A bicycle odometer (which measures distah3+b ic,yc(zfv 01yksnce traveled) is attached near the wheel hub and is designed for 27-i+zyc fi c1(b3,k v0syhnch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulc5m/mq j0u0 asar velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accel0/a5mm q0s jcueration change?

Could a nonrigid body be described by a single vqdx2of:s,mjn18uo m0 ,+on yqalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaim*c d9hy7(ig m/mflp ;n.

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 head than whhng,cq( 5sudfs 8,qd :en your arms are stretched out in front of you? A diag 5 d(shqcd,,8usqn:fg ram may help you to answer this.

A 21-speed bicycle has seven sprgc/sc bk/ zok4q 8-u,mockets at the rear wheel and three at the pedal crankqkgc8 sokzc /b um/4,-s. 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? Why?

Mammals that depend on being able to run ht1 m+0k*7bb5lbcx*g7tqog e r0fq. efast have slender lower legs with flesh and muscle concentrated b5*0qbxetbm q.g+hf et170kgl *rc 7ohigh, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lon 9wzchmz9 xpkjew 2nv918t g7/ -sf7yng, narrow beam?

If the net force on a system is zero, is the net torque also zero? I)s:-g j xuuy7r 0dnxp,88wo drf the net torque o xn7uo0dsw jr )u-:r8ygpx, 8dn a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontaa+ptf 4( eu,j(-gkdlzl. The same steel ball is rolled down each inclin+ft,zu(gdjap(4lk - ee. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fmeqv ad760z9o2 :b* pyn3hvorrom 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 hy0a3 qv odz p*o69n:h2bm7r evas 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 same mass. They start frov kz+i*u m3odrf(m,er (w ph1)m rest at the top of an inclinehdorp( mm+ek u,(zvw* )rf1i3. 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 greater rotational KE?

We claim that momentum and angular momentum are cm5fcvp/,(jqa:6x f 4 q.cebyy, lgv7 lonserved. Yet most moving or rotating objects eventj, :g.75lfmlq py/v,4acc fq6y(xvb eually slow down and stop. Explain.

If there were a great migration of people towlf8 bbi 8/ 0kar0mpgs5q4vsbc78ve*m ard the Earth’s equator, how would this affect iq7 b 8r50ave4pf/*scvb s8 g mkbl80mthe length of the day?

Can the diver of Fig. 8–29 do a somc: l pkzbtx 6tr3cl;lq +kx053ersault without having any initial rotation when6lkx33 ccqzx+0 b:rkt pl5;lt she leaves the board?

The moment of inertia of a rotating solid disk about an axis th0 9w*ti6nylwg*sg y/g+rvtf 5rough its center of mass is 5gn sygylf w/w t*+9* gtv06ir$\frac{1}{2}WR^2$ (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 holdi d036a yx)7ypdngsk0 ang a 2-kg mass in each outstretched hand. If you suddenly drop0 x na60)pdks3ygad7y the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mawm,n*t 1h s ,42vphihvss. However, one is hollow and the othtis4nwmh2 vp,,v h*h1er is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating rsa 1f-ts+je i9z xcby6:y;0 xon a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular accel:r-zaxty1 xfys ;ji9+0sce b6 eration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a y 9epgo(wf(d8i :r .nelarge freely rotating turntable. What happen9dp ieg(f( oyern:8w .s if you walk toward the center?

A shortstop may leap into the air to ca5t4v4c6zyeta* w r1ubevm0u del1w /2 tch 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 hips and legs rotate in the opposite direction (Fig. 8–36). Expl utw6db421l/em c40*ua1 vvrt5 y wezeain.

On the basis of the law of conservation of angular momentum, discuss why a zv+ffq 5pkm/o0 +in 69klz5rihelicopter must have more than one 5iv9q zkp0f k6+/fozmi r +n5lrotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following aa j ;am7cruk6+ntl 43yngles 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, usia)dma6f ow (03p8tq)r ,j vbehng the information insio3 fje6t,a0av)m d(q)hbrw p8de the Front 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 Earth. The beam divergex6n*w(1eh fmfk g1nbc+q 5gj1s at an angl nm*fen+5bxf1c1w(jgq6 kg h1e $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Whe9) qr 6rbt9a0w4qa-9 mcrfmee n the motor is turned off during operation, the blades fa9er-9q 946w rm0rc mbqate)slow to rest in 3.0 s. 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 makes0qz3 zlw jek9cbr-og in4z1)q14 6jhw 15.0 revolu4b 9ncr6i3 k-q eg41jzq zz0hw)wljo1tions, what is its diameter?    m

  A bicycle with tires 68 cm in diamete o)r0byh1yafi c l94l-r travels 8.0 km. How many revolutions do the whhb fo0yr-)lalci1 y49eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its ani sv 8,sf-uq3hgular velocity inv-hsi3s f,u8q $rad/s$ $\omega$ =    $rad/sec$
(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.mb(wp +:2dm8nukweu: 0 s (Fig. 8–38). (a) wkup+(w:8mdnb :m u2 eWhat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around cp,n1tvmmf7pbol qj/)5 qn3 6the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee fkl27cs)yo, yd 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 c u6z+j* 8aoqddd- 4 )igirzq1pentrifuge rotate if a particle 7.0 cm from the axis of ia48*d +q jo1q 6rigdd-p)uz zrotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itufo,6foo ym: )s center from 130 rpm to 280 rpm in 4.0 s. Dete),uo:f ofmo 6yrmine
(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 radiusgcj 4a6eanc12s 0 j0hf km5tq5 $R_1$ 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, 6z:a4r:h1ak f) wfu koastronauts aboard the Apollo spacecraft put themselves inkh:aoa ukrf4)61 w :fzto 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 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 gzem/76u6-lo qqpxf* rest to 15,000 rpm in 220 s. Through how many revolu z mqo f7l/6xpgq-e6*utions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcuko oq l(v o/-n*smc 68u2,1feuymu f5xlate
(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 flyioraw(5 ytg . mchaa,a5ca;k4 /ng highspeed jets in a whirling “human centrifuge,” whic5t(m,kr;/acyac4 a o.w 5aghah takes 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 uniformly from 240 3em*: ioaap4um o ,)x(tdxg* trpm to 360 rpm in 6.5 s. How far wgi*t:m oxpxo( aad4,*)uemt3 ill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned o8xf,; z ar)mikff when it is running at 850rev/min It turns 1500 revolutions befor, zai;) rm8kxfe it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its sp,3dh dkd. 8,uh;(n3vqqk rzbueed uniformly from 95km/h to 45km/h The tires have a diameter of 0.vh3kzdd nqrdhu8 ;.,u, (3qbk80 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 itsk/ his 55r3rb 3nnkgb0/ lgs*e speed uniformly from 95km/h to 45km/h The tires have a diametsn3gni/rb/ 5 rs0bg *kkl5e3her 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 put5o11j piiygw tk+4g9aw 7x ua+s all her weight on each pedal when climbing a hill. The pedals rotatj g45awi x+wg91i t7kp uya1+oe 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 force of 55 N on the end oa;j +u6 n,ulgnf a door 74 cm wide. What is the magnitude of the torque if tn+u ;,jau6glnhe 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 togx2jrdz; mr gn604w/crque about the axle of the wheel shown in Fig. 8–39. Assume that a friction torqu6 d2rzjwn0 /gg4 xrc;me 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 ofy no am9*2zrq2 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 on this system.m9o2 y* zrnqa2

  The bolts on the cylinder head of an engine require y7h :knm43k p2zr *loztightening to a torque of 38 z*l7pkh2oym34knzr : $m \cdot N$ 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 ine4ax x sta799my hts cm;6.7 .66lfawzddbusi7rtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. xlsy4 7z m;76tc .6atw9dahism.d7ubf9x6s a    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 8ls u ttwq7y2(66.7 cm in diameter. The rim and tire have a combined mass ts28tl 7qwy(u of 1.25 kg. The 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 in a hq6b1n.md* t( rruk5qyorizontal cd k(*b5r6 qq nr1.myutircle of 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 wtab 6 5rz7fa;*h tcx-bq*l,va bowl on a potter’s wheel rotating at constant angular speed (Fig. xb w-*67;lzh,*cbr fata5v qt 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 of 1d-m )yvdc(pm point objects shown in Fig. 8–43 abov-mc(y 1dmp)d ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoml +ho6sxb 0o9m- awr(zs 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, z ccj5wm59 os;engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg w;s5 mjc5o9cz and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a ma alw, (r 0rto17mxq5kgss of 0.580 kg. Cr, rto5aw70gq1(l mkxalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fromwjlcvv(xak l .yw2 51(iq,h8m 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 tangenzaj14 ft/uyme( /hu48tqoc q)fs5wa-tially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each w(qtu msh) ufyc/aft5wa8/4o1z 4 ejq-ith 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 10pz2 *-ejd+whdqdwm+dfe .a387amx 2h ,300 rpm is shut off and is eventually brought uniformly to rest by a frictiona* +. d awfa3 zx2dhw-7mh+ed8pm2e djql 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 accelerates a 3.6-kg b qnta+.nxh5aq o*:; a0t+cnhlall 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-kgqwtgtf6am 8 khejy,;g * q ae;c-)lx65 ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformlyc66 g;ahextakmqg- yet8 lf;j )5*wq, from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin z8o)el)5v7b :dwhypv(2l q mm rod, as shown in Fig. 8–46.(lm) 7hq m) l5dpzv8y :ow2evb
(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 mas2dqvfwh.j49n ses, $m_1$ 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 fro ,xbyl.b8e 2l4iq q+asm rest within four full turns (revolutions) and releases it at a spee,bqbs ax+y 24qli .l8ed 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 ha b)r26lqik8*5 6afcycc l *juws a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ,y, vr quzq:ws ts 2ch+-83n gg5pjt6xof 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 without3ymoc2 ej1epz; k.sbez(i0+, / hhyux slipping down a lane at 3.3jhzyhko2e3y1e/.cm;p(eub0z x+i s , $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as yash*1 v +,+u3vwdb ak -pqf5vthe sum of two term h1+ k*d, yp3v5u +q-wvabavsfs,
(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 5s8 y)w;og7vip*b(mzla16ubt- iai u a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution perab;i lv81zyub6uit7mwg) i5ap-* s( o 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mas:eq4*s hk, da2k .vatus 1.80 kg starts from rest and rolls without slishdq aea, k v.*t:4uk2pping down 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 itscce km4oke-91vv r +skw2bp1mk*i5; o lower end does not slip. What will bevpowec*r;eck 1ks9b k 42kv+1i 5 mom- the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin :2f). yrjwm9n/5hmh4 z ew ukt6d-8vg-cqkwastring in a circle of rw. ackgv dk6tju4w wq--/renh:y f9z8mh2 5 m)adius 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 cylindricss0nb+m1 hr4mad mh)8al grinding wheel of radiu+sr s41m)mh 8b 0ahmdns 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 is rotating at a rate of r279*pwio1x s) 0a w5e e rowwvy3vx)u1.3rev/s If he raises his arms to 2vesx*wi y o35)rv 0eura79wow xpw )1a 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 moment of inerti6w aqob2sf(mzt;;2j da by a factor of about 3.5 when changing from(;;zbof 2sa wt6 2qjdm 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 r/ e9slhnrot2 jtplel71 ry3-;otation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 -9lle hr2syjel7p; /13ttrno $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical t5 r:9w *zsyk 6oapnf9axis through its center at a frequency of 1.5rev/s The wheelr sno:p5fw6zk9 9t a*y 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 momentum of a fhsvg 3y4iyq j6ws/da2x,,6, wp lcdw3igure 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 Etqosxzq4tz) xy mni1 1rz5-;9 arth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped 8pfm / n7qq.6t uhdz6 dmpg4k9to.6xkonto an identical disk rotax.9pm dktq npomuf7dt4 .k8h 6/g66z qting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.1e whzs j76gr 9a )7tv2piuj6e4 $rev/s$ 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 y m-5leq4pg( mmerry-go-round platform of radius 3.0 m andpml(e yg5 q-4m 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+benw (:kl:jc angular velocity of 0.8 k e+:(jlcb:nw$rad/s$ 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 whitu-wrjb(h x 0d4e dwarf, losing about half its mass in the procjh r4udb-0xw(ess, 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 integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exuxrf;47xrx5 /wayd . jcess 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 ;mhv por ;/aik+4-t- gs:rz yy$ 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 ro5k)q key:x rlui91t+ etate freely without friction. The moment of inertia of the pl:5q)iyekex utrk+ 19erson 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 stand yot)3sx y5;yru*)h jis at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings andy3ru)*th)sxy ;j5yoi 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 ent:fdd -,e,yqh d 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 unwinds from the spool? How far does the spool’s center of mass mo,df: -t, qeydhve?

  The Moon orbits the Earth such that the same side always faces the Earth.+)o 2 bxuy: ml.7b+*i0llvdmshzs 7fx Determine the ratio of the Moon’s spin anguls27zfbxh7ld b mo sy0mv u++lli*x):.ar momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates gtvrqg0:w0 +hp:,:f - qqyx amfrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m a5ntca39ronzbsy *y4p7v uq( 9 cquires a rotatr5b97 n czyvqpt43s u9o(yna* ional rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrybgvw5(ui ,me *j03 ndical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, ifb*jguw m 3y0(die5v,n it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is thf34ayi+pi1gir :1 b* 1waox7zmpxl3ae angular 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, bn 4 m9pirk w:xr6(goj3ut with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse tmir36x r9 w(ojp:4kg no 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-popdo:)o9nt (3 +td jvy9v,vk fht86lfyllution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 140083)9f+to :6ypnt dt(ho dfvvyj lv, 9k 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 illustratexudkc8er l7 ,0 vy44aws 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 on a horizontal surface at speed u wmso8/)ap0uis/4f dv=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignor 9-65n:fpo0wnctl nn ge friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment o n6wlgon 0- nf9c:nt5pf release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. viv;sb6d 83 nye7/ikdIt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. /67sb3; nikdveiyvd8 8–55). The step has height h, where h

  A bicyclist traveling with spee4gkvo5xv 7j7 ta q.b;jd v=4.2m/s on a flat road is making a turn with a radius The fobtav; 7gv5 j4xjk.q7 orces 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-kn/pvh4 mwz t. -bwymbtve+,kl r20x1(g rock into a sling of length 1.5 m and begins whirling the rockr2-4l.h e(k/0twy1z ,mtpwvmn+ bvxb in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, 2pq6kwi(dy9m h0 io) emaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinnioyp9( )idw qm k20ih6eng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plat pz*7s qv83niues, of ma3zvp*7i8 sqn uss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8idlrp.up(;:cjn:jzj cl(pg 40 x, hp6–58. What is the minimum value of the vertical height h that the marble must drop if r.p :ip;:jg0jlj lu6xp(hcz (nd cp ,4it 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 do96i4zdq5yhsspbsk*51ngfoa s t59/g not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed gy/:ju/ 3nwakaa,6azk0d -lu uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleratu/zy3n6lu 0adj-,: agakw/k aion 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