A bicycle odometer (whicl57 *ul)*cbrqibfp8n5.f vd;6ry mz0vsw1o v h measures distance traveled) is attached near the wheel hub and is designed for )6 n*.ivwq *md u0rvbl lc718;z5f5of r vbpsy27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poi*w4 3.119xzrnm)ju wuwqho xint on the rim have radial and/or tangential accel11oj. wr49 w)qhzin3 wum xux*eration? 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 2r3hab; ag 3dvkv g;i3the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lhdlw9oquo5 s9: o+;a asi8qp ;arger 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 with your hands behind yclttd;tl3310 tah6uv our head than when your arms are stretched out in front oltdh3t3ltc 061 v ;atuf you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whes+f2ddt:ly 8d 27 qx*jei4smz* ,y wqzel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In whichyjd ez7* s4wsxmqq8t*zl ,2fi+d y:2d gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bei*5pnp 7+t dyp ;o1f5u:czktpo ng 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 dp o 1 :cun;ppdopz*f7y5+tt5kynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.4z)mj1iv yj vo.j*f, o 8–35) carry a long, narrow beam?

If the net force on a system is ze 2dw-0b4p ycliro, is the net torque also zero? If the net torque on a p d 2y0b4w-licsystem is zero, is the net force zero?

Two inclines have the same height but make different angles with th aq yj s(2,0brease-xap80qe3 e horizontal. The same steel ball is rolled down each incline. On which inc3(2es ,-qaarebj xq0spe0ya8 line will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest4lg hc:9( 2.mtvhsx ( g5)g7wi rrfirq) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches th29 x.gl)rw7mth(q g :rivcfs( 4 rg5hie 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 the same mass. They start from y ( ggfmja/ypv**5,hdva 934kg a/idj 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 kineticm f3 i ,5kha/g(ad *j/dva jg49ypgv*y energy at the bottom? Which has the greater rotational KE?

We claim that momentum a6 v4zx93zhkm qnd angular momentum are conserved. Yet most moving or rotating objects eveq4m 69kzzh vx3ntually slow down and stop. Explain.

If there were a great migration of people tow7:,i7ox r-n6rhqvuj/cx)sm lard the Earth’s equator, how would this afjvx xco)isr -6unr:/7mq, l7 hfect the length of the day?

Can the diver of Fig. 8–cpaox/pr8lhcp + 9fi 4qh7s+ g9y.d3q29 do a somersault without having any initial rotation when she leaveq pcrqgsx3h.8apf+o/ l+4hc dpy9 79i s the board?

The moment of inertia of a rotating s(fdfe imjb7 5p2hu2 1eolid disk about an axis through its center of mass is fie f ue17hdj(2m5bp2 $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass hvbhc13q4ql5 in each outstretched hand. If you suddenly drop the masses, will your angular hvl3c5qh 1bq 4velocity increase, decrease, or stay the same? Explain.

Two spheres look ideskj7l67hya, djn w1 9l*crd of2sk1-intical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determr91*o 7kld,wdk 71as2fl6hiyj s jc n-ine which is which.

The angular velocity of aqt+. y)dqugs 0+g li+i3uk+8)uu fst w 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 accesd+ gu 0yutf s+iqi +q)g.u3+l)u8 wktleration 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 standile(s,*kldxtoh4 p1b 5 ng on the edge of a large freely rotating turntable. What happe s*,1d ht5lb4 pkxe(olns if you walk toward the center?

A shortstop may leap into the air to catch a ball and thrqy/p r 7 xwq,lh7a.qyc4rt95zow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will natrypqz 7qchy/wql . r 47,5x9otice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservap24 .+-up idd/ uvngextion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to kx4d/ ipu+v2p -gd un.eeep the helicopter stable.

  Express the following angles e6)jxeb-y0t1 rjc b0jo 0c x5zin 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 becausav ce0f;4zp3 pm a45doe of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular ma34 of0 v4 a;pdp5eczdiameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direcxu17y z(jg:jtqn 7w ;gted at the Moon, 380,000 km from Earth. The beam diverges atxq jzn (u7yt7:1;gjgw an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When (oji kj; qjumk. 2lx x0m sty;b;(rq06the motor is turned off during operation, the blades slow to rest ilk;0;ujt(qi ;x(2kmxjs b. qjr0o 6myn 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 mkmd1 cs0 0k)k4yxo*vi to another child. If the ball makes 15.0 revolutions, mvk o*kxck s01y0d)i4 what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do tm;91f4z jh yuwhe wheeu9j ;y m1wfz4hls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotat cl c n*ing y59e7r5ee9iuf-qp tz4p js*vq,:6es at 2500 rpm. Calculate its angular velocity in7*re9ystpipe cf5 z5n 6 cvi4e-qjnl *,:qgu9 $rad/s$ $\omega$ =    $rad/sec$
(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+fl.* lh)hqdiw03m;/uodcc/r thuf * 4.0 s (Fig. 8–38). (a) What is the linear speed old*fftu+ .r3 ;lduhqc*w/h0i hc/m o)f a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the lb*jr3uly2: kl7,r wx Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of gwe7q 6sz dzgji.p9 cu:+09tpa point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the h c5.hjm(3gm,ttdjk evk7ir+ eb0o5z (v(2ne axis of rotation is to experience ah e,c n37d(0b ejto5(ve+vig5 z2 tmk.rkjmh(n acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a u4qe3 r511crc/2 y 04uotgedy4xll sqbout its center from 130 rpm to 280 1s4y ru4rl42y lgqo1e eqx5cu/ 3dt0crpm 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 radiusac,5j uolr 825;j uihv $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put txd q6 q/185dd.cmfz ujhemselves 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.dfdj d6qxqz .c8m 1u5/ 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 re8oegzgr-v1e*+r7 z7nqt ji.-gr9vns st to 15,000 rpm in 220 s. Through how majge7q9te-r no8-zgg s r+.n*1zv vir 7ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm ib:q(i6ck1t(kp 1bk; w*q pgdg n 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 highspeed jets cn(qtxych8i*6 p: l:9e2vmuuin a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete rev(x 6p ccmu:t *9v28iyqunleh:olutions 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 uniformlyzjnaxjqr .zd :2g37o+vy1/g r from 240 rpm to 360 rpm in 6.5 s. How far will ar7vz2zoj+gjxng:d /1. q r y3a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is d;- jj.cui v-jrunning at 850rev/min It turns 1500 revolutions before it comes tid ;-c.v -jjjuo 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 reduc5y.eww zm;: /tlgc .k02ri vhnes its speed uniformly from 95km/h2.m .nety:5vz ;hk0cr/ liwwg 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

  The tires of a car make 65 revolutions as twyyewwoj8bt 1i/ 5( i,he car reduces its speed uniformly from 95km/h to 45kte i,(jy81w5ww ob/yi m/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 on each pedal cfma e 83n.-ey7 rpo7na:.b ibwhen climbing a hill. The pedals rotate in a ayrof. ie-ebn3cm7bp n8 7.: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 of a door 74 cm wide. Wha nck -fp/g;v lu939wz9dcus:pt is the magnitude of the torque if the force wcdf nul;9zk cv-sp93:p/gu9 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. 8–39. Assume e.lbxuavh.v/ 7j+y h8 that a friction torque a8.lj+vey.hh vbx 7u/ 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 mas9;d or9lbnnyu ns)z2 2sless 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 d2nbry sndn9 9 ul)o2z;irection of the net torque on this system.

  The bolts on the cylinder head of an engine ) lny+yap 2, 3z4wsc.xjhh g4erequire tightening to a torque of 38 y lhy+ cw4h2ej4g xaz3.)s,pn$m \cdot N$ 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 sph+)w: fb/tfgc w8h-9y lk*y yrejp1v 4yere of radius 0.648 m when the axis of yt+)k-v 1 :y fr*l fb/w4egcypyh 89wjrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamvku b9d(bk7n4 l a1fu9eter. The rim and tire have a combined mas9u9aukkf 1v (d4 7bblns 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 horizont8wl p,,g +knn fhvh+t/al circle of ralnpv g+8nf /,h, wthk+dius 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 +2.mi10 ;avuez nptb zbowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay ist1 v.+pzum0naz;2b ei 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 thz/1txi 1fc7p-.k voj de array of point objects shown in Fig. 8–43 aboc-1/k1d xfp i.zj otv7ut
(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 ox7g gr0 b.:ycaet, kza*ygen 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 ratei4/je842 rtoy usbs +t, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a rjtyt u+48r/i s4ebos 2adius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with aqgyklk2n 5r/lq wu ;ed:/s0w9 radius of 8.50 cm and a mass of 0.ys2l0:;/ w9k l5ger/ nqkwdq u580 kg. Calculate
(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 fre/) 9clcgd,c aom 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 sm fjw7gt 4i.:xiall hand-driven merry-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. Calculate thef x:iiwt7.4jg torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventuauwg0,:-;cslyf -ihz zwk 2/ 5g lavuu1lly brought uniformly to reszi-w0h l;c- g5:ulgzu 2auvwky/s f,1 t by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 acceleratesrdc;mkc 0(u d( 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, which crsqbj1 d*afpq8 5no+x ,*expm7*6ne rotates about the elbow joint under the actides6f+bx mp 7ap cqx8r1 en5oqn,j***on 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, as shown u6a goapnd jr,jh:t,xz1/4 *ub x9p(x in Fig. 8–4x :9 bx*arzoh p6p d,x(tu un/jaj4,g16.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses s tsd yt;y*0+sc.q*wjg+ ng1e, $m_1$ 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 res0blo) 8k7w, ah5oy9r aybrl-xt within four full turns (revolutions) and releases it at a speh)lob5 rawr y,ka0o- l9 yx7b8ed 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 inertia ofe 8+j e *fyjmjq6gn6m) $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops 6q m:9i+j8orl dkq bup*s9ga)lsm74pa 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,4 ac9u coqu)02 zku-poo w,az rolls without slipping down a lane at a ,kquz ,zou)094uoo -w2capc 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the ;3 yuuaqo2hqm3 ( utt8sum of two term mou3tq quaut;32(y8h s,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net woelck/*8-qvm/lur +spqtjv;3 rk is required to acceler jm v- sclr+pq/l8;ktq *3/uevate 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 20br3wofus amz2 ,i90r20 jv0eo .0 cm and mass 1.80 kg starts from rest and rolls without slmo 3irjeau,0sf0 2zvr 0 wb9o2ipping 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 starts to fall and z zu2rrq2u.w8y upcw.)wx*r2fyq i+4its lower end does not slip. What will be the speed of the upper end of thezw8u p f.w uq4rry+ *.2)w2z qryic2ux 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 rotatin40r) j/j5xe.v1ko trs59wwzo qi 3pcog on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.4 93sx 1 50/wqrvrj jt)5ci wzepk4o.oo$rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylex+pf u),l)fc indrical grinding wheel of ra)+e f)pfxcul ,dius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is 5lcx as:o3x9f9syq-4z cm uk ,rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decr9: u scy9 -5af,zls3ox 4kqmcxeases 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) cvy-ws) e4/k4o7arn iuao p :x+an reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positionwx /u4y-aai )k7rvs:4 onpo+e. 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 ofuclozn )3 pfv99q 77ex)a8ig m 1.0 rev every 2.0 s to a final rate oq9iofe )7n7u8m3 z9vplcxa g)f 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 ta.l1 xvegcp8:eys +o1hrough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, 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 afteryge xv+.1c :pl8o1aes the clay sticks to it?    $rev/s$

  (a) What is the angularfd vj:dd6p 8ok o(;a moi8md:8 momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular moment jn(v4atdkn76um of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped :;mv2fc.f 5bw4bxra l onto an identical disk rotating bmwvb5xrlf2a ; 4fc: .at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsdld xd hzf1y/*. ur/n 1cs*ab2,if+ah at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the centt3yl 0)j( q0pf6rg euc04 eirxer of a rotating merry-go-round platform of radiuq3(ir0g )0pjc6xe tl yf04r eus 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 angular velocity oug x:7b oksb13f 0.8b7g: us1bk3ox $rad/s$ 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 int0a1y z a3hbm-wo a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new zbhmy w0-3a1 arotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exina1mzobo9r4 qwgo c+)n (q-- cess 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 d)m+i; 03*qtk psad gd$ 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 free.qvax8eluu1 y e3(q*y; dj r5h vyt+0uly without friction. They8quq 03r1d5 (hue yx*t+jue;vva ly. 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 edhju*1 yajdx+fo 9l md6h)4+ncge of a 6.5-m diameter merry-go-round turntable thafuyx mjjdla+)nhd co h+*6419 t 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 ql4gttau e078w81 pxnthe 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 slippinn7 ael4ut 0pqg8x w8t1g. 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 sambvn8 kd,,su4b e side always faces the Earth. Determine the ratio of the Mu4vdkb ,8 ,bsnoon’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 rest at a rqdv*s0 j:dii-;qzf y3ate 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 0.2 itudm- *om.u8mr8c -k0 m acquires a rotational -iudm 8c*8-o tkrmm .urate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made o w:jd xlfkp-1dpjb 16 gach56p0*d ;pkf two solid 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-long string, if it-p0xd p*;6 1f bpdwj1:pkg5dlca hk6j 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 owmn+ 7oem9a. 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 masbr-*z9 tn0ioy*l98 evy px4o ss 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 neut98yz*vbl*nxy4tor9-ei s p0 o ron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution a1v +qlb zy tw*l9fd- * sqzzzcj7g(6q*utomobile is for it to use energy stored in a heavy rotating flywheel. Supposeq*q qlt fs l*z61-wyjv7(z +zdb9gc *z 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 flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an 5 e2 cjzko0;mj$H_2O$ 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) abn9r) a0kk-:eai1 asis 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 and length L can pivot freely (i.e., we ignore ae:euq5 u6m*t cdu w2)friction) about a hinu ) u6mtew:*ed2qacu5ge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thdh-3b 5pl1+1 ofna 3zdcoy/ pue floor, and we want to exert a horizontal force F at its axle so that it will cliap-o/b15fdzh l 1dyu3+cnop 3mb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fsgc8 da34lo d;lat road is making a turn with a radius The forces acting ond;3 saolc gd84 the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengthyv*t- 1nl:6e s/ wyfyi ysba(6 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 torque come y6n/bf1a- v: w s6syeil*yy (tfrom?    $m \cdot N$

  Model a figure skater’s body as a solid cylin2 a5ozypytzh(/ 35ehtder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a phyz(e2t 3oz h5ty /5aspinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cpm s (1gb8f,jtonsists of two cylindrical plates, of 1( g ,8stmfpbjmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rougmk15hswbmjq9 t 7 0z6sh 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 high1j5qzmh b 90ts6wksm7 est point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buts1e ab;9j5wzw *. b kj+puc*re g-nmj1 do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its ,fb;s v80kvduz yvt(2ec)-b hspeed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angularvzk )v -v f2ys,ucbb;h0(ed8t acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s