A bicycle odometer (which measures distance traveled) is ati48 86tued9q5bcbf*ch so 9/df4 jhlztached near the wheel hub and iq8iotdsbbj49df c8hcleuz f*/6 94 5hs 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 velocitych 7cdm of*k:li9m rw7v,*e t:. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accelerati 9,fr*m7ic:tvk c*lhm7 o:wedon? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of tkkr9su cb9m19he angular velocity $\omega$ Explain.

Can a small force ever exert a greate ep xmjr;c d nw0+itzk;80t4(fr 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 difficultd tbe.fq9v54 i;++ewsramx . a to do a sit-up with your hands behind your head than when your arms are stretched out in f e+. as5xi9;rvt4aqebm. fw+dront of you? A diagram may help you to answer this.

A 21-speed bicycle has seveit ,0l+sogu,iq6ws 4 fn sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front spto6+, sigs 40,ql uwifrocket? Why?

Mammals that depend on being az7 ( z5r:imqeoble to run fast have slender lower legs with flesh and muscle c(5 q 7oiezr:zmoncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam*od gyj:ko:4hyo-; dlyp .7 vp?

If the net force on a system is zero, is the net torque also zero? If thez0pe4u6 iq/du net torque on a system is zero, is the iepu q /6z40udnet force zero?

Two inclines have the same height but make different angles kjl) .h1hxg+fwith the horizontal. The same steel ba fj1hgh.lx)k+ ll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fr s5o)hm4b b;ruom rest) down an incline. One sphere has twice the radius and twice therm;5 )b4souhb 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 the saf0czpt.d 3 lq9me mass. They start from rest at the top of an incline.. 9pq tfzdcl03 Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most m:zodai7h a+5 d.,mecnoving or rotating objeh.5+: izaa7ec,ndmodcts eventually slow down and stop. Explain.

If there were a great migrad8h(),/a/ymb.k-rrrp xuwx w tion of people toward the Earth’s equator, how would this affect the length of the da w dr/h,yk)8x.upb -/x (mrwary?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation xt i-u:au-rw6whtu-a ux6 riw:-en she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cena eta fx16l7nd5ov7j; ter ofx1aetn6lv d5oaf7; 7 j mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretw5p. laa4 :tliched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 4wa5l:ap i tl.stay the same? Explain.

Two spheres look identica0u h7r(hz. kpw kv,gb14ucn3 nfc3i g5l and have the same mass. However, one is hollow and the othern. v3pzc34cnu50k,1 7 (g bhuhikgfw r is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rot p;vhnr91:lftq2 y/yfating 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 acceleratilhpn ;yff/q19:yrv 2ton 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 tur -osrq jf7nyb21h0 f )o7)qebsntable. What happens if q)fobj71ys )fneh r7so02q b-you walk toward the center?

A shortstop may leap into the air to catch a ball and throw itbh:pvfb uc/8kg5rm q06 g:di- quickly. As he throws the ball, the upper part of his kb:mgp-/uv6qci0 brf5 d :8hg 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, discuss why a hghn)60z 0d jrhelicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellerr0 z6 dnjhgh0) can operate to keep the helicopter stable.

  Express the following angles in rad. f7irc3iz r b3czk3d:ians: (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 o,ak,pu:.w mjeb zp .v: s*zt7vf an amazing coincidence. Calculate, using the information inside jvk, bt:p eusp.. maz:z*7w,vthe Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamyi ;m gm(nl8v il+k80a diverges at an an(+lyaimi0mlkg n;v8 8gle $\theta$ (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 o q-/ebzdcjn9xxo -,.ogb s8/p ff during operation, the blades slow to rest in 3.0 s. What is the angular accelerax x , 8-so/poj-.cbbzq9g/nedtion 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 5 qvxed0c4:e -e qtpt5ball makes 15.0ptedc45 :-te q5ve q0x revolutions, what is its diameter?    m

  A bicycle with tires 68 cm i(lp vk;(tmh:(enmphj 7x) . s ec*ypo7n diameter travels 8.0 km. How many revolutions do the wheels make? c h7p:p) jey*(t emo.k(lvhpmx; 7n(s    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rdkclwv .+ 9x1ord bkvx4p;*k7pm. Calculate its angularpk14kb xc.r;l* v v+d7d 9oxwk 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. 8–38)y j1zy/v3v:vr h2 bj 6jt1 ublpwr4i4+. (a) What is the linear speed of a child seated 1.2 m from thywjb r/i4+26t jjl1y:h4vbzv3u1p vre 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 j4kd guwtm; 94q:xjbod9cl 3j1+ctm1orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of wy(9s) e.cr fta 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 centrid -apf7i;dp6mi .) htufuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10fdi7aui ;6 phd-)mt .p0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to0 lv:+-bsnttq 280 rpm in 4.0 0: +v-ttsnblqs. 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 radius ,(r ib ytf2i4i2t1 jse:4njef $R_1$ 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 thaj+u 4+whd 58xblb yzpef :n44emselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. edw b:uh ljy4z4a5xbpfn8+4+ 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 3lov/2quc, lzy91m/)o vs gbx6o8wz bto 15,000 rpm in 220 s. Through how many revolutions goz,c/s1x b /l8w2ov)3bv6y lmoz9uqdid it turn in this time?    $rev$

  An automobile engine slows d/b8o)o 0hyvd6 ,p ;mhc(kk.k,cg wwqvown 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 highspeed jets 9tht f/qars;t+b -9qkme :x9bin a whirling “human centrifuge,” which takeeqx-ka t:+;mtb9fr 9 9b h/tsqs 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 : r .iv:jri156 uhlauefrom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in ij1aerli.6u:r h vu5:this time?    m

  A cooling fan is turned off when it is running a7 x gx1zo 3yb(s741d0fur )fmf2-gmjb5sj pzbt 850rev/min It turns 1500 revolutions before it comes to a stop.32bxgfz ro 7 -ssf)bbdxmpm g(fzj0 4y57j1u 1
(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 -sc.3+ka n :jeies te7as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ofie37tc -e ejns:a+k .s 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 mazylxrc6p6vsu4m,9nm n( nyj.jn 5. -q ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 46.rnmj, xyy- jmn (unq.s64l9vzcp5 n5km/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 /sq8 joq:7y tkdx*ym1 bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 ctsm7 :/*xo81dqyjk qym.
(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 th:h d+zvv5 id6ze end of a door 74 cm wide. What is the magnitude of the torqu:z+5i d6vhdz ve 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 wheel shown in Fig. 8–39. ly iozwp(7e0+ Assume that a ywoz0+i (7lpe friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the end)aw9mki/x o- b1 1jfvaexhsrd8h9o 6.s of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the maoo9fa x - 9 vadhs/j.8 er6wb1xhm)ik1gnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine re6 dj ygm:9qift2 v (ac9jgj2c)0n8dosquire tightening to a torque of 38 y68c2f9) 0so va2jmj dj9d(cgt gi: qn$m \cdot N$ 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.d*bu4e1 gcz(x/ca5x ftbm /3p 8-kg sphere of radius 0.648 m when the axis of rotation is through iex cb/ gu/54da1tpxfbzc (m3 *ts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim18qj2/ i ba jspmh-3cxws0l 2s and tire have a combined mass of 1.25 kg. The mass of the jp 10bsaq-x 8i2mlhsc/s3w j2hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, l3e;:sz f1z0i b1 ao9ngeizjn0ight rod is rotated in a horizontal circle of radius 1.2 m.ona001e3z fgz s z9e1;inb:ij Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s whee .qdtx*vd.3il m8cw 1yl rotating at constant angular speed (Fig. 8–42). The friction force between her hands and xyqvc.3id. ldt8wm*1the 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 q40d 3vagcp- bof inertia of the array of point objects shown in Fig. 8–43 pd0vc-3qa bg 4about
(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 consistovo 8) g,ebg0ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sasj/zi0: jxq+9 q ya2use.ct+dtellite spinning at the correct rate, engineers fire four tangentde2+qtqi+xys0 s ja9cu z.:j/ ial 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? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius ob,(w06 qtbqs etudvl.s xh+ix 7 9h2;af 8.50 cm and a mass of 0.580 kg. Calcusbul; h6exxqs20 qb+itd .ht (v, w79alate
(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 sweux:)c0y8 ae rings a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drve3v krbt/g wm/wgi ydm439 8z(3v 7zaiven 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 oppositey dik3 tabgvvm3g /3vz 7e8(m4w/9zrw 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 7rhtw (dk7)nv rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 thvk7dn7( rw) $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 393 edybzm 9- fki;mosf )o59ed.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, whict8s k;:;l*eod5r6 p n:zk bpomii5c32s 5yyda h rotates a5y k2 a6:oekssir ;;nd 35py tdlm 8*ozcpib:5bout the 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 ro(z:zemx-f ,6pw:3 lb kj -1q,hgzalx gd, as shown in Fig. 8–46zlgqbhzpx :3-mkw g(xl:z , , faj6e-1.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of tw. 9e; me2r) iblsq0dhzo 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 t i6 jbi+(md8xhammer from rest within four full turns (revolutions) and releases it at a speed of( j8+ x6mibtdi 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 momenh,dwp f3knjl.f5k 3cst;r*ltk h ,(n(t 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 deve g (c dt84az)t:4jqky38e xdvjlops 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 slippi7k ;zlun*7ozg ng down a lan z7 uz;ngk*7loe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th d7-lm,4bo3shn/i;o*ddtsg+uc h (j z e Earth with respect to the Sun as the sum of twolc o7*;h(szjtdb3 m gnd,hdou/ i4+- s terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How 1b/6wauar-py much net work is required to acceleyb-6/ua1ar p wrate 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 1.80 kg starts from rest and rolbd a4ef73vt4k ls without slipping down a 30.0adv 4k4f73 teb $^{\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 sta +px6hma/619 jlb i k:ve0bgx kinsi65rts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use 6xgp iai5s6jhk6v /bx1m:0e9nil+bk conservation of energy.]    $m/s$

  What is the angular momentum of93m1c-gae;4-kl g u;sn wqaju a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular 4l-9ua n 1jgg-qeca ;;us km3wspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of mt0t 7npk01wia 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whenpmwi00t7kn t1 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 rotg+t, (i u-yg) li5mc mk*kie)iating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spee5 iiic(-,)t ig*e guk)mym+ lkd 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 5fpskw4fhzez7,fu 53 vdvcy lg,;d*0–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what4z7dw5, p y,s3vdkvfgzf5;0c e*fhl u is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate fromu2)s9z wbtl a- an initial rate of 1.0 rev every 2.0 s2s )w zb-a9tlu to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at aid 6t,znw s:q7 jz)xvsu3:nk 2 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 7ivsj:2:zq,t wsn)6nz3d kxuafter the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinnyry22pov5f /a4 g+i 7ema (2q*5mvht qz7wvyring 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 m;gcm ;qxfg 7he0y)qz0omentum 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 46s, cltnznm g9njar8 l6-+ikI is dropped onto an identical disk rotating at angu 4tij+6,gln 9rnmnl-a6 zkc s8lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns vntz)dhmm98n0z k5c*i(z* r sat 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 center of a rotating merc byqt+n*0q p b:23dejry-go-round platform ot3qjy qnc : 2*0dbpbe+f 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 wioe(5 xmiedta*bi6 d8r q kg.5g7-u)azth an angular velocity of 0r*(6gba eaeq-k 7xu)t. dio5ig m5zd8.8 $rad/s$ 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 collapses into a white dwarf, losing about half its mp o 7z/*;6pf2noxyray ass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carypp ox2zay7f/6n o*r;ries 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 wm hi q/(n5ntu0inds 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 w9z0l7 ni9whk$ 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 rel(1gz:epx ae 3-075sx1altmu : jcmbd e(e*nest, that can rotate freely without friction. The moment of inertia of the petee cl j :1l:e7*d(3ea(1sm enmb0ax5-p xuzg rson 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 tcz6,. m+tpe szs /z7u tn:)yxbhe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of tzcte+x.7)/b6z:u n,s pzm ys 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 n;:irlxzh 64 ithe 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 rollsxi4 i;n:l6zhr 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 samzrn - x458(sbn9 wqapqe side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum.9b w8sqra54xn (nq-pz (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 k-d zt k8b7i-u rq,0.igpygc+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 o;pv8t1uakg (s 4c(hwuf a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelerat4pc18v ( hwut(g u;askion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two0c; -zcdiu,9l wkj )w1e7h fov 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 coe - k;lz,7ciwhv1)ojf d0cwu9nservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of 5:t:gm j(fud9yrygz .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 ourh 0fmi+k75 +tgln 4b):xtoyxo Sun, but 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 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 t m5x4th7kn+o)0 ogfi y+l:xbangular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile ypsr( zw*x( df0l*1mb 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 flyw01zr p*(( xl*sdbyf mwheel “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 illustrate 1 .5 6; iddikkhq-6vpivinmyqb.0hd( s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface 6jljcq*a5tt7at 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.2 rmt7zs.pt vdige4 + ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horr e vm2p .itts+z7g.4dizontally 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 theah2px zyadueaa3 )q+l 7 kq:5;q r.p9f 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). 3xq dzp u+qaa9; hq2e yaf7 pa)l5:r.kThe step has height h, where h

  A bicyclist traveling with speed vse sj fmtlny*:-vjexb i7fy+.-va(k 25n ,fc+=4.2m/s on a flat road is making a turn with a radius The forces js b(:meclvsv,ny i2 t- ek- *f++fy7aj5n .fxacting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a s.ijj)t+,g dqo ling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a raqo+i, d)jg .tjte 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 and her arms as thin rodd5k nur39 3ksox 7)mg- usqn4hs, making reasonable estimates for the dimensions. Then calculate the ratioux rd3g nssmu9-34q7 n)kh5ko of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemb tp4tkc3uy q2wi2bo +37gz *caly which consists of two cylindrical plates, of mtzoct2bw g2 ya4pc3*+i3q ku7ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alopgwg 3q ox:uuzk *u-y)( mn/*dng the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest poing*xw3dn:omqy() *g -/ zuukpu t of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do wiw2sh1 7j+ga v9xgl 2not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluty5g/7 h7js zki7a4u utions as the car reduces its speed uniformly from 90km/h to 60km/h Th/ kjy57tzuhsug774 ai e tires have a diameter of 0.90 m. (a) What was the angular 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