A bicycle odometer (which measures distan9a6 vdzr rict3t0+,p;fwg7*vrj jc6 mce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you usizjcr 9fd,t0cv6pw j 6tma+; vg37rr *e it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloci b4q (db vrrbx6ua::yfxf3v/f67ngj.ty. Does a point on the rim have 3abrv (qfr6fjb:guv6fnd x/y7 b4 .:xradial 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 acceleration change?

Could a nonrigid body be described by a single 9)-*(orpw+m6h, sroy 2 qdiri tfhtjw b78y,dvalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a j4ujwus nu,vj;ec an :);l6c/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 witd/tp bizd/ e 097r0fe* sbqkn-h your hands behind your head than when your arms are stretched out in frontdsr *kp endfqb b7 z0e/0i9t-/ of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and uo:y5 s)sd9xg three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Wh5osyd )9gsux: y? 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 fast have slender 3 dmrqq:ui*;l75 ozcs lower legs with flesh and muscle concentrated high, close to the body i*com 7 u5 l;zqq3:dsr(Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cu,fea /38m(w4pi6zhrmyi3 v narry a long, narrow beam?

If the net force on a system is zero, is th nz +p:w3aag1fl a+3sie net torque also zero? If the net torque on a system is zero, is tn+f ag1s: zw3ai3+pl ahe net force zero?

Two inclines have the same height but make different annj7.yyo9oi3b m svq67z ;n :lggles with the horizontal. The same steel ball 7vjb;9l mg.o63noiqsy:z7yn is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an,*ycg3is5tc,n wio4e( *lav i incline. One sphv5ocy glae*s w4ni(,i*c, t3iere 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 has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same 2am)y2 kt 9wsv9jac5g radius and the same mass. They start from rest at the top of an incline. Whicy t2v) amacj29k95swgh 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. *dotrtuo v/33Yet most moving or rotating objects eventually slow dow oo /dttur*3v3n and stop. Explain.

If there were a great migration of people toa9zfri5i 1krs 4f; x1bward the Earth’s equator, how would this affect the lenfa4;x9 z1sk ibi5 r1frgth of the day?

Can the diver of Fig. 8–29 do a somersault without having any ini;6u p v5sdlc5btial rotation when she leaves the board55slpv;cu 6b d?

The moment of inertia ofi rm wkk86qohhd30,)r a rotating solid disk about an axis through its center of mass0km)r3hdkr6 8wh oq,i 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 a rotating stool holding a /a1:fbeboz 6a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the ez1a6:fa/bb o same? Explain.

Two spheres look identical and have the same mass. Howevebmewmz qvl e g7 5b9ef+2.6b-cr, one is hollow and the other is solid. Describe an experiment to determine which is whicgevwm 9.z+e 6 7qlbbfb-2e5 mch.

The angular velocity of a wheel rotating on a horizonrrl5zu2dc*b b7. e6xe u;pe .g x(pr*ytal 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 tanc6ubg2 eppdb.(y5.*er *re; r7x luxzgential 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 freely rotating turntable. What e8mg ag3vab3+ happens if you walk toward the cenb+ga8m gv e33ater?

A shortstop may leap into the air to catch a i3zck* ; bj+pd9+lb2nx nwus,ball and throw it quickly. As he throws the ball, the uppniu9;xs +bwpl+ c ,d*2bjkz 3ner 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 whyj wx)ur:*aw 26*nfoqb ib((ul a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second proulqao (2)fw xrnwb(*:6 *buij peller can operate to keep the helicopter stable.

  Express the following ang m jcqx* t6c/mny:.odzz/agesy)nlpf.: d;-0les 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 amav-d l0wz g*sd1zing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radiad -glvds* z0w1ns) 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.tb-cj,0 xe ,e4tnh-v-q l/cxp r, n4kr The beam diverges at txv /rlh-j,ecnxq-t0-c4 b,pn ,ke 4ran angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Wutpfyc r9f+d( au:(d z4gufj;2c37d mhen the motor is turned off during operation, the blades slow to rest in 3.0 sc2dpfuud39(j +g frtma7;4: (zcy du f. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor e ert d vbog bt,0,ea- ,rje2fgg.7z8*3.5 m to another child. If the ball makes 15.2.g8- v ,gjz*bebda ,re t0r7ee,fgot0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revols rm51, gwm75ijs-i3m s; zvfkutions do the vkgs;5 m-5im sfm,sw 1j3iz r7wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cm w2,aoo(ww-.lmx v,ealculate its angular velocity , x,wowvmla- e2m .ow(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 6zvujegyd(2,r me4 n/(Fig. 8–38). (a jny/2e4( mg,zdvrue6 ) What 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 aroundulct) 8d ej;ub0akh x y(t6,+cm+sa n3 the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin6mhrl/a2 abfv9 21 nlit
(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 axis2 615dv7pxvyjo 0 lbtgr-sb; x;fxq)w of rotation ixob xgf-x7s ; 0dl5wb trq;vpy1v )j62s to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about ineo z2f ju kl3o88moa-n4 doq+2i j*n9ts center from 130 rpm to 280 rpm in 4.0 s. Determi of24kjo+*3iq- aueomz nl o98d82njnne
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius m8nkc ya0 -pmg5h/ls1$R_1$ 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 Ay2 bv 66ehn4bhpollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cyliny2 b64vh6 nebhder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm 5 d*-nw,vhpjm in 220 s. Through how many revolutions did itpd* j,vh n-mw5 turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm o1x3v,frkk o+4b1 nnato 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a wh5*md1 ks l;kcwirling “human centrifuge,” which takes 1.015 ;d*cw kklms 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 rpm to 360x wrpzh: w:,syjuqt2 - )kyp1: rpm in 6.5 s. How fpr-) xwkzt sy2p:hq ,uyj 1w::ar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rikkgcm )v4*z4 2e n(pzevolutions z ) 4(nezpc2k mgk*i4vbefore 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 ca3sl0u r1 ggxyfe6rr0 w8 izq)9efp7*ir reduces its speed uniformly from 95km/h to 45km/h The tires have a diame01frrrx0 q7weu6 3f gzgypi se 8l9*)iter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions a ;u+pbsco cn4p/-.rjks the car reduces its speed uniformly from 95km/h to 45km/h The tires ha .pr;spjcu o- cb+/4nkve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbin)a *75x5-ma sfwb xxiig a hill. The pedals rotate in a circle of radius 17 c * 5 7xmxaw5s)afxb-iim.
(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.7 e 8)nce y ilf4;.bqeh.xlol end of a door 74 cm wide. What is the magnitude of the torqlyn ei fe 8qhx7)l.4obc;e.l.ue if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of 3twaouw*pg6gc, a5dxo8;9 gi the wheel shown in Fig. 8–39. Assume that a frictioa*p9ww8u g,cx6og odg ;t5a3in torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a*q 1pfgy. as*f4n- rotttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this systemr.tonayp- f*g*q4s f1.

  The bolts on the cylinder head of an engin/1( 8y qfdcpikfimwmo3 md8qu j4k-):e require tightening to a torque of 38 3f1 kw88(i / :mcq fuoyjmd)-4kdipqm$m \cdot N$ 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 radp1lte d*txv9x4jdqx475ia q+ius 0.648 m when the axis of rotatiotviq q* 5dx4x 7x4t9ea1+djlpn is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rn71 no2oh8*zsv6ltjsuvx)n1t4 makec 6w7- bim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).b71 sms2c6 et1oa*o8ulvkv7 xhnznw-4t 6nj)    $kg \cdot m^2$

  A small 650-gram ball on the end of a thf/ btq3 fuy3/lin, light rod is rotated in a horizontal circle obtqfuy3f l 3//f radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating av7vk o+ (e.lv;d 2c-bfnhfvxa))t bo+t constant angular speed (Fig. 8–42). The friction force(aool ++fhk-vcvfnd2 )e. t v)b7;vxb 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 point objects shown in Fig. 8–4 r : 6p-hm4/9gxpzkife3 aboi9m /6g: fp-rhkzexp 4ut
(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 atoms q0 m+dl;bwgj, +yv m+ 3-llvhlwhose 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 spinn. r6c9jjdem hh5 ryt,*ing at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has aj yjd ,m.9hr6tc5hr*e 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 cylie-mro p9p)4jx( 9zzvpnder with a radius of 8.50 cm and a mass of 0.580 kg. Calcu9pp moj9 pz zvr-xe)(4late
(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, acceleratbomy tdh) 02tri;1 .kcing 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-round an f g+8kbm8swg8* lwy+mml9j)a+ty 6eyd 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 other on the edge. Calculate the torqug+)9y6 lgw8m8*ya8tw+f mb emsl jk+ye required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotati9ce xp h/vnix5bwf-n p49611v.v t chdng at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque o6 x 5h9p1bx 9.n1tivvh vcn/wd-ef pc4f 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 ball at md-2clmpzy 1 4z8oid/xo4cc 17 $m/s^2$ 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 va9b+ k0g yn;xlzs)v0 by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fi)y x09 kag0vbn;slv+ zg. 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 thin rod0x(pnjx9h dab 65k7kn )sxn:wlx+lr 9 , as shown in Fig. 8–46. (x+njbx sdl kp)n0xlh 7x56:rk9n9wa
(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 ow1jnc t34f/d 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 full turns (re )bbxe *wpiec/7kz)l-y 6id0cvolutions) and releases it at a speed of xiz6) y ekep/*0 cbd)bwc7-il28 $m/s$ 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 g/./.i7v clct 7r:wq 5iyb ux7+zgt fhof 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/: lg8*a 1eckwis gg-k 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 ry7z0fr q8 le*nf/a 9pgadius 9.0 cm rolls without slipping down a lanefr/ p znqy89*e 0f7agl at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the E, wyf 3jf.2ojdarth with respect to the Sun as the sum of tf,jf.2jdw oy3 wo 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 1(e)cm8wl-hu5u (b e 0wb) roxu640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotahe8)wl (buruexb) m 0o -wu(5ction rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from22;bfdpp0 l/6tyl 4 vnik3ce7 d(f nwv rest and rolls without slipping down a 30/v2wb;(72ld4fd6 lefynvnt ic pp03k.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 25gh7 rw erg,o pw4v:wvertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the groe7 wg 5ghr2:rw4p,vwound? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a tpfzdh6k 1,+ rshin string1fh +s,kzrdp 6 in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylingkt.nmhk5t. 0 drical grinding wheel of radiu kt. .nktmh50gs 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 rota;z.o 1by wemn:ting at a rate of 1.3rev/s If he raises his arms to a hoo:1ymw . n;bezrizontal 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 ql:)wg ajw)i+xmu p. dt.t 0y7moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. u)g:t+a7twlj).. d x0i mqpywIf 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 fi;kn7 i7uwc/k6v v*p rotation rate from an initial rate of 1.0 rev every 2.0 s to a67pv;*7vifwk/ c unki final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertic6 n01g xwf+ ep8fuf:jqal 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 thr1p6fg eqjf x+8w0:n fuows 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 figure skater spinningbp ac;wz6 ;2zj 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 lw as; iq6zop;3w8p6 c:zly.k momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of iner0;9e+ cz7+wj/vzgzcz -tvj sjtia I is dropped onto an identical disk rotating at angular spgj zzjczv9vs+7/wtj0ez+ ;c -eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsy,r(wp/xkyc 2 m:)ea v at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-tsf98y,k.uj rn i36qkas*p.e round platform of radius 3.0 m and moment of inertia 9skjk.e,3 8np u6q traf i*y9.s20 $kg \cdot m^2$ 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-roundiyrw/nnzjg b 7 q-ctqd h 0+zm15v2+wfw*7i+m is rotating freely with an angular velocity of 0.8 z *5tq iniznwv+mm rj7+/dg1-h+wc02wy7 fbq$rad/s$ 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 dwar)d;+jt tviiv*f, losing about half its mass in the process, and winding up with a radiuvt+ *jviid)t; s 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 iop;(9tkt-6 budd pc3s n 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 iusoly dglemmx3./3( rmd**gsr+;6l $ 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, v7nan, klk 521kly f,mhuw1gk1w(oz 2 that can rotate freely without friction. z w5(12v,alygw2,h1lmu k1 nok7fknk The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diam.mj: xpu gm uls38,yx5eter merry-go-round turntable that is mounted on frictionless x myjgxs.pl5:3 u,um8 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 thmu x-qfj(tcd3w6 hz10 e ground 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, holding onto it, Fig. 8–50. The spool rolls behind the person wif3cxwz6h10(-t qmj duthout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Ear c;fgr5p0aifm* 22xn 04wmlub th such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treawai2 puc lm0x n2*5 fb4f;rm0gt the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate0 .x6tnup azd2s from 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 08/ gjz33o qr .*fzwaoi.20 m acquires a rotational r3jz/rz iqfg 8.ow3oa *ate 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 soliv3c ye45xfowb9 .8q uad cylindrical 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 0.01oy85e9 3buwa. xq f4vc0 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 thesp1 9q2- hbvnqewq792:r p dce 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 gre65vso; uu nk8mat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carri56nvu um;os k8es off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored inte/,n +cg4:nsm,qneqd4 )cz:cghip 5 a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 khc), +nqn,nt4 zg :c5g4eci/pemd:qsm 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 ;af.2lyqrk -o0qgp9tr 4*n ez$H_2O$ 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 hor:.2iicy7oeuoa z6gs3pz 5 l+tizontal 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 ignorp t-5 n4ghat,ee friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the ang e5-4ttph, 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 the floor, 8gd rdo.* ogxd7eq/*kand we want to exert a horizontal force F at its axle so tdko8ger7oqx*dg/.* d hat it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed va/abhnym- )s;zn)6 km -+zeyq =4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and ) hb key;nn)-mzy- a6m+z/asqcycle 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 rw60xunt iao j70 t) z:nd;hvt0ock 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 the torque required to achieve this feat, and where does the torque come fromi6 o0ttv 7n n0:j)zahdxt;wu0 ?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her5t20 og1 -daned+qvo w arms as thin rods, making reasonable estimates nt+ 1ag- 0 w2vodeq5dofor the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly 8v9*;b7egyyl qxqf3e(i;u f ;(m jgckwhich consists of two cylindrical plates, of masguqfq bk vy;x g;lei* 7;( f(myje3c89s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rou5ry2h ) wx7axjgh 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 highesthxrj 27)5aywx point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do u:w w; h7dr8o -oqr:wctx9l6cnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a5 n5nh(+(c budil(bqgs the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was t(uqd5+igcb(5(nh nl b he 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