A bicycle odometer (which measures distance traveled) is attached near5 kpj2qq oa*r- +zi0q/ld;;jcse jhw lbn4 2q; the wheel hub and is designed for 27-inc lwjhbi;n52arzp-cos q ;j420qjl k*d+;/ qeqh wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant ang2au9k6sw+.w+lyl 2y5 ;xmi o3qiu3yfwti- l hular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increasesi x+2qi3wlwayiu s3. 5yl+wm2uy6 -oh;9tk lf 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 describp5d8 jmcy:r ;red by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exdoy7 sdkv q 6*drk8.w;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 do a sit-up k u .b8g3euvi ;qq.zfv :i.ef7with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thiz.q kb .f: fvi;ig.8qeuv3u e7s.

A 21-speed bicycle has seven sprockets at the rear wheel and thady(r8d3qfa qq-ac2fgt (n3 /ree at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprockqcn -gt f a23rdy(/qqd8faa3( et? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beinszfv s)245xcbg able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). fv 54cxb2 ssz)On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.ky ;,zzjyrk *zhq4n,ia-8tf 4 8–35) carry a long, narrow beam?

If the net force on a fwq64c0 72ai9jfgb ig system is zero, is the net torque also zero? If the net torque on a system is zero, is the nfgfci7wi9ab 20 g6q4j et force zero?

Two inclines have the same height b-kp7.d;8sedb w(yn + rykie :uut make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at ther i+ u7k(dpee sdnwyb:y.8k-; bottom be greater? Explain.

Two solid spheres simu.z8xy hwhb9j*)0tgb;kxs rj -ltaneously start rolling (from rest) down an incline. One sphere has 0;y*xshxz. bkj j)8t wrbgh 9-twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and *yne-4w h) ub)u3w 1:zppj l+k1gbudethe 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 g:-g)epjz +w) 41 1nkuybbhwpde*ul3ureater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular 1q0p4 kbf 1atkle,g5rt5 d .lnmomentum are conserved. Yet most moving or rotating objects e 1egrt .lkk5q,t51pbnl4af0dventually slow down and stop. Explain.

If there were a great migration of people toward thesm +4izlt, xm: rat*fq54axw)8r vat e p0+jl6 Earth’s equator, how would this affect the lengt:048++5 txal*rmv ezm xr 64pw)qfjl,ai sat th of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotati 6ilxw5nj8)h/- r3onwip(ne von when 8vhri3(wnn p5)ewi6 n/l j -xoshe leaves the board?

The moment of inertia of a rotating solid dis-a0 uj,q yj 3k24enymwk about an axis through its center of mj- ,w02e yj ayk4u3qmnass 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 s b:ep b h2y)*8glf1v+7z;u xzvsc/b9 8ctvwr ktool holding a 2-kg mass in each outstretched hand. *r;:)fxeubyl k1 2hv9z /c+8svcp 7 tgvz8wbbIf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.l-j(r: c(a+gj gwufe( 4r dc-b However, one is hollow and the oth --guf(dr 4 bj(c(rcwal+e :gjer is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointspgq t) a)h)81l l.fay(pqlo(liax0h5 west. In what direction is the lia()it)l5a.lo qfg8php0q lx h1)l a y(near 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 decreasing?

Suppose you are standing on the edge of a large freely rotating t.bwtownl z;*n l94b3oe7c4 teurntable. What happens if you walk tow4ol4 *;.t bo zewn cwn79let3bard the center?

A shortstop may leap into the air to catch a ball and throw it quickly p3 owxwh(ykv:4m q8v7. As he throws the ball, the uppew 7v( xpv:oqw y34mhk8r part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dnn r1+86h hg no6engxa5x + q,k:kwz8eiscuss why a helicopter must have more w8+zgee+h:6 kr6nnx8,hak nxqng1o5than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 ps7:ga;4hww i$^{\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 coinciden 3(w 3ig,u:*obqv ).x.renjqp lee :soce. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earj) *pqiu:goov3ne.rbe.3 es:(l qx ,w th.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed atkm, -umevf ;eyf cx*4; the Moon, 380,000 km from Earth. The beam diverges at an angl c xvfe;y u;-emmf,4*ke $\theta$ (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 is turned oi*365 i5yijpdc ij0w* wqhle1ff during operation, th0561*5yhqi i wl*ijdijwp3ce e 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. q mrat-c9 /,61jvhl tuIf the ball makes 15.0 revolutions, what is itaqtt vm/6-jhucl ,19r s diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do4+(fqji .ciaamu*b a 0 the wh*if+ua m0(c .j4ia qbaeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at4z8x j5sn)qz jkjm4 neib/1s: 7 )jdif 2500 rpm. Calculate its angularqz4k)x ) 8nmbz dij17sfjs45en/ijj : velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig.:la kgaa.fu8 4gn .lq bi8e,/k 8–38). (a) What is the linear speed of a chil4 l8q./flg,.baieka: u8ankg d seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (adl2(xlzi wx.khv -.7zko (k *cdjtx+:) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pt 77zgxb28dmz3b6 pvy oint
(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 cmwjf:j.,qa7 -znnaxsrz t190 r from the axis of rotation is to experience anj-0z fn:jsa.t rw 1,ar79xnz q acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its z(7ymidyju*zlv2b o5 8n/gx6center from 130 rpm to 280 rpm in 4.0 s. D5gzv62 yi(ozm ny7ud/8jxlb* etermine
(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 radiu9 bhi,k y)8fzeo/6r ku4raa b/s $R_1$ 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 Mos+v8hfpkaebf4r - - f w*tmh:/on, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’f *bf8a-- 4+ fvre:m wpkhhts/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 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.d7gf4noc mlo4a n (f,.e9m4da Through how many revolutions did it turn in this time?d9amld nm a.eo4nf4o7gf 4(, c    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s fmwo06/8p dgsiuc /n2. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirljn 4u/ wtbuw;fp li;/ch1a5k8ing “human centrifuge,” which takes 1.0 min to turn thrnw p;8;ulf4j1c it bh5ukwa //ough 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpn.r7x9qc: sjgm in 6.5 s. How far willjn9 gqs:c7.xr a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runni0 akoo5r/ sgi*i 7ts1o*tst6 lng at 850rev/min It turns 1500 revolutions rtt0sk/ 5 o*oili a*o7g16 stsbefore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reducj )rybp7/e8yy b(p(xues its speed uniformly from 95km/h to 45km/h The tiresb ube8(p x/y jy7ypr)( 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 revolutionso5dg51 ryi2mo as the car reduces its speed uniformly from 95km/h to 45km/h go m15iroy52dThe 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 aahhwoi8+ d+rzn ) p+lg.w: l/yll her weight on each pedal when climbing a hill. The pedals rotate in a circle of ral:/w+d+ h+g8 laz)ohyp.irw n dius 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 oflha8s9:t a87.grxdz l a door 74 cm wide. What is the magnitude of the torque xt sa8h z.7l8ga: 9rdlif the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39mr:al cdh -lfppmw2f:v +) 1xcq3wu: (. Assume that a friction to (h:w3)q u12cwff + :cavmlxp-ml:pdrrque 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 rod which 7wtjp,v r6 4x 6wt5sxfpivots as shown in Fig. 8–40. Initially the rod is held in the h ,f5vt pwwxr4j6x6s7torizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteniq tal3,r)2dqhng to a torque of 38 2a , 3l)trdhqq$m \cdot N$ 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 s(b*3hthz2uewt7+acnwb4h*ui* u r) of radius 0.648 m when the axis z*tr3)ub h2e ** thcsub7 wuah+(i n4wof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamz( q 6kdnu* dftxg9h-(eter. The rim and tire have a combined mass of 1.25 kg. Thednug6( x fdh-qzk9 t(* 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 horizo mk *zvnf4mumj9o kmqk1q1-o)9 ca;9 hntal circle of radius kf )m1uqvk; mn *mqc j-ok9o1zh m9a491.2 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 bop ha6 rd264mg gfr;k(gpe henya07s:)wl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force hg7 gp62er)pd0m fsay(er 4:an;6khgbetween 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(8 +dvspqbddjlmf:hi0,vaq o0146- kwoh6z g point objects shown in Fig. 8–43 about g jqvfp i d+ 1ovsbza:-h6,lkdh4q(w0 dmo806
(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 twoog ;;.p bopsp2 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, d ;/jhu mb:xmoox,ir jfu;*4 y9f(qe(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 each rocket if the satellite is to reach 32 rpm in 5.0 min? 9biejo(fjdx(r u ym/4, :u;hmofqx*;$\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and sx2e7pb*u;ok s8k u 4y6rtuo( a mass of 0.580 kg. Calculate psx ;(r4 ob7 ukoy62su8ke ut*
(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 itjzcgqkz:2 al / ( +emc,juaj38 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 m5pu(h +xcsx)k t*y g1zerry-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 7y(xsk+ 5zhu 1pxt)g c*60 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm,z4ahntjd n1 1 is shut off and is eventually brought uniformly to rest by a frictd nzthn1j1,4 aional 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 ao4a)u2jn 1 bi-zo) ofpccelerates 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 thrownvr/oglrth 81pria :r6+p22 a8bcm ma) solely by the action of the forearm, which rotates about tt2rpambrm2g +61cipala : )8vr ohr/8he elbow joint under the action of the triceps 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 be considered a long thin rod,p,cp3k zpj4c ) as shown in Fig. 8–46. pjpkzc,)c p 34
(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 )ka6x8gb3 fmtb bp *5z2:i dyeconsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer fromoju1k ((vony)f i0 bf+ rest within four full turns (revolutions) and releases it at a kjob of0+n1 if(vuy()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 inertnhn5v ofho (w9/2e k(0xicj m,cw,lv0ia 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 enginek athdslfvy h9/c / 3dy2:h7v( develops 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 slipping down abk bl)a6r.zo4 ;,cc hk lane at 3.6;, kb.4z r)a olckhcb3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the z-vb: ex-n2+)zskao gSun as the sum of tk:bz2- ea)ons zg+xv -wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg ar0+ ;rh +d66bozl nsmznd a radius of 7.50 m. How much net work is requ +6lr+ or;bdsmz z60hnired to accelerate it from rest to 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 t)vm cw gfy7(,fdvs8df9rpg,nit847 1.80 kg starts from rest and rolls without slippin ,49tvm8rwcvf i7yf( 7gstdg)f,8 dnpg 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 vertically on its tip. It startrf3)r o,:ek(scau+bem,nsp-6 4xewcnx( e7 os to fall and its lower end does not slip. Whax 3 nrw(exf+rka67nceo :),,e o4e c-bss(u pmt will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a3-j-b;nedk*k0n.azc l np6 ko thin string in a circle of radius 1.10 m at an angular speed of 10.4 *n.z0 3ldk-oa k6bkj;cp-nn e$rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical g 7zkfr uvcz),3+rxyqn1p1l/k rinding wheel of radius 18 cm when rotating xzq fr ,n lryp/ k3)+v7cuk1z1at 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 rkfii )+i9vhd3 rtc. ,yotating at a rate of 1.3rev/s If he raises his arms39 i ,d)khrc +iyv.fti to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment+y+mgpt* - 7kdg9vu/ vow8d 9c0lpbhb of inertia by a factor of about 3.5 when cpmuywv-/ 0obbh7lc9dt dgk *p89 v++ghanging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 (hyj.e f* 83bsziyo 0vxn0j/lrev ev.ojhy 0l/zyxbie0fv 38( n*j sery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating aroundf0v/gjxp ga6 p0c w k* 9sk18d8cn(zpj a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and *kwdp gc /xvjs8 6fzaj(901pk0 n gpc8diameter 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 skaf j8xpntox exx)6*s ;*dd)gy0ter 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 Ea3kkp j(ocq 9jmpxn27 tmy) h5+rth
(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 droppedjrn.- k 9. rrr:icd(lo onto an identical disk rotating.rr-d. 9 rcl(kj ro:ni 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 )odni b l(;bi*$rev/s$ 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 plv5c5 nr:-vnr4hg 4s .npntr/ patform of radius 3.0 m and moment of inertia 9 4gr-t.5nhpv: n /4 vcrnpsrn520 $kg \cdot m^2$ 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 angular velo8xtcr uo.ua/fa;a.9wwut5 ,g7dy2hse + mh f-city of 0.82m-ah w+gcxwahrt8ed 5f.9 t/u. usu,;yaf7 o $rad/s$ 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 collaps( lngzw*c ;9fo;hd (wcogo:go1x :-uyes into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assum* 1dg( og(;:9o uwwxnzfcyc-goo;l h:ing 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 -q+kwae :caqpm/ 3lps /q- s,sin 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 2 4 r4hdc-:qfhzzla8 y$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withox3xru u. .q- q/nalrkfl0.;zd ut friction. The moment of inertia of the person plus the plax 3rl nx. ;0qfduquak -/z..lrtform 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 elu in .iaqo7 -f(1lq04 y4bfvadge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of u a4-vf.bl4o71iyq(filq0 an1700 $kg \cdot m^2$ 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 ron6nkdc 3lq0mpzm 5,,cv:zs* e lls on 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, hold 3lepmcv z,nsc qkm:d0, 56*nzing 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 spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces th 7kzb+ y0ewey zf(c p2l:h/k8de 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 particlb p kf+80lkw zh2:dyyzee(/7c e orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratbm/ . m : :e5qa.eu2h:tooxdpmgxj5 v.e 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 ra::+cd:ga e-u h ie 2c:gmy*rardius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular ac:rg: m-::auaych2 +ceri*gde celeration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eac 89tzz4l*qf ych of mass 0.050 kg and diametfqzlzc t98 y*4er 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-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how ypnf;tk2 ak-mm/8 ffh) d )gux)9n9bpis the angular 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, bu h-9m;f/pxn*xci w+o3 g8nylb hxtw: 9t with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the processg 9 cfht i83 o-/hw:wnxp+nxy;9b*mxl , 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 lov5 v6)owrzjz 4w-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 without needing a flz v5jr64w)ovzywheel “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 illustraocl9hfk j3e g0 k.r11uxxf+/g 2ewi, kzbv0a9tes 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 jlxcl4nxz l*+5 i.d:y) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M a*8e8nfkwp vz me:c )-lnd length 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. Avk*m 88leen:w pz-fc )t the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing dtg4,(wq c3h 5.aks5h qeaxl2vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step hastq k w5.hahldac qx(,e52g34 s height h, where h

  A bicyclist traveling with speed v=4.j ntt8bz:fljl 8 p*6+g2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle arelj+b j :gt6p*n l8zf8t 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 ro/oaee 1;9f p zaba9oi0yls3w( yvlq.:ck into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to ac0;qoyslw :z ae( 9a13/b9 of.plevayihieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder a 9*isaoh80ooiph +og8nd 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 against he o ii haoh8+o0s9*gpo8r body.   

  You are designing a clutch i6f o a5 n4datan;qy))assembly which consists of two cylindrical plates, of ma 6nf5 n;ayoatd)aqi 4)ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiuslb;+i ob+6yce x*hcpv+f t 5u+ r rolls along the looped rough track of Fig. 8–58. What is the minimf e b+lcu6opbt xcy;v+hi 5*++um 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 v; du+l,ksy g4assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spem3b;;aouulv 7tf diy1:-5 qax ed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of xmua5qa3 t ubvl:y71o ;;f d-ieach 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