A bicycle odometer (which measures distance traveled) is atta+y3bo2h km6;wc+j w qdched near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-bmww26+hdjoc +q; k3yinch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim ha,o 95a .h:a9bl x.+v hkh3mkgulsnlb 0ve radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude o vbb0glhk+9 ,oa: hnl .xu5.9malhk3sf either component of linear acceleration change?

Could a nonrigid body be described blsqo0t(hkk/0m)2sm j4hn u4 ny a single value of the angular velocity $\omega$ Explain.

Can a small force ever exi5q2jhx 2lf82tul5 ka ert 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 0s i5/w vjy23 ammjq-mdo a sit-up with your hands behind your head than when 0a5qmmw 3m/si2j jy-vyour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sevfr qf293wd os6en sprockets at the rear wheel and three at the pedal 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 fronsdr2 f93wqo6ft sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with o 7zw6+lsnuhqq:8 w d2flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamiuqlh 8o d wqs2+76nw:zcs, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam- )wrog*2p/p 3w osqqrlx *g(xzxr)a 4njz4q*?

If the net force on a system is son4j vg6s62s zero, is the net torque also zero? If the net torque on ag6v2n4o 6jsss system is zero, is the net force zero?

Two inclines have the same height but make different angles with thkwbw 6q:;9cl *v sbll2 : d6ftnasud.9e horizontal. The same steel ball is rolled down each incline.2bldld:9laf wq:k 6ts9*cb w;6nus.v 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 incline. k qbt d*(nk3) zuh2,pvu91;b,ljhiahOne 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? Wh1unq h(kbjaiub;2p v3,, *lh9 tz)kdhich has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and th0is1 gk1/)hrzvtc v4,mb g e4qe same mass. They start from rest 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 rot1vgtbg)vim c,h1s e4 k40/rqzational KE?

We claim that momentum and angular momentum are conserved. Yet most moving o gsch.b/syf- 8rv qwaqgs1 9rh;c)g8+r rotating objects evrs8;f9as yqgs/ g)cc 1hq bwhr-g. 8+ventually slow down and stop. Explain.

If there were a great migrati uri7b3 iu/bk6m* m0s5wkv :wion of people toward the Earth’s equator, how would this affect the length of the dayikmmu0 i3bkr7:isvwb*/6 u5 w?

Can the diver of Fig. 8b tvp5lca;))t –29 do a somersault without having any initial rotation when she leaves thelc;5ba p tv)t) board?

The moment of inertia of a rotating solid disk abouk wiql8et68;u t an axis through its center oft8;i l6euwqk8 mass 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 hr6,pho+)d)wch dv: 61xpic9 -n jdvv rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, 1r)hvd :dvc) 6o6,ivwcxp+dh -9j h pndecrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is ho/my*w7k-d b qcllow and the other is solid. Describe an experiment to deter kmbd-*y/q w7cmine which is which.

The angular velocity of a wheels of9yceji6xf6li 83zbg9e 30 rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top 6y o3lfb 0zj 9g xec3fe6ii9s8of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge w9alwdqj:u5bj,8( g m of a large freely rotating turntable. What happens if you walk towarj:(w8q 95ba mg,wdlj ud the center?

A shortstop may leap into the air to catch a ball4iawu j76vmfpj. o2a) and throw it quickly. As he throws the ball, the upper part of his body rotat)7vf4j.o6i p awja 2umes. 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,uqb1r*8cgo fqmph( em+r;r;5o +/p tu discuss why a helicopter must have more than one *1f (5 u/epct m+r qhruqrmgo;bp+o8; rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following a.-0 l5x do ptrsk5 nhsj/xm:.l8x fab:ngles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculv;4,vhu y8rek ate, using the information inside the Front Cover, the avr hk yv,e;84ungular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 k,d(7nvq sqg z-s9 6f7tor fak4m from Earth. The beam diverges at an anq7 vsat fnks4g,r-o 7qz( 6fd9gle $\theta$ (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. When the motor z8;zr umof:0) i;gsho is turned off d mo ouz08r:zhs;)g;ifuring operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes 7x d6momhj6m (15.0 revolutions, what is its diamhj6d 7om 6mm(xeter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions1s;,aidca5 c s do the wheels make,sa1;5 cic asd?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcub ;hy 6kmza+- m)jirl3late its angular velocit+);klr-b3 yaimzmjh 6y 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 (j:sj (1w oiz5fFig. 8–38). (a) What is the linear siw:s (j1foj5zpeed 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 th yfrfcg*g3u+n)ui m67e Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o wz*1*sa v4vcjf 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 aj,1 g t6v cwr62sjegup3hz:w 9 centrifuge rotate if a particle 7.0 cm from the axis of rotv6g:w6,jwe 9zt 3chgur1sp2 jation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cenp jdl:jn8/ z7juo0r jrw kt ,.wf)45lxter from 130 rpm to 280 rpm in 4.0 s. Detkn80 p zlrwdr / ot.wj)lx:jf,57j uj4ermine
(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 radiubx.7r8u 0fof)2 .ae i l,nvojjs $R_1$ 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 pu dzfoi c.a3oxe1t 3frf(e8/m* t 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 cylinder with a diame1c *oe/fz(dma3f8ex.3 ftoirter 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 in 220 s. Thri2..kyt2 tlw n0py/ewvzw ou;b-.k4j ough how many revolut;b2k yyz.j.p nl/w-ue. 2i4w wv0ttokions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C;qegvcu/qp6 ,alculate
(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 teoq;y*. 0mqfdlf 8cu *he stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final sped0m ; *ouqqfc.ye8fl *ed.
(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 unifopi3jq6f1nes3bz y4 d.tyk)yyec1./ormly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hdnctey33/1yky oi 1jfp6 4es.bzy )q. ave traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev- mvl un:e 3k5)s(qcfv/min It turns 1500 revolution(kn f:svl)mce3q 5vu- s 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 a;vt y0kx;wo 2ah i+ a5i+v2hars the car reduces its speed uniformly from 95km/h to 45km/h The tires haka o0;ht2;vwxrh+ iay +2i5avve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fnd v)d,5u8h1y; aj): xvuexlfrom 95km/h to 45km/h , 8fuvx:e5 1ayd) dujvnl hx);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 b.3-juqvh 15cg )hkbd uike puts all her weight on each pedal when climbing a hill. The pedals)3jh-gk.hu 15cqb u dv rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force tkt 7ffb394euinj9t:2 +qunfof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exe9kfn2qi3feu+f tt974jb: tnurted
(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 thmmb) .3cost 5snr /fjv9p 0ys9e wheel shown in Fig. 8–39. Assume that a friction tosotrpf9. 35vjm) y cn9sbm0/srque 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 ro9cje+m c8tg.4fd yl4 kd which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direct cmf+t4l89k egy4 jdc.ion of the net torque on this system.

  The bolts on the cylindem0ht9c- jo4f-r0stq-katx2 m r head of an engine require tightening to a torque of 380s trqhaxo --k4j0cm2tft9m - $m \cdot N$ 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 10.8zp/e ;y4 a9e/celznf,w2m2qd -kg sphere of radius 0.648 m when the axis of rotation is throa2/lezp;z cdy 9,mqe4/2e wnf ugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rk6i yq, rff12vvpd+y* im and tire have a combined mass of 1.25 kg. The mass of the huy1rfvp*,qf y +v62ikdb can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of gud nh:,**eg(;ce5(s3 p m nnghbju5da thin, light rod is rotated in a horizontal circle of radius n( m:,dggcp5;uh* e3gnbs h *n5uj(de1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant nb c(;l8c5 i- x:dydr8r-nivt angular speed (Fig. 8–42). The friction force : lid idnc;y8 r(c8rn5-x-btvbetween 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 showwl4qu 1cfd8+l n in Fig. 8–43 abou lq4wc+l1fd8u t
(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 whose total massh:.x)gh8w dvyaa; 8 na is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corre 2qbby6:luo)e7/ o6 dg4dreuffm5 i:fct rate, engineers fire four tangential rockets as shown in Fig. 8–44. I:r6oeif)f yqo 4ugleb2 : 5dm/6fbdu7f 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 cyb3 yvo9y q:mw 3rqzc)1linder with a radius of 8.50 cm and a mass of 0.580 kg. Calcuv3m3y9 cr)zq1bw q yo:late
(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, accele,a ,xopw .*0yrjmw7g v) atyk:rating 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-g3bv + qr;hhl-/j4e yelo-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate thq l4lh v+;hj yere-b/3e 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 i3. r y)/6glws,uswg)j. tegses eventually brought .ewu wtl6yse,gs) )g.3 /sgjr uniformly 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.v1bnqwlp* /z 8 +gzu2o6-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 bv2i j64 4 )d0oxeln+4mvbu wesall is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accele0b4 i4dvlwo)ee2n4j x+um 6vs rated 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, as shown in Fig. 8lpc9oxj- h4p*n06 ui75 liy zm–46. yh iu97jl46lpoxzimn c 50*-p
(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 tw,a g 6 m0* qvski(:ezlhu*gbh+o 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 withh/c+jx4a5u5y lxq , omin four full turns (revolutions) ao5l4 xac+ /5 ,hyjuxmqnd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertifqk9 8kf9 :rar,b+qojnm+-;d emnhz .a 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 deo1l tcg8uon / *6pzy4:si*vtp velops a 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,4pb53is1 r.shqvo:u :c.lms 0yl huav*w7ta lipping down a lane at so wpuis m::l 5.av.70a4hyq3t vhs*u,c1 brl 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofu7y3iax );zev the Earth with respect to the Sun as the sum of tzaxu 7e;v yi)3wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. H5;gfh ;t - t7gkbf1nwaf )b.aeow much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Asa-;e;f )agg7t. wbfhft5kbn 1sume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest h-chg,hed04 ; (yyx u;qxmg:e and rolls without slipping down a 300umy-;e d(xeg,g : hqxychh;4 .0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically z :km d7y4tinch - v*jvln+3v-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-hn c*vt -4+d7 3lv:ykn mvizj ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mo6eknwr// nwp4qm z,s x7+0oplmentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an w nnmqel7+owkp64z , /xsp0r/ angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m4-3appqejn (d +k-dtromentum of a 2.8-kg uniform cylindrical grinding wheel of rjrpnp-ta 3+ d-(4qek dadius 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 pla(z:qh:tij2eek2 k agy*,f9 yv tform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of 9ky:2 j(eaig:q z*tvf 2eh, kyrotation 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. 8–294 2o2-kzqe8x3 zbbr fp) can reduce her moment of inertia by a factor of about 3.5 when chang4zq32be-kb8 ro 2zxfp 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 f73j5i 3 4 m +5kcz.vuynmmtbxorom an initial rate of 1.0 rev every 2.0 3 mtmyb j5vo75i+u mzkcn34x.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 vertical axis through izxsna )* i fof,t;ouif)- a0y5ts center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel afte)-toiu0 fo*nz x a ;fi5)ysf,ar the clay sticks to it?    $rev/s$

  (a) What is the angular momentu, 8, vhu0ejt3i csgr,fm 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 angulafks y z*.bv),x5(auu0lli* mnr 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 inertia I is dropsn5ufcxr16iqf1 4s u x* 3fux:ped onto an identical disk rot xxcr 3n511ffu:fi6us x4*squ ating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 /3;u b8uaoruhgn6v 9 v$rev/s$ 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-round pl7g 9:su.egna fatform of radius 3.0 m and moment of inert9gfea:ug 7sn. ia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an ang p *h0leafl8.5nnfh co-;jgf7ular velocity of 0.8 f 5jhlpag.e; c8n-ho7ffn0l*$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about .zh0x 8y2rre lhalf its mass in the process.x8er z0hyl 2r, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds jvz;ngf i*r;qb f( ,.uin 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 ;1gjxd, o q m/qk ljfqm3q62o8$ 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, c83mtp6dcs+ldx*t+ e8 ajiov/9s; bw that can rotate freely without friction. The moment of inertia of the person plus tho+db x389ed6;s+tcs ilmc *j /vw 8pate 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 persvfte4j. k790pwa:f f eon stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment oetj pfv9fe. w40akf:7 f 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:pcyj;6v p2sh)mkz gdq7qh/ 0 the 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. Th ;6 vqj7:dhps0zkg2c hq )py/me spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always t;26f gn( 7ok d1btuqvfaces 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, treat the Moon as a particle 6o; q(v tdt n21g7kbuforbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates ,/-c)zt uo: di ql1cr yfp.j9gfrom 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 radboy m) r,s(ew8ius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcul ym(s8,bo) wreate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eace)1xp32n- dwistc 1ak h 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 cs)3 p-1k t 1n2xadewiconservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheelalrdaws 6d*i7; qxe:xs07fg ) ($\omega_R$) 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 Su,klyw wh,jfmwz 6u *mr 4u.l65n, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution everf r 6w4uzu kj ymw*wm,,65.hlly 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 carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lolc-k ;a1 dv;bow-pollution automobile 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 able to travel 350 km withou;cobd k; v1l-at 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 l0yhyl* . t9gqc/ p79*onrk zx$H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal suuba .mo1rbic8w j +(y*rface 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 fremzb )bov a ,ycm)f5xw*8) b.seely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center*,asmbzm)fv. xb e)5wy8 )ocb of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertica2h lyl lokp.p19z-wm/lly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which 92lw .h yp1p/ loml-zkit rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelindofj.9s)aber ;7kvwg4 bcu 5 ,g with speed v=4.2m/s on a flat road is making a turn with a rav ofa9sbukeg.rc)b7 4 j, ;w5ddius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a slh.ol(wkr3:/mx1eccs ing 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 achis: m1lx .cwkce o(rh/3eve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms d,czg86) m qu,rn4ve1m2a bmfas thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a zcv ,mef4m8aqb 6grum d)n12,spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,nyl) -w w 1;hoq9h 4: +xawws/btg;gpy of msg;:4owb1tn- h;y/pl+ x)hwa wywg q 9ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alongrf;e+7gr6-wz -fj d cq the looped rough track of Fig. 8–58. What ;rrjfgw-+cq d7e6zf- 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 leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dw. rs1y t0ip+ko not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed sx 4vmc1bx r(8uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the anguxv 4s8(r1mbxclar 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