A bicycle odometer (which measures distance traveled) is atta e5ybnouyx .4n3txk/8 leq6 7gched near the wheel hub aq lnyuyenex36k/g8b 7x.54o t nd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. tgveuxb9wjo z.k;j 83 q3t91zDoes a point on the rim have radial and/or txz;wo1v qbt9 gjuzj38t.k9e3angential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the an -vf mo/5bev6 zb- y,6q38f aivyxnga(gular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than obs ukom6r- jljom)*.;3z1cl 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-uy zxhw :x)sqajp 7h,5rt3g:+ gp with your hands behind your head than when your arms are stretched out in front of you? A diagram gj5st h rh,:x+):g yz awx3qp7may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the jnde: ,p(5xzym h8;ck 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 haz(8:xnejkm p cy ;,dh5rder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bejc,/wnqfx;- o 8 has k8,cly;d5y7jyzing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why tq8lkjwc- y/xj, ;yady5ns7 c8 ;,hzfohis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) nm 31bz*ccq9scarry a long, narrow beam?

If the net force on a system is zero, is thft11 z1f.ujhj6 ginr2+u j2rk e net torque also zero? If the net torque on a system is zero, is the ne2frnku. z 1 ri21hujfjjg16t+t force zero?

Two inclines have the same height but make different angles with6b1 y/4tylbw q the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the boty/4bytl6b1qw tom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Onelysu0 t5;pvtdeebzh ):3;n b 3 spt; sb evy)de35z3bp;: nhlu 0there has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the 803x4ckm2) uahhxth59tp epnsame radius and the 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 x)u4 p8n5 t9mkta2p3xc0 hehh energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentus7ctxlld7l:7q gxw36 7sz9qfp 1vt ) km are conserved. Yet most moving or rotating objects eventuaql7:6lcfs t7x39lvt 7 gp)1k7wxqz s dlly slow down and stop. Explain.

If there were a great migration of people toward thz*nubi4+ n5-h l +rg2scb9 isne Earth’s equator, how would this affect tncr9u54*sb+hl+ 2zn- is i gbnhe length of the day?

Can the diver of Fig. 8ylv7r(kk3nu r js5g/ +–29 do a somersault without having any initial rotation when she leaves the board?s3jylr+k5u /7rn kv (g

The moment of inertia 4r:)oc s m/ow1tes2lgof a rotating solid disk about an axis through its center of mascr4sm ow/g: e2 )t1sols is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdingm loo+ ,0a7uvv a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreaua+0vvo, 7om lse, or stay the same? Explain.

Two spheres look iden 9 wmy*i ym73xri*3gzatical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine i mg*7a9*3yw3 mrxziy which is which.

The angular velocity of a wheel rotating on a horizon*nrtb863-yk4fd 5d 9vwkgkrlgb/xao af 72q) tal 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 increasing or *nx-74tgvb orqgf6y9 )akfk d5rk/ 3b 2dwa8ldecreasing?

Suppose you are standing ohhr 37aftb.i9 w7jy ezqzv50. n the edge of a large freely rotating turntable. What happens ib ezwfh. 0.t9 jih3ry7q57zv af you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As hdpltm2 3xoo,/e throws the ball, the upper part of his body rotates. If you loo3xt /od,pmo 2lk 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 yo 2yqx7c2i3chhdm;0/) r jqamomentum, discuss why a helicopt/q2i72yo;h)rq jh0ccm3 daxy er 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 anq1 2uqkchpos5 (wk q,)gles 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 amaf j3u 7nl,4a sv7tlfy*zing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of 37f7fl*lu,tn vsy aj4the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam d5vjh/t)-m qjq 5jqrv*im4(tk iverges at an ajjhrt -*5)/ vt q4v (kmqmjq5ingle $\theta$ (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 ishu g9pn(ci s91o 9mg,o turned off during operation, the bla 9(9p19 gghmcn,isouodes 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 bd;/4 wmga br;sa7n wy7all makes 15.0 revolutions, what is its wa;7nw4 r 7mysab/gd;diameter?    m

  A bicycle with tires 68 cm in diameter travele0v lmob-75zls 8.0 km. How many revolutions do the wheels ml7v5 -e0l zbmoake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angulrp a koh/,tan-- vg r;3u4yr(rar vro -vauy;nptr/( g h3-ar4 ,krelocity 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 rev,* gymo/wxu5mmxh us.7z a*1jolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from th ,. 5 /jxmuzhy *s1a7xuwgmmo*e 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 Sun lnihbd4 jgd: 6((*mpbeev 1 4j   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p,g-x ow2f1d6cot0 za ooint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the ax.le 7m5r3, hmoboh +qur4jme-v(y l*pis of rotation is to experience an acceleration o5+.m-o7ybr 3pl4 mv(urle o,hh e q*mjf 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceln 0o)4qfi oaiv1 *ce8ui;v1gperates uniformly about its center from 130 rpm to 280au ) oneof8vgi*;101 vq4 ipci rpm 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 radiul:w/ dhpbkd;5e: :yczs $R_1$ 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 c,q:aefr g9*sds/m+ uApollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thoughsecda/9fq s+mg ru *,:t 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. Through how2gj h)e+8 n7ay wu:ul /6ums. wt,kpqk many revolutions did it turn in this ti.)akh jwmt2y ukn/ wpu8s,e7 6 :lguq+me?    $rev$

  An automobile engine rcs*//y )o b*q yw3hf7 *jtopislows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying high0xlc6 g;s+9 g8qxox,xof c9ux qp(0qospeed jets in a whirling “human centrifuge,” whiqof6(u9 qgg +9x;cp0x0lqx,c8xxoo sch 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 from 2(v/am:eyn pt 8g5k* yd40 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra ya5*yep( 8 t:ngmd/kvveled in this time?    m

  A cooling fan is turned off when it is running at 8506ww rwak:m(2j gzt;fe v1+ dy0rev/min It turns 1500 revolutions before it comes to a m1tfje0d gv(r;w:+yaw 62wkz 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 r3dj0yb k7sym;nrh/ o l0 15smtevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tm/y;jr0yds170mlobhsn tk 3 5ires 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 revolutione( icd9bhu05 b 8lg8rss as the car reduces its speed uniformly from 958 0hbul5 r(cdbgs i98ekm/h 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 bike puts all her weight +cp7(mp uueq9on each pedal when climbing a hill. The pedamup e(79 c+uqpls 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 for 7iu; )on8uv xokng3;yce of 55 N on the end of a door 74 cm wide. What is the magnitude of 8xv); n gu7k3io; nyuothe torque 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 whebfpo j3j,4d q- n*pm(t+ fu,9o,pujb eel shown in Fig. 8–39. Assume that a friction torque o(tbpj o,p unm -fqub,dfjp+*oje943 ,f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attache l(phk,*ub8s:t ri ck;+ 9dmrnd to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position a 9u+i8k mcrdpn blr(: h*s,;ktnd then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenin d-: asnc5;-l+v-ew3bt1 mgpx hr6l cig to a torque of 38 xl;-m-ceb1h3 65-sr:gi lcapwv + dnt$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere w5 v, r)ql,txlof radius 0.648 m when the axis w ) r5xq,vtll,of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rii4mz at+6 oz(eoi58j sm and tire j6zot48 5+azioes i(mhave a combined mass 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 rod is rotated in a hori:g jdk,3( 5gd2zloohi zontal circle of radius 1.2 m. Caldd,35olgg(:ihk 2o jzculate
(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 am 1jl,vl v58skngular speed (Fig. 8–42). The friction force between her hands and the clmj,l1lvvk5s 8 ay 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 sho:l-m.pvrrz mu*2dqzc k /*hg9wn in Fig. 8–43-qpu z d2* m *h./rclkgz:vm9r about
(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 bjy b ec ,tgn5r4ky*gi5 /zj uf72a6dvh3:sr8mass 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 r a-7j,vxhxzo .ird y- 0f,o4dyate, engineers 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, what is the required steady force of-vyr0-, 4x f jz,yaxdio ho7.d 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 ,kvrk0uk- (84bdi8zyu v rwb2a uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cr08kwbz8(kurvi, bkvd u2y4- alculate
(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 fromp6ie6v0 ngi o- rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-rouni:krs5h oe)b:d 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 the torque required to produce the acceleration, neglibho5 :s: ek)recting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is sf*aq sf81rzb+vwk 52.f4w s pwhut off and is eventually brought uniformly to rest by a frictional torque fw*bfr12sw8a wpkf5.q+ z4 v sof 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 accelerate 4 k2i gnt+x,*t:8rfnpump)fns 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 by the action of the forearm, ,nnjg5 1j rti,8ix/ww which rotates about the elbow joint under the action of the triceps/,,x j8ir t1njiwg w5n muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can becxnqbz+i 9lqa w4 3ps 98q0-lp considered a long thin rod, as shown in Fig. 8–46 c8-qb0lx4 qa3s q i+lwnz9pp9.
(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 masses3wopw4 o(q,j5 vurt 9e om4+gm, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fhsz/;ki hu b-0our full turns (revolutions) and releases it at a speed;iu/shbh- z 0k 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 ha-fq yk4, ycd6js 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 a z+y2*oh4uu /dj2 ujmv6zv y6 htorque 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 rolzov 3ly gc 8rw +:w/)b)nutnpy70+kxb ls without slipping down a la8tyk 70w+blv yr g+ /)op:n3x u)zcbnwne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eas6fqk apjj jy2 /(1vj+rth with respect to the Sun as the sum of twoj +p sqaf/j6(yjk1v2j 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 h yrnmzvm u .4-x6j-tg7:-sdz7.50 m. How much net work is required to accelerate it from rest tomy6dhvm-uszr x- 7jtn:- 4 g.z 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 k mhmg*chue2t 2ig g,d bg*4t)-8, ypkvw3bz3w g starts from rest and rolls without slip8)c23m ,- w3gu*gd,h2k p4 wgyht mzibev*tbgping 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 vdi (dy fed2( v*-gu(uyertically 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 iu(y d*ue f 2dd(ygi(-vt hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating b +nu1rl* z9j,o xglm:on the end of a thin string in a circle of rad zg1j+bo,*x:mnur 9ll ius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentu-hiy4xu5ih wey ce3o;f/ (iz6ln1 w 7jm of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatini;/f-wz6hec ie1u5i lw o yxn7 4j3(yhg at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at at 7x;xachxf)v8 /,jp(auy3 vq rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreasec xt/xayfauq37hj8)v;x (,vp s 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. 8–29y/+/tq ere)k-fl 1k xg) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to ek /yeq-/fgk+xt 1l)rthe 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 rotatiz )p), u3e6gc:vc aamdon rate from an initial rate of 1.0 rev every 2.0 s md :g p3avz)c)e cu6a,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 rotatce69sgg a/yf6 ing around a vertical axis through its center at a frequency of 1.5rev/s The ws 9acg/6 e6ygfheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figu px44mezqfw fyy:5.n4re skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth s t nrq*j oq,fey0p-u )t477qp
(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 baqd +*6*ggadof inertia I is dropped onto an identical disk rotating at angular s qgdd6*bag* +apeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4nk3 s+jngb3(vf+ot cgdu. 7(n $rev/s$ 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 tkb+7 yuhrp*w.i u.w0rotating merry-go-round platfor+bwwtyu 7p.h*0kui .r m of radius 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 :ie.b. s fj -5aot0cboan angular velocity of 0.fa j.b-ct 0bse :5ooi.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually colla1g twn fb1z5oh8e7vv ms34 p1wpses into a white dwarf, losing about half its mass in the process, and windht7es v4vnwo1m p wzg 8b315f1ing 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 involve4cx ct 4m7*rlq winds 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 fuwzeuwg35lp+60rl 3y u1od 8$ 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 restu6 (yu5vllbv*3- 6vx)hn qruv, that can rotate freely without friction. The moment of inertia of the person plus t-lvyr) 6vvbxl6u*(n h3uu v5 qhe 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 di ;lbnvazp(ey.(c+c/ zameter merry-go-round turntable thata(nlzc (e+p;bz.vc /y is mounted 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 sv 3 6tualiov6v8l) s1end 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 unwinds from the spool? How far does the spooli1tv683 )vov6la ul ss’s center of mass move?

  The Moon orbits the Earth such that c :sokth29*c* nb7)fiphoos ) the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter casnh) phc t2*9ssobfco *) ki7:oe, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist acceleratejt g58 5ip 1zobuwae5:yrb8g,s from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape0ge*+;7 e zjom-eorez () wtj3y2tdpk of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcdomkjpy+rg70 e; z*)ew-2(3jto t zeeulate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mh9t qi7 wjg5/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 itqhitm wg75 9j/ 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 anguqg ,v,/3vh)*qarzijnxw .i -glar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timese3q(-v84 s dmi)ztqkt 2g 4ebp as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to g(4t- t)m4zi pbqedks3eqv 28undergo 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 lowjcnyex7vk)-qh ri6,iuk5t xb3 3 g7r2-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a 7h 6nx )c ,3g5uryeii-v3rtxkk7jbq2 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 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 ad2hb+( 27wk b pc:e jwj5uzq9wn $H_2O$ 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 f7cj8k8):m5 k w8uwcmqj . vbson 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 lenaz ak+ u89se3dgth L can pivot freely (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 of mass of the rod, as shown. [Hint: Saakd9 seu38 +zee Fig. 8–21g.]

A wheel of mass M has radius R. It is standj jt:gn 9kw/has 3g7-cing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against ww kg37/aj h s:c-njg9thich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mak(+lppk5v*+ :t,j8c/esmz kvi vt vqh4 ing a turn with a radius The forces acting on the cyclist and cycle are the normal force 8/*:qp4vk 5+ tk si,jv telvzc+vh(pm$\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 length 1.5 m and ,.x:*2k- j(l)rdk5hpjggmw e pn p8e sbegins whirling the rock in a nearly horizontal circljmrgh2.)-(e:5 pl*xk ejks,p g dwnp8e 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 d /):ybsm m:upjuo+ i)and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agp)ou/bmjds +im y: u):ainst her body.   

  You are designing a clutch assembly which consists of two cylindri,ouds8ym+e3 5-less vcal plates, of ym3d-lo+sev 8 s 5e,sumass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped r fbb o:p )mbm v*f ;+.1i6 yiwsd+(rt(bm,upsqough track of Fig. 8–58. What is the minim16*d + mfwyvp (usts;( :f+,bbo .i q)rmibbmpum 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 do not asa.5x1ibagmhq+ ;bvu e6 7ck1ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as kx 4pitts:6 2mp9te3jthe 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 eaik3mx6 ps j:pt4t2 9tech 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