ *e71thin, light rod is rotated in a horizontal circle of radiusdk s )h 9 7of 1xm*vrykz+/eb4o5thpcu;6f d)r 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 rotating at citdt1xk8 2 ;gp4t-fkwonstant angular speed (Fig. 8–42). The friction force between her hands and t; t -8x t1gtwdipkk2f4he 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 of point objects shown in *2lac(s198 mt j2n(vuvnerea Fig. 8–43 abat mv9rjl(c esu2e n182na(*vout
(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 oj7l 6bkxx g5j:f 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 satellite spinning at the correct rate, engi+:fwy u1yi7lv/ lux b7h0v;abneers fire fou: ax h1lyv; +0vb7u/ i7fbyluwr 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 is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.r.ahobc9gpuh- hjzx + .rzbq.v7 2n2950 cm and a mass of 0.580 kb2+c2x9-ohz 7nzrr hhb.vu q.j 9agp.g. 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 swing/jf-tq3o *slb -ek/hg hitri+i -/jd)s 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 hand-driven merry-go-ro /hk(9 vkjj:gf+rxe ,t ,wrkv:und 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 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each othxk(+kv k:9he ,gwvr/trjjf:,er 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 i*8neu:xj8jdxo7w wo 9s eventually brought uniformly to resn8oxj *w7eu9dwx 8 oj:t 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 accelvb w i9)/jq+tn;1n3f)b ymhtterates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whi93 (mcea xmsf8ch rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleratedse98am f(m3cx 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,y(7qw-rqzgj p 4aohqsu)(c( can be considered a long thin rod, as shown in Fig. 8–4((rz u7 -cqq,qw4a s)yj(pgho6.
(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 consistb ocakd *1w .;(nt7scns of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withi-k x; y; s(j93uzp+vir qi5lcln four full turns (revolutioz-yq;li3v5;ljc 9 s(xiu+kprns) 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 of inertia of l2zx p 6t xwoqi.yt. h7b3v(g3$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2 c2bm h2ild5m380 $m \cdot N$ 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 slippm-w 6ulxx( c/25*fs 1tfvmfju, 0yndhing down a la5/j*ds (f- x0tffuvm c2lymhw16 xu,nne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t wa ;jum3qo:2xd ,acp/z3je;ihe sum of t/a2jaq w p3m:o e,z;xuji;3d cwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.ksqvm *w, 6z 6cdymt+-50 m. How much net work is required to accelerate it from rest to a rotation rate of 16c6kvzw ms,t-qm +d *y.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm c zmuz6: fgt ypevw22 k/svu5r+05ze.and mass 1.80 kg starts from rest and rolls without slipping do+ft/.0zrs emy55 2kpuuz:v26c v zegwwn a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its 4ruv svlm8iwk(1e l 49vm.e yu1z1k4qlower end does not slip. What will be the speedk(i8m m91k4su.vr lwey1e v41quvzl 4 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 0ba l x uz/h/ .p.5nf0frep.r:(zwji91bd4lt c.210-kg ball rotating on the end of a thin string in a circle of 0 bxpezu.. hpztadncwr(r4/f:ll 5 /j1 .9bifradius 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 cylindrio:mw+m v 9ayu/cal grinding wheel of ra:mvu9o+w ym /adius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that go 1+ys(rsrmc)go.r 0jg-t r13z3keyis rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation dskr jg.to(1ggcye +zmr0 s)13 3r yr-oecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one s z7 ) a31yjjd,7sh1zzgiqg 5afhown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she a iz ay 5fqj z3zd,)hg1j1s77gmakes 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 fy67zx5fea*ju rom an initial rate of 1.0 rev every 2.0 s to e*5afyu6 7 jzxa 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 rotatingzs ;cr5k tlz/- around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk5ct;z -rsklz / 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 angul5oa-nloh:mxmhy e,. ) ar momentum 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 Earth evkd/klvtm.0vi7 /x,) r5w +ejq2orw
(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 myx k. ekazl9y*e uk0.)-wz.jmoment of inertia I is dropped onto an identical disk rotating akeyx0.z* ewm. yz u9.k)-ka ljt angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2)lj+mc. * hz*0mhb xq2y0 6ac qbofg4ax l5:dq.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-gov(8 hny25ufp/n f1- wuuvzv0 m-round platform ov0nf2z( 518pnfyv/vuhw-uum f 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 ofj)a hoctruz70/cx 15z 0 h/)c5zzco u0r7a1txj .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 dwarf, losing about 90ma*e n*ckgy,npa0 ohalf its mass in the process, and windiy*9 og a,n0mn*aekcp 0ng up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve win:i1q(y nxzqu :ds 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 .gekbsq97jt g4.kmg($ 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 atkmqpy8s0h /m2gt/ ox vn0 20xa rest, that can rotate freely without friction. The moment of inertia of t2o/tmq 8 a0g20h xmnk/0vspyxhe person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-rou39+yiyy cifs;d*,ep dvkti8 8 nd turntable that is m y*i8+ciesyd9ifd8 ;tv ,k 3ypounted 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 4q9 edefh l/n+(avf/ le3pn8( ,pkftnrolls on the ground with the end of the rope lying on the hkqev lnf9tp (3el4 f8/an p(ed,n+ f/top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earthtbq/c- .dd.8c/ol liv v fci5(. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the(8fic/d bid/c q5v.l o .-vtcl latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rjio7yfb 80l lz3+o +ncest 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 enbbnt9n5c6 0nnrx4 o1a/6ys+w ir-y 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 then5ebn o4r-ynxin61r+c6n 0/9s ywab t torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.+p u3.(e zijmzr.ogx 6050 kg and diameter 0.075 m, joined by 6(+ 3 ijoemxzrg.p.zu 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 angular speed of the rear fvmn ,s9+g-v*dfm) ++v yiwrjwheel ($\omega_R$) 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, weresqtjj/:x l2 c12tj/uf 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 allu: /jfq 2cst1j2/xjlt 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 energcortf 3c*/bpdpd*- sc;tn.l 9 y stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 mtflo drc c.*pnsc-d93 *;tp/b and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an f jzp3e5l4+kch0(1j oy *gnyf$H_2O$ 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 surf7zf ,k-wj7yxbk 2tp -)u)ugtuace 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 frbvk.yov 27 ur6eely (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 acts2u.v76 ory kbv 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 vertically on:xa o6or lpe:5x,m 7kn the floor, and we want to exert a horizontal force F at its axle so that it will 76o: lnxa5,porexk :mclimb 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 fatr;)9ldl p r o egwv64+ddf(2lat road is making a turn with a radius The forces acting on tdld2lvg+od(;4)r w9 pf 6tarehe 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-kstk(m hmx zr,h05t- 8lg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is mkh5(t8m -x, hzrtl 0sthe torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solidgd2wrg 4:*dc5vp rl s1 cylinder and her arms as thin rods, making reasonable estimates for tr4p*d5dgc vsw1 lg:r 2he 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 cylindrical platg(10r d;z,t sywii9 ikes, of mass gdy w(1;0i9,rkiz its$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loopjw94,d. ya na j8dcr9ut;ly3 ted rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the high r.y8j34da nwaj9;dt9ct yu,l est point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu /yo*ll6 zak+hd t1;6aprqa2ys fq; +ime $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions bynfk:;.u o m1f22hxvas the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whkf.u2vfo2:x1; nmh ybat 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