A bicycle odometer (which measures distance traveled) is attached near the wheel3/ vhzyk o/+bm hub and is designed fom + /hv/ozky3br 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. *x,tid:dk 4l bDoes a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, d4l*k,xid b td:oes 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 deslgzp1 . n2pxeqkh2-nv*p l-b4cribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever ex++f 7mym sc0ap 7atvi- 0b3tjsert a greater 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 wit ,pwo gqvbc.4tqyki;-f5) 9 ceif,r+q h your hands behind your head than when your arms are stretched out in front of you? A diagram mq e-4r,5 bgikqcq9,;yvfo c.i+ tfw p)ay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pyxvv z 5:9(ry8-4uub6ray p xwedal cranks. 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 o68 a:wbxuvux(-v9yyr5yprz4r a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs wi55hc- cvd0fqv th flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain whydh fv5c-0cq5v this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bez833bpmim(l uyghq8/am?

If the net force on a system is zero, is the net tor yi e(c;;otn;wque also zero? If the net torque on a si;;noce yw(; tystem is zero, is the net force zero?

Two inclines have the same height butbohyvb+:qk5 zsz9ka,r (u1o/ make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline willobrysz(9b:z1 k,uv k/5a+hq o the speed of the ball at the bottom be greater? Explain.

Two solid spheres sihh fo9a2 e4n-vq-qg;c aj 4d0imultaneously start rolling (from 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 0;v4 canih4og- f de -j9haq2qhas the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same raku1ajncifg33/*d 8ut dius and the same mass. They start from rest at the top of an incline. Which reaches the 8/1tjkgfu3na d i c*3ubottom 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. Yetbp;weo3s0k c f ):hhc4 most moving or rotating objects eventually slow down f4 cwpc o;):3e0hkbsh and stop. Explain.

If there were a great migration of people toward.+4*rnuwr caw f 05jkz the Earth’s equator, how would this affect thn4 wa 50z jk*f+rrucw.e length of the day?

Can the diver of Fig. 8–29 do a somersault without having an 6qt30zsf t.ruy initial rotation when she leaveutr6sfq3. t 0zs the board?

The moment of inertia of a rotating solid d2m9+ (yuzdjglso;p, q isk about an axis through its center of mass is gzl, j+o qys;up2m( d9$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass -ui:aflh0 )ub6/ ma fjin each outstretched hand. If you suddenly drop the masses, will your ani b f)6a ujhf/uma0:l-gular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevero 8fl/z(i1a9 wb .guua, one is hollow and the other is solid. Describe an experiment to determine which b wg/a.(iazu198 uofl is which.

The angular velocity of a wheel rotating on a horizo*4fki h da+a+dntal 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, dea4+k+dhfd a*i scribe 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 large freely rotating s43ou zch )dw w6/a(rbturntable. What hapcur/h6s o 3a4z( bdww)pens if you walk toward the center?

A shortstop may leap into 1o9 :w4 tc6x z -ebt)wii+ioed9f,mg hthe air to catch a ball and throw it quickly. As he throws f,1+tom-o 6: bwhii exw9ct4zig d)9ethe 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). Explain.

On the basis of the law of conservation of angular momentum, discuss why jfoyq5 oa,6x831e vu(tcdv )w a helicopter must have more than one rotor (oef5ct6jyq u3x wod81 v,a (v)or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radian-9 ugfn.t-z)k jpq, wos: (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, z okz5syb+)l, using the information inside th)5obzsz k,y l+e 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 directcrw r*cn-b6 qnv.-n*xo2 q5kqed at the Moon, 380,000 km from Earth. The beam diverges at qr* qr c6-nnxq5n-w2kcb o*.van 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 blendus9 iqaok0t81h;ir0oj :k y,ier rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? 0au:oh0tyi9;krkoijq , i8s1    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If d obmb; 8ue1a,the ball makes 15.0 revolutions, what i81 bob ,uame;ds its diameter?    m

  A bicycle with tires 68 cm in diame o37n 08 lzbq3kks9g-erjqm5jter travels 8.0 km. How many revolutions do the wheels mak s ze8l073jj9q3r5 n gbm-qokke?    $rev$

  (a) A grinding wheel 0.35 m in diameter ro4*zslm.cecn pb4 4zl;tates at 2500 rpm. Calculate its angulacz4m le4slb .pnc z;*4r velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes onez *muq8 qjyregd:cr69gbf-; : complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from theq:jymr68:*; fz rd9c- bgqu ge center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velq ht.7e jk):knocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe8+l1 dlq+dsxoed 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 cml* cacn7bkq/:sl 5tj) from the axis of rotation is to experience an acceleration of 100ql5 lns: kta /*bc7c)j,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its c:.yoq nn(ii 9tenter from 130 rpm to 280 r( on 9iqt.ni:ypm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius xib ,4g -ykrk4$R_1$ 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 spacecraft pute) 3shwdis2: vad/tk3 themselves into a slow rotation to distribute the Sun’s ened:s iw hst3)3eav2/dk rgy 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 rest to 15,01oy1 0 vh4tbf h xbxoweu8m(9/8ndb9k00 rpm in 220 s. Through how many revoy d uf8xvbtweb1h/b994nx1k o8( h0omlutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calha myl*cj shgncfqn *4t 1hzq(j0( 87i.: kk5rculate
(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 testedf mmwz50c8 5dqh; 3yfj for the stresses of flying highspeed jets in a whirling “human centrifuge,” w5 fcw y;m3z850h mqdjfhich 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 unpna6x/(m8b7qc7xl from 240 rpm to 360 rpm in 6.5 s. How fa/ al8c7x(nnx pb7u6 mqr 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 revowxi1n8rsvk;c,hc.yt.vk( 8 clu8hv1c.ckxtci wn8;r.s ( vky,tions before 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 5 dmx(z r5t3-lj/g:vi)f gwipits speed uniformly from 95km/h to 45km/h The tires have a diameter orzjiw -( v/gpi)lmxf3d 5gt 5:f 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 65xxd + z1iphyzqisp0yd(0v6 q/w):(e y revolutions as the car reduces its speed uniformly from 95km/h to 4 d(/yxpswy6qe 1i:dqy+zh(i v zpx)005km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight ovj;e e6ndp3v9j:l, j rm,j6 lrn each pedal when climbing a hill. The pedals rotate in a circle o6ldv jnle6ejj3,rmp 9:,jr;vf 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 of a door 74 cm wi 7uvqy t+*(kmp ,ln,u(i/quk yde. What is the magnitude of the tork(/vpu,qmyu*n,l ity 7uqk+( que 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 the wheel showp v9p68dfp x(l27ipxs n in Fig. 8–39. Assume that a p isvp x6 79xpf8pd2(lfriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of massx,l 23z6sq zxf m, are attached 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 directx,2fxzqs6l 3z ion of the net torque on this system.

  The bolts on the cylinderm,j+(mdw6j zm head of an engine require tightening to a torque of 38mz d+ m,(mjwj6 $m \cdot N$ 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 ofxi o;ad.om4 z) inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centemo)i z4oa d.x;r.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ri,vsx3a))9c tekf dl/y m and tire have a combined mass of 1.25 kg. The masstsf d)),3y c/k9lexv a of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ligtn2ispoq- :qfy 2 yi)osc (,e2ht rod is rotated in a horizontal circle of rad,( 2oyt)i2so2-cqsep:i q nyf ius 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 rotn f9go0d8*c dqating at constant angular speed (Fig. 8–42). To8d d*9f0gqn che 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 point object 1 xfx h4gz;*b*a aeikwoy82/es shown in Fig. 8–43 about4o*hexzg28afb a*ek/w ;ix 1 y
(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 tvmap8h ;2z( wi wj6)p0qbbju.wo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sateixb/k5u:;5xb ah ese8 llite spinning at the correct rate, engineers fire four tangential rockets as sx;hb ix8uk5 ea/bse 5:hown 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 qlh)lu3z .*ma +a*ckuof 8.50 cm and a mass of 0.580 kg. Ca*lh uazq am+ck u.)3*llculate
(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 2e5i3 fyw ltm34l,r zi8.p nduto 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 anhjjrem5r-)an)i* dgh; i3tc s6ylw9 0d 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 torque required to produce the acceleration, neglecting frictional rn 6ms daciwj9))i5 y 0lj-e g*h3;rthtorque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brougsrdrf q-p) 4z)ksg;l 4ht uniformlyqsg4ksrpzr) 4; - ldf) to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball dhaf+:p/x ntkl8gg 4o) si4,cat 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 iszs 21c /e mgeuho zw7/ixodk z14r*/rm):o0ca 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 acc*rgeco:co 2)0s/zu zw od k/m1a /7e4x1hrz imelerated 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 rod, mpvsd4 /:d fecwq+d.esey-/32: w r ghas shown in Fig. 8–46vqf d+: w sw.d/y/:e4cse 3e-hdp2m gr.
(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 masses,ihsr qf -w 71e zcm tt:o+e;c0bhg0/n; $m_1$ 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 restg 081gly- mgrd7v*:ag gwlea+ within four full turns (revolutions)wg7d-g*g:lrg 8l1avgmy0a+ e and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hae 2woxw:zwc)gd 0b9o 5s 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 ( ns1gsp1l 6dhim 16pma torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without)s 2v(lm.;j bkde9nmd slipping down a lane at 3.3 m)(2;sj ebdnlv.d 9mk$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suw;d dayrw;u jl.8c 5;vn as the sum of two rddu jc; vawyw.;58; lterms,
(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 est0vmkl:o2- cv 0 x2s 4g9 p.2d(ze:yubfunfa radius of 7.50 m. How much net work is required to accelerate it from rest m(f xgsb v:eofk-d4u20tpvl.0n:2 u y 2zesc9to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stab.lr o4xcw; kfgq:;lpm,eiy55;i w;irts from rest and rolls without slipping 5ig;l:pwqfi;ry,c. b;eo4;5 m xw kildown 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 balkz36jc46qr/ ke +o oyjc64eifo 64updanced vertically on its tip. It starts to fall and its lower end o46jp6oe4co6c/yjf 6 +k4uik d 3q rzedoes not slip. What 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 0.7qe gyfa4-1u 8bx/ 5gjcjcg8n210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.4 ynaejgbg5-887cg c/ jx4 fu1q$rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whq l.vwfz df0fp1;q)k 7a16ugdeel of radius 18 cm when rotating at 1500 rf1dg7. w v 0a)1 qqzukf;d6lpfpm?    $kg \cdot m^2$
(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 ra8rov1ho, 62 ods;q uerte of 1.3rev/s If he raisoruhr2 6e8;o sqvo,d1es his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in7 g(.ux wng;ifn6jgq, Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when chang;gw6ijfngxn, q.7g u(ing from 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 rotation rate from an ininp3o /i2l;p - , 7b5cnyfiswaxtial rate of 1.0 rev ever bsowa/-;p 7i 2lxycip3n,f5 ny 2.0 s to a 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 verti07sjfqa 3zrczq sc::5cal axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approxs 3q:czf rq7zc0j5:asimately 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 momenu )sge-a3 yw5e)5ucjutum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angula.r2 vku:)j-d qzs os 0c0z) tcbkyvb0-r 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 di6ej2 qgu )0allsk of moment of inertia I is dropped onto an identical disk rotating at angular speedu) l0 ql2j6aeg $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atykxqf+3m0l -si dooyf+ ; .2ip 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 +dy3 .smqp4tia rotating merry-go-round platform of radius 3.0 m and moment o ismdp3 q.t+4yf 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 f:42xjdtc lpu6/oii .i reely with an angular velocity of 0jci ox:l.2p6ti4u/ di.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 eventuall8zzk uzg 255m,zwd6f cy collapses into a white 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 gzc z2w5d m,86kzu5fz 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 excess ofby827d0 iqf8w2) q dkf :bocyw-aiq:p 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 p n* y)y w-o/wn r s9,ln69gd(q-f-whnvn(fan $ 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 rotate freely jr6 p2j*aijmr oz69 m ;kxer6-without friction. The moment of inertia of z6jp*6eo6m x ri9; m- rjrj2akthe person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-ro7 b4cak-lb -/a ii.g: slppis4 vch0/aund turntable that is c l 44a7apls bv: b. chpi0i-g-ai/k/smounted 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 ground with the end of c3;vde4-*; yex .qdo 04r0dtkumbmof 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 w.fvdt 0;dd 0qk mu3 yx;e 44mobc*ore-ithout slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always f wb0rchmham022 /: v5fmda ph-aces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axihb0-fava pmcw2rd :m5m 2h/h0 s) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest l+se el9;(a no8tew.bat 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 cylir*l73 9xvvwz;n.xxkl f0i3jvnder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate j3 l90wi3x nx *x;v.vz7vlfkr the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so, g*r4*o4/igyg qgu9u n1k4x ( bgayhdlid 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 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end o/4,ybg u xkgr*n g94*hodg14qagy u(if its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angu)/e tst0ab zp.lar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size o.0f6 hadz9bki;hic 9pf our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gra;b.a9pid9 zk ch0f i6hvitational 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 carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobil2 8c,mr+ f. e-2ersrbfe9ifq te is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be e.9 e r2rs,cefmf+r8-fb2 tiqable 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 anl0c6 vvb+,x ipsk0h5 aqb7 e -0zg7ehj $H_2O$ 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 rp-m c pgye7*csq.5c( yolling on a horizontal 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; d8 2lh49 rpgm4fk;cdgkta9r can pivot freely (i.e., we ignore friction) about a hinc29pdrra ld 4 8gm;tkhgfk4 ;9ge 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 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, and e115xwadkyy po ,ryn+(m6e4tkd2 c j7 we want to exert a horizontal force F at its axle so that it will climb a step yd42jayxn 7e1t+ y6mw(5erck1 dp,ko against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speef+y:2n/zep xmm -t., (ccw aced v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclistme/t , ccw-+x .ac(zmyfepn2: 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 sling /efp t.r61b amd*c;keof length 1.5 m and begins whirling the rock*d;m6e /1rp b ft.keca 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 armf*b6jg0x8dmg s as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinnjgf m b68*gx0ding skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consistum*pf /o:x*s 5p/s mnug pbr33s of two cylindrical plates, of masspbpsp3 nru3/5omgs*:/ x*ufm $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roll f1nbsp (h57 i13aavcv9ld 5:jq q,scvs 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 loop without leavspn:c5 j,iqvadl( 19v3s7bhc1q v5f aing the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuzq0v s+xdg ybj+1 0c7g m,frjb0*qi5ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speef6h iu5o vuq.+d uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each thvu+u 5. qofi6ire? $\approx$    $rad/s^2$(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