A bicycle odometer (which measures distance tu .7xug4l bn4iraveled) is attached near the wheel hub and bu4in7xu g4.lis designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant an 3.1ln nm8ksc/gf.t jd0 f*sw9fi+lgsekw,: xgular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incren0lgk.. f mgsc*1fse,/dxkwnliws:98+j3t f ases 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 valuev +xfrcx7/6lq 7 ljs -qu1c5fu of the angular velocity $\omega$ Explain.

Can a small force ever exert a greain3sjhpe+7kl* de 7v6 ter 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 difficui-o8 j ofzmlk a/ygyksr ;,w;mt391r,lt to do a sit-up with your hands behind your head than when -8 y;3ar,,yt9s;/okf zmwm jrg1oki lyour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel andsuarq b w9z3uv4b rs: g*m5(i sw)jf6;6jyq,h three at the pedal cranks. In which gea rwryq6u ;(z s36)sabbs5vq,w*j9jiguf m4h: r is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legj4.f ,/ s7:xugfyal bvs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributionbvg j4:ff.y xl/a,7 su of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) h nzzuld 0d)-u7+ y3cneg.;y ncarry a long, narrow beam?

If the net force on a syst6a4eks* pt nvwr(8xc -em is zero, is the net torque also zero? If the net torque on a system is zera-t6 8*kc4vr xnwsep( o, is the net force zero?

Two inclines have the same heightnlp6/dek;k)/say 4 x sji xlmr/:vy.8 but make different angles with the horizontal. The same steel ball is rl /l / i ks;y.4rvj6 x/)aekmp:sy8dxnolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollig*n3p/j q:jot ng (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 has the greater speed there? Which has the greater total kinetic energyj3:*nqgo/ jt p at the bottom?

A sphere and a cylinder have the nd/ozah(y*(nb/e3u z same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sp* (bznhnu/od3a e/z y(eed 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 moment uw ubdzn)qi027nn. /zum are conserved. Yet most moving or rotating objects eventually)bu nuq d2wzi z/n0.7n slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hoh6kex0mn* ) pjw would t*xjkn 0mph )e6his affect the length of the day?

Can the diver of Fig. 8– pfy.l;-9 s kmd:r/pit29 do a somersault without having any initial rotation whp:sk9 /d;.it lmryp -fen she leaves the board?

The moment of inertia of a rotating solid disk about an axis throulw*w5ewcuy-v xa 4r: .gh its center ra w-w u5w:vlcy4*.exof 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 sittin7w rihc:l0e7 g hifl4 l89cyx26f 9rbqg on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity g ihcyefh l7: 6ib9f 07xqrc2llr 4w89increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howet d81i)-,7ekhy *gwc5aor*2fvv hv oh ver, one is hollow and the other is solid. Det* vyowc)odh7h1 g-8vf,h*a2 i rvke5scribe an experiment to determine which is which.

The angular velocity ovz r )qb/3p lncbu)g5bn 6k*qbkiq0)/f a wheel 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 of the wheel. Is the angular speed in) b05 u)3l/n/zn bk)6iqp kcbgbvr*qq creasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turn4ao/aqycn.j++ua g. i table. What happens if you walk to/4o++.n gcuj i.aaayq ward the center?

A shortstop may leap into the air t 9 0lorqkk ako/1(1kvmo catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you k11 0k(krl o/m9okvq awill notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of condo,nzkk . a 5fhkonvf,d2.h-bh mi821servation of angular momentum, discuss why a helichodkn 8d2a omf 1,hn2hi-f, vk5bzk..opter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiano/q ,um0zp-tn x(hd i;s: (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. C:lx 5j9 gb:qlbg zv qnmet7: i85e-dg0alculate, using the information igglt:xqnem7l0v8zd iq9b 5e g -: 5j:bnside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direihu ;vjuegf9 4ynsj9x0 ,7w1jcted at the Moon, 380,000 km from Earth. The beam diverges at an angleeg uvxu7i ,jh9f0 ;nys 4w1j9j $\theta$ (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-ll jr)9z+4ki rq1aq h 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 d4r-laqjrhz +1ki 9l)qown?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If87 k. m4zley p3qdy .s/vz.ra/n7m qhc the ball makes 1v7yq 4k8crlz./ zp.a.dn 3h/eqsm 7my 5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many v de i6ns1mqg 98/t7fprevolutions do the wheel / sfi7968gpvdqm1tnes make?    $rev$

  (a) A grinding wheel 03s 3gmvb. no9j.35 m in diameter rotates at 2500 rpm. Calculate its angular velocity ismo3n 3.bv gj9n $rad/s$ $\omega$ =    $rad/sec$
(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 inj9la,t )ef-: 0agg xei 4.0 s (Fig. 8–38). (a) What is the linear speed of a child se:xaa g0,gt9 )fj-eiel ated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Su;joejy.qahh2*, 9 56hwp qwlf 3(x tenn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s y)/k-2xrfr)fgo**ai;rye empeed 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 48e o:tpke r5fcxmp,;7.0 cm from the axis of rotation is to experience tekxpep8: ,r o4mc;5fan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13 7az39 -bf/xyrezlun o+4a mw*0 rpm to 280 rpm in 4.0 s. Determin47y a9-+znxue3rbza m w*lf /oe
(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 (z(oi+g / 4ot+,ozsa+cgk evc$R_1$ 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, 6 nu nr)lnss2s(6b0tj astronauts aboard the Apollo 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 revolutiss2 nnl0b)tnj r 6us(6on 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 15,000 rpm in 220 s.ovs p/(zro --t Through how many revolutions did it -v ozt- so/(rpturn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.d lf -mk5-b2jt poc.6e5 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 stres*2 dq6hrrzmyblr-,x 0s( nw6qses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachin hq* mwd2brrx0-sn6,r6yqzl(g 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 accelmin yzfvig9 k,w8,8twi0*i h 1erates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? nv8,k1 *z fhw , wimi9gi08tyi   m

  A cooling fan is turned offy*;jh3x3ouf7 qx5wba 8rgi,l 3y. pna when it is running at 850rev/min It turns 1500 revol .8nha,xluyb;xf3y3jo i5a7 qp* w3r gutions 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 ca4*, v4fpokmuir reduces its speed uniformly from 95km/h to 45km4 u4 *oi,kpfmv/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

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fu ( ufak9jkt):rom 95km/htja k) uufk(9: to 45km/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 bikeg9t77awbjzdc,2mp6 /ie tsb1 puts all her weight on each pedal when climbing a hil1bd 67 gbttz wjic /p9sae2m,7l. The pedals 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 of 55 N on the end of8q0dv g) /wbdjyol/d- a door 74 cm wide. What is the magnitude of the torque if the force is ex dvdq/obj8w/d 0l-)gy erted
(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 tor1pjrp0w +44myqwjyhz*+laqt2 d : 5ll que about the axle of the wheel shown in Fig. 8–39. Assume that a frictiow2+*yq h m dj0l+ly15:4z4jtw rppaql n torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends4qvyn-dr 0ke9 0kv pf2lwp73x 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 t fpwyvrv 2l -nx d090p47kke3qhis system.

  The bolts on the cylinder head of an engine ng( nkd- es*+orequire tightening to a torque of 38-+ (ds*ognnk e $m \cdot N$ 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 1qyv1l3 2dzm;r 0.8-kg sphere of radius 0.648 m when the axis of rotation is throug3m d1zq2;y lrvh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of aqulsm-bxbyy0 5:0e7l qq-0ew/u tkc3 bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mmuqcybq 5b- 00:luyk weex3-0qts 7/lass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light ,nbhf1fmuo;2 yd .b3g rod is rotated in a horizontal circle oyb h b;1d ffnuog32m,.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 rotativzdh n-uc5/u ;ng at constant angular speed (Fig. 8–42). The friction force;nu5-zhdvc / u 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 mtt8gp t5*xg h9 6j)5orh9vxuobjects shown in Fig. 8–43 abv5xu 58t)* gx96h t9mtjohgr pout
(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 *4 nwy8 ybizm9of two 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 satellite spinning at the correct rate, engi * 8xncgv;si, uzz/do,neers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m,,/vziun; zo *d,g x8sc 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 ofox/;ggram ,:(p j vul8 8.50 cm and a mass of 0.580 kg. Calculat8uapg x( jo; :/v,lrmge
(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, am rnleu3 7or8w a:gnqw cj.. 2gh)5fh2ccelerating 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 mkq bl-ql7w 5v0erry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge.bq5v l 7wlqk-0 Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating a)ch hrep)4d; kt 10,300 rpm is shut off and is eventually brought uniformly to rest by a friche dkc4);h )prtional 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. 8–4a m8vucv;o bv0t an(3;5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely b ,bd,hsxw* m/fs6zjm/y 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 sbwx,jm/fhs , dz6*m/elerated 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 ro/cnxks3iu 4. -zcbs7n d, as shown in Fig. 8–464n .xcbi-su7k/ s3zcn.
(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 066.fluadbnr /shtt:2uw hjv ;-uo)n 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 ham,,5l exndua ,umer from rest within four full turns (revolut,5auedlx,n ,uions) 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 has a moment of inerti8 bwuwhg3dv5)b)ptp5a 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 devlo nze ul8x v-2ih/+79wa ylj7 +0xltaelops 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 ma oq hb2sn(6gp5a/t6f/ qzb9 5.gqsfbu ss 7.3 kg and radius 9.0 cm rolls without slipping down a lane a5nb/tg5z9sh./bufas2o g q bq6fq(6p t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun ,tx.oz8j1:- i2nsd8dc:3a zi qawskvas the sum of two termds :vz:x8d. asq21 zwn,3ji 8ai- okcts,
(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 n+ pv;4+ez6t nyj8-alv.z( sfinsr q.1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solidinvq n s ;flj4ea+(6..r pz8stn vy-+z cylinder.    J

  A sphere of radius 20.0 d a16o rfxel/z1whp 9cdx.7- ycm and mass 1.80 kg starts from rest and rolls without slipping 9w11dyp7 zcrd-eaxfh.x 6 ol/down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertixtqwt (3-z dyvp (iw4h;2+u lncally on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper e4 (lqivh3zn-p(; xtu2dy+wtwnd of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg--+e kyi.*ka)avlvin4hi;i y ball rotating on the end of a thin string in a cirai)haiye4*v nikl vk i.;-+y-cle 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 cylingz y; bu,1.,aepd7 pkvdrical grinding wheel of radius 18 cm when rotating at 1500 ey, 17uppzv,adg;.k b rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating 5da)xe 43erzk at a rate of 1.3rev/s If he raises his arms to a xe) 4edzk3 ra5horizontal 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 ax fb6zexc;dc qjd5g u)9kkyw.. 1 6g:her moment of inertia by a factor of about 3.5 when changing from the straight position to;kd u1x9 gaxyk gj.c5e6bqf)w.dc:6 z 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 ratek3b-8 icbrf;f ra 8-b8)lqnjr from an initial rate of 1.0 rev every nlfb f8ikcrb8-)ba;38jr -qr2.0 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 vertical axis through its center at a freq) ,,+gcxh wpv)u hzw r/zpt-x,uency of 1.5rev/s The wheel can be considered a uniform,g)ztwhxp x+-c,wrpuvzh /) , disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spin 8ypd1n1xb-essq (lx .ning 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 8;f1 u,i5 js)guzq0vcrw tgs0r 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 teu ftuww6* zs12j 9xy2kz 4e/I is dropped onto an identical ditteuef z* sx/ 9 uy2z2w46w1jksk rotating 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 vu- 4eg 3izx4urer ,6 8ba:lsm$rev/s$ 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 merrkrp+aw iws:+u )brz ,8y-go-round platform of radiu)u,+a k+swrwbz pr i8:s 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angi*4,h -bzkscz ular velocity of 0.8 sz4k,b* z-chi $rad/s$ 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 half its mu tcgx;dj( /deqd 8 r/)a0k cqezh+(i,ass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what tezk ) r0qai +q8cdu(;jeg,dd/ (c xh/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 w(vv s(.qeo tm+k)hqc f,t;i-yinds in 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 p e7dss 4q8n1iy3q9 q.uez4pk$ 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 pl7js.ytwpo ge mk a6e f-(s+k+.atform, initially at rest, that can rotate freely without. aoftesejg skw(k y.mp+76-+ friction. 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.uhc,kb3cj:nx i e9 6,q5-m diameter merry-go-round turntable that is mounted on fricthb6n3xucc ,ek,9jiq :ionless 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 tq*l dzr6qb)3 )tiq5xf nc8fow3u 6 ow+he 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 without slipping. What length of rope unowu )fztic6fr6x)dq5bqnlq8 33+o * wwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sidf6hpa7trz:r0ssz9 -d e always faces the Earth. Determi: hs9- 7zszpr6a0tdfrne 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from riuk) (n6vf3ar*-c g qoest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquir p(1.a hwsh,3fzc3lrres a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d3.c a(p hr3,lh zsrwf1elivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two siheb imr:zi2w 8linqq ( 3(e,t486oic olid 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 of its 1.0-m-longi,l4mi6zwoq (i3i:h( tq2n8rce bei8 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 whee0( h9* 0oomw;vn uhikhl ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were ro ujd7)3ff 5ytv fbjp7:tating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 k t f7jjfupdvy53 7)fb:m, 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 automobile is for it touw 0.y:lwn b zo,-xmi,( 4hdxx 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 :,xo 0( - b.nux,lwdxyhmwiz4kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratg; ((:a keixyees an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling ozcyf sy76d+ ,tn 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 can pivot freely (i.e., we ilgd 9 rv b(k 59sz,k,f:zzpxt9gnore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment oflvkdk zp bz(x9f:,s, 9t5r9zg 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 jq9v )w0h aqlj4c5e86mcw,e0 sm8rrs we want to exert a horizontal force F at its axle so that it will climb a step agamcqq rw0485)sm 98vra l0ejh j6e,scwinst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on aqzvve +lr h*o6w,:, zp/ist 9a flat road is making a turn with a radius The forces acting on t zqiw /h resvtv6,op*l:+ za9,he 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 sling of le;e3 xw vq c2bxbvu*m7:ngth 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 tv2vcu; w:bex7mq *x 3borque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin 58o orpjrmlxwkc,r)6ww3bo)sv ); 7g rods, making reasonable estimates for the dimensions. Then calculate the ratio ofrg5k)bcorxws;m 76)8ljpw, voro3)w the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a cloldr*z/0ws vjoy8l8 ;*y dl 5gutch assembly which consists of two cylindrical plates, of w y lo5**d/jzdrl8vs;8goy0l mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo l,)(ki)l bh0 wh5 +d )-i1fwvuoplopang 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 phwa) ,))+ h-vowbii(fpo15k0pd ll luoint of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notj1 h;d .k85.vlrzyte6 xsjhwri- 6wr ; assume $r\ll R$ h =    (R-r)

  The tires of a car make 8i:a m wqxnud.su0+ ;7w5 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of s7 q0;a wmn+d :iuuwx.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