A bicycle odometer (which measures distance traveled) is attached near tg0, p2b w0prbdhe wheel hub and is designed fo0 gp2rb p,wbd0r 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates cf0 q g2byy2d:gm:yg0at constant angular velocity. Does a point on the rim have radial a m gc0yf22:yb0 :dgqygnd/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 acceleration change?

Could a nonrigid body be described by a single value of the angular velocity (9vr.x*, aj 1ohkeeuwe2p,m x $\omega$ Explain.

Can a small force ever exert a 8ty(p do3-catgreater 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 hands behind yo zhgsa4r7 k ou;xh alh2-wg8;-j s8z:bur head than when your arms are strez bho;l x-ahsgjw:4r- 8k7g;h as8u2ztched 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 the7phmb*qgy; 8g pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which ge;p hbggq 87y*mar is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower7a3z 8 vqtnkb:lu;5l h legs with flesh and muscle concentrated high, chl8vz qbn7l5u;: k3talose 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 long, na cyvkryh1 ,8z8rrow beam?

If the net force on a x(hr)k 4t4,ine y4enk system is zero, is the net torque also zero? If the net torque on ax44ht rn 4(eik, ney)k system is zero, is the net force zero?

Two inclines have the same height but make different anglva tw3,3rqzsms* 0(8kqur *xljt7 +w res with the horizontal. The same steel ball is rolled down each incline. On which i wxlqatqz7 rs k u0,83wsm+* *r(j3vrtncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fromaofs c j -6e34gu--gok rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which --g-jesoo3ck4f 6ua ghas the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mxh0v5vz2lss.q;:rfmh bx or- y5, z-uass. They start from re ;x,5s -ru.fmz-:oxvb2zh 5q s 0yhvlrst 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 greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving.wj:4ela/xd ymy gaics9v1 (2 or rotating objects eventual/ wmx s2i1j:deycv(aayg 49 l.ly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, howwbt x w* b*9d,tyb+z)scp(5m v would this affect the length of the dayz), bbw(t5vsx*y* b pt w9m+cd?

Can the diver of Fig. 8–29 do a somersd 8(-bvh3 sqteault without having any initial rotation when she leaves the board?d 3(qetv -s8hb

The moment of inertia of a rotating solid disk about az /6zfd,z0 wokn axis through its center of mass kwz6zf/,od0 z is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on ap9hbc d ,yoq.mho 9:;t rotating stool holding a 2-kg mass in each outstretchedc9p; ,yhhd:ooq. 9btm hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howtpr7l/q ,b8 :/ha xphwever, one is hollow and the other is solid. Describe an experimen,xt7wa:p8ph/bql r/h t to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. c8fwpqp -y2gdnyywqa1 ) 3g 25In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangent q8-ad yg2w q3fny )y1wcgp25pial 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 large g ba s-py9sfsn - j74iq)u8p,ufreely rotating turntable. What hfg,qbs7 -pyu i8)j94- pua nssappens if you walk toward the center?

A shortstop may leap into the air to catch a ball and tcwhy3c/n :eolo 7wuedl 3:h3 yow/8q ;hrow it quickly. As he throws the ball, the uppe37 w;h/eneq/:3lc hu3l d8owywo y:ocr part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why oh.9q6hzkn3 ma helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep th.9m3hzk6 oqh ne helicopter stable.

  Express the following angles in radiansl 2jbt t vxo)(ksh41(u: (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, usingm/y n )sev:7vw;em4p e the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth:vm e4m ys) new/7ep;v.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamoze 2n. zhnsi8 ;0i,*alhnxg3 diverges at an alx;ngz i3o n a8hhse02nz.,*ingle $\theta$ (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 650+nd c799nlw2ol /teyk0 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What 997l2ldtwc + o/ nkyenis the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another4hes-(oa;hbmwgw4+j child. If the ball makes 15.0 revolutions, whajasw+gh-o (4m w;b4eht is its diameter?    m

  A bicycle with tires 68 cm in r o76iordfh)+1puqtob .cj293ny g ; ndiameter travels 8.0 km. How many revolutions do the wheels mak7ifhucr.n)o3d+ 6rtq9 np oj ob2y;1ge?    $rev$

  (a) A grinding wheel 0.35 m in diame1bh ;0qhgg4(.pplk meter rotates at 2500 rpm. Calculate its angular v04 h(lgehpgkp q.m b;1elocity 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 s.r r6-qaxad fy6r-j +complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated r ds ar-rxj.y6 qa+-f61.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sunk:ct 7v k2jjg4    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed k zkm3-2vgw7w 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 centrifuge rotate if a particle 7.0 cm ,ak :qr 500xpnj,i.mb m6htdg+h (agfr :fz0dfrom the axis of rotation is to experience an accele.,pr:qd a zamitffx: nj km(0gb0g0, 5 hrd+6hration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboun;yx j n9liek es()j5cn50s*jt its center from 130 rpm to 280 rpm in 4.0 s. Deterxne(9c*y ;ins jl5e50 jj)n skmine
(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 radiu x4 n+m c)wa/lyr(li)cs $R_1$ 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 Apollo spacecrao 1w9cg t,6udift put themselves into a slow rotation to distribute the Sun’s energy evenly. At the stg ,id9o1cutw 6art 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 rzs c -:hl/n1y2wipr4aest to 15,000 rpm in 220 s. Through how many revolutions did it turn inilw2snhc ay z:p4 r/-1 this time?    $rev$

  An automobile engine slow,pyl do)7c +.mby lr 2h5qvxm-s down 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 82y*ib;eut i /h0jtqqthe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn thro8;*uih20 t eybi/qq jtugh 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 u8p o*coldg*(hq2su7 aniformly from 240 rpm to 360 rpm in 6.5 s. How far will a poiop* gocd7s8q2(*hl ua nt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Itg ru. 0.7aps5f lqq.jk turns 1500 revolutions before it co5ar..lg 7p0.jfqqk sumes 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 revolutio5a*8 emp mg+wml(vf0 xns as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamete *5x0m(vm g8l+ mfeapwr 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 re.)2-9 rmb kwihg6/vne w(*lqfivzc2 ovolutions as the car reduces its speed uniformly from 95km/h to 45km/h The2n zg*wrlqc)-f(kv2 vw/6hbo.9im ie 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 .1jpbmy-a y( fputs all her weight on each pedal when climbing a hill. The pedals rotateym.- b1(yafpj 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 s h36vr,wv,xx 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exert,r6 xvv3x,hsw ed
(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 ak0y/yg-wojy+w xrn j2/ie*s0xle of the wheel shown in Fig. 8–39. Assume t0yk0 xsnregyi*woj 2/ -+/ywjhat a 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 the ends of a massless rod which f)8)mg a)dqe v48m)d7cdfute 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 nf4e ufmt)8)ddmeg)v8)c a7dqet torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque ofg8 td)v/qgfxb4*h j 5k 38 k8gjv bxd )5qg* thf4/$m \cdot N$ 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 0.648 m when the *eus48i3aq)- nun9k ap s q5qmaxis of rot p84am )ssqkni53* ueqqu -9anation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 2+m zq ybz(n:dcm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the q+z2ydb:nz(m hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the eueb (c/p gw4 2 n;f6/ykwkvn* i1-qrgwnd of a thin, light rod is rotated in a horizontal circle of radius p/ubcrgw; wfyq (2wg-i4v n/1kek n* 61.2 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 tq7m7b2j/0q-3g xe evou+ rnp i;m7po b8mi(hon a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N mmpuio te 7xqgb7e+o r ; 2-0nb p3mj7qi/h(v8total.
(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 inerts.1++h)7 w tnpp / gjs6ws3+uya tqm dborw40via of the array of point objects shown in Fig. 8–43 aysm anqwosdru j bg) p++47+1 0/tw s6tp.v3whbout
(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 oxyg9b16vuu drqmb* ;y9o2/qbksbv z;zosq jk16:en 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 dg-ujd.4;82rh ) p zbe(zy0gu d8nwk mthe correct rate, engineers fire four tangential rogbzm-d 08;gzj.ud h8ky ur2en4p dw()ckets 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 8.50 cm and a mass h0qabhp*i: q/5 m;okz bl0j2zof 0.580 kg. Calculathbq p/ka*2io:b z0m5zjh;0 l qe
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to vg60z/,e zgwv t/4za d3 $rev/s$ 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-round and is ab1m (l2q0czz l29ae fie*y2,u7vti z zble to accelerate it from rest to i 2 9zze2(1a lm0i,uvzcq*by 2fetzl7a 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 other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and igs35f5/ykdhhi xfzv ), 1y4q hs eventually brought uniformly to rest by a frictional t1 yk/ fs)x5vqyi 5z f,4hd3hhgorque 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 accelerata/g 4fbuo2et*/ dqrd1es 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 ce4a1 86i0fj.xxns f8mni tpj a*g,/kthrown solely by the action of the forearm, which rotates about the elbow jomj xsf, jxiti.n60gk f8 *apn8a4/e c1int under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long think6mj/on1 iyb2i+uboql52+l i rod, as shown in Fig. 8–46.iikj2/ 5 l+oqu+ib n1bo6y 2lm
(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 o0l- vr :f9obkdu9/gs pt2vt2 lf two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fuz 1e 6kh.hsu9kll turns (revolutions) and relea16suhkh . 9ezkses 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 momou6h2h6bp9wlbj 9.37hjll : me)hm) kgg kw(ient 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 of k9s8w mhhgheht0d.+4/ zfj/a ghwtn- ; lo48q280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and rz1kw:s9su-3t 9-mq r-geeh r radius 9.0 cm rolls without slipping down a lane-ek:tzrh3w9e --u r19 qs rsgm at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of x/qm 3owfe:j 5 ub67lao-/ maptwo/am-ewou:j7bl f/3qm x5ao6p 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 radiuso)- - jyqwf:6nec xc2o of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a soli6we-qjnf: )y2c oox-cd cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kggt l3ma fxjp+l-h pfqj ,82g;*,/irvo starts from rest and rolls without slipping down a -qla,;pph/3m8 v,og*i flrj jf+g 2 tx30.0 $^{\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 balance)i.4f 6;+xama,z o l;zczn4 ndre dx8td vertically on its tip. It starts to fall and its lower end does not slip. Wn4r z8m )td. n +a6e 4i,oca;lxzf;xzdhat will be 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 thi af;(1 isoqje/0,l() wjd mine end of a thin string in a circlmo)l i ;wii q,dj((anfe/s01je of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angularwupk+jx ..o d;+( pcjr momentum of a 2.8-kg uniform cylindrical grinding wheel of j(x+ .wu;.oc pd+kjrpradius 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 rotatin o-t5f1zs9 sztg at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreaszs-oz95 tf1t ses 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 redw*mee(ihh, g b5e g-t o5xc(2auce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck p wam,h(ge(ei t ce*bo 2gx-h55osition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can ity q9 b3j;(bsclts 8p:ncrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a fit p3tsbql(b9 s;y:j 8cnal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axigij; 7ouhnl37ijimck:b w w):w*a-a 2s through its center at a frequency of 1.5rev/s Thewo*:mh laau;2ikbi-)7 w:jj w7g3n ci 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 momentum of a figurenkt rsyt7 k .jrej +:/:,tjp+k 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 Earthynyti ml. m:8-
(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 ofh3bk( zxa;qxl 5zxr+ . moment of inertia I is dropped onto an identical disk rotati;b.z3xrq ahl5+( xxzk ng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4glpm xk u047lu4fiw4-+ dao o0 $rev/s$ 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 ta hp0i zssh210he center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920s2z1p0h0sh ia $kg \cdot m^2$ 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 rotatinge 3hp: w/ *col3;qwxyl freely with an angular velocity of 0.xwqw ey ;*3pol:l hc3/8 $rad/s$ 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 whitjap;l/z8 su*ve 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 l/; u*zpasv8j 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 excess of 12sice -912jp ws+bh /w9 vpi+c,ybyuj 20 $km/h$ 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 k+h . adi5e7s3t* g.kivu)w fm$ 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 ar -4fz31pr4pjvlj s-y at,vv+ platform, initially at rest, that can rotate freely without friction. The moment of inertia of the persf-j l s4+4vrz v,tap1-3jvpyron 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 diam/bjv )+air(tcf 2wkr 7eter merry-go-round turntable that is mounj+ 7cfriwv) 2rtk/(b ated on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the gr ;9( qfj1,dpzxqtpduyc.e55bound with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, hold(fqu e5t9,qp pz.;jbcd dx1 y5ing 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 0wp5 ezk(p+lxthe same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momenz+wpp5x 0elk(tum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratej0nj4rs myx*oeq4ry8zja;4 . 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 aic42 ey;q0 7icij be23,ximgy uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered 7ybq e ; 0xi3ce2c,yii2j4g miby the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each :xbor ad829 kv/(v py8qpgz c)of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of massbkzx ryp g o8d82)v q9:(/vcap 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, hcz/xuwp7a01hip6:c m ow is the 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, but with mass 8.0 times as great, were rot/ ynu.1 xfb:nlz5w)+w med4ghating 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 1d nxy/)u bgnm 4:l5fzh e+.wwin 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-pok; /u-nkau6yba fvaw3 )woy, 7llution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, usf7/y,k-6 kaauvn ba uy w)o;3wes 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 z wigri7q yx;46+xf9ban $H_2O$ 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 h+chlgve54f v9 bssu0idg., tixs65 s/orizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igng ; kd3 pv5ta(qkta-s3ore 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. Asgv5-q3s(t; a pkt3 dkasume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticai+(p v1w;fl; at p9vhylly on the floor, and we want to exeryp;l+ 1(;v p fat9hivwt a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with s40pz ajg c6g*a radius The forces 4sgp*0gaz 6j cacting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a slin) 7ae,v,kzbc b ,a/nhig 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 z/ ,caeb),hvbi kn7, ato achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a soli5 u,d5zt pe2bgdf3ze 8d cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly ag5t2zfg,budepd3 8ze5 ainst her body.   

  You are designing a clutch assembly which consists of two cylindr8q 1o7eg qr:yrical plates, of m qqor8e1r: 7ygass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radw:f /c: ij3vzvius r rolls along the looped 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 highest point of the cjwvv:z /:f3i loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu)9cy s,) yd(bi,qnf1vqkhg k1t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th*bl *cds 0a- /fq8u tv8;dmim*uo9rwze car reduces its speed uniformly from 90km/ a;z mu8dorc9**- l0iws* fdqv8t umb/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