A bicycle odometer (which measures disdo 3u5,+-6eyhs8;q ywhw pd rytance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch;eo 5wyqrdyd+,-8 sh6ypuwh3 wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the yyl5l1 ,laz- v6kefvbg4t 5s1rim have radial and/or tangential acceleration? If the disk’s angular velocity incl,5l6-t5kveb sv1fyz a1ylg4 reases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angulau : l3ht5qs1fwb2hks. r velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger bmn3jd5-ps0xy +n7uv -*fpdx 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,u-r*w- zr tjd.*xuvh to do a sit-up with your hands behind your head than whr*juwhurz*d v -tx-., en your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedashb0amug36u ,+pw 3c al cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why?c0+m3 aa3ugsw 6p, hub In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs wit+j.iv1/p wa zih flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass iwv .ajzi1 p/+is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, ni.(p hrvz/m(7e-lc gw;jus w1arrow beam?

If the net force on a system is zero, is the net torque also zer ynjuhn 2;oq2fvu9,l 9o? If the net torque on a system is zero, is the net force zero?ol u2qj n9 n,9huy2;vf

Two inclines have the same height but make different angles withobv-a i)c6;yv sg 09e riy2f9k the horizontal. The same steel)rafyybi6e kscg0i9; - 9ov 2v ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simu04+2.dmmd gklgi.k zxltaneously start rolling (from rest) down an incline. One sphere has twice 4zd.xkl.mi0 k dg +m2gthe 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 samqtq kb)w18o(csnfa 8ia7n ;4 ae radius and the same mass. They start from rest at the top of an incline. Which rea78qit awfo;cna ) bks8 (na4q1ches 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 angl+) 9kj lpifst,0vav 3ular momentum are conserved. Yet most moving or rotating objects eventually slow down ana0)9 tl3p+vvjl skfi,d stop. Explain.

If there were a great migration of people towarde7ozfv+f./m 3 w jt*hv the Earth’s equator, how would this affect the ovj*h.tw/ f73vf+ mezlength of the day?

Can the diver of Fig. 8–29 do a somersault without havm/l7t0enoe(m+ gjyah o+ 46uqing any initial rotation when she leaves then(7+oe0au6q+lygt m4ho jm/e board?

The moment of inertia of a rotating solid derw3 kh m ,rc5f;nr)n*isk about an axis through its center of mas*,5)k rhnrecrw; m3nf s 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 holdtpv m jp+dig--i12j8o ing a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocom 1gp+8i2pd-i-tvjj ity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the + en*dmyvdmk6g1-8 xtsame mass. However, one is hollow and the other is solid. Describe an experiment tkem dvxmgt8*-+6 yd1no determine which is which.

The angular velocity of a wheel rotatin gwsak7uu+ol; py42h4qku/ k/ g on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed incpy lgs4+k24 uo ;wqku /7a/hukreasing or decreasing?

Suppose you are standi x(g3c6 :ibuxsng on the edge of a large freely rotating turntable. What happens if you walk towardx b: (u6cgis3x the center?

A shortstop may leap into the air to catch a ball v9csdsf v:h ;nkd:).mand throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quicksvcd k.f) 9sv:dn:m;hly 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 momed3x os1ryr5+ untum, discuss why a helicoptr 5+s3ro1ydux er must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following .8h z-jrin r1cangles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because o7 mh0hhp.u ;qqf an amazing coincidence. Calculate, using the information inside thm pqhh;7.h0 que 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 Eartdxfw(+7 .gsas(cl v7z h. The beam diverges at an a(fz .x7g +s sw(dvcal7ngle $\theta$ (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 rpk4ljl2cp 5 ok*v- l8vhavo( 8tm. When the motor is turned off du24-c*lkvot8p ( kl lj8va vho5ring operation, the 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 a,stt b mjq 1(ydo.4:ul 5ziiam12wc3level floor 3.5 m to another child. If the ball makes 151 dsw1bi:qmtyi, 4az (.oc2aj3 l5umt .0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many re sk0ftd w.uz8ubr;- 1vvolutions do tb.ur1w-vsd ; u0zt8kfhe wheels make?    $rev$

  (a) A grinding wheel 0.35etcs cb5 z6*/p,e;vo,0gaff sal 6 1np m in diameter rotates at 2500 rpm. Calculate its angulal /ab,;6sc ffa ps voe01ctzpng5,6*e r 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). (aab *jc 3.wzwg-) Wha.* 3gjw-a zcwbt is the linear speed of 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 Earth (a) in its orbit around the S,t 2al yl3rgp+*ju:adesrz1i2 d( w(jun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point 27 sbf:hwa4pp
(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 centrifugey*-h9 ut*vhy x8ddhs 0 rotate if a particle 7.0 cm from the axis of rotati9- 0xvyt sh h*dyu8d*hon is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fxxp4 r4(v h)md:cb.bhrom 130 rpm to 280 rpmbhv x:.p4xm dc(r h)b4 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 radiujelevw) ikk/jt (p .oq:9. 6shs $R_1$ 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, astronau 4s+ ydau ytm:)cwvq/3- )wlbuts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revol u-))v+w :wuacq/ bmy43 dysltution 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. Through qlapy*p , p((v5 p5etlhow many revolutions did it turn in a(e ,ppty5q(l*p 5lpvthis time?    $rev$

  An automobile engine slows dowdqlb6o i y2hu6x0ym18n 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 iw nn on ;q(d9n ka(kk a)6juhsd:;18umn a whirling “human centrifuge,” which takes 1.0 md ;9wjk)nk h;k(u 8un(6d:samanq n1oin to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 24l cm,ra,ac*w x9*o vol h6rn1)0 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in,cl * r6 mr)ovc ,1hn*woxaal9 this time?    m

  A cooling fan is turned off wcw( d(1lw;ttlyzw4,ahen it is running at 850rev/min It turns 1500 revolution tcdzwl1 waytw(,4(l;s before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniziod3/ sdsi zm3j(dth 8*6aw,formly from 95km/h to 45km/h The tires have a dd8 h osm*i, aw6zztd3s (j/i3diameter 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 aoylo ;.75 kazja1jubwb 2*nq)s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0jl1kn7yaob2.j 5a uqo)zb;w* .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 weighjf7 cp. /jp,xgt on each pedal when climbing a hill./jjp xgfcp,. 7 The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whato) 61vbcion4j cg a(* 4-szjpz is the magnitude of the torque if the force is exe(b4vnjg izj4 -1)cpc6 o* oazsrted
(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 dr*5tn .qdjoq i68 /q,xm1 hfqwheel shown in Fig. 8–39. Assume that a friction torque of 0.dr 5,8mi6qox/d t*n 1. fqqjhq4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends g9sig*l/h:1wu*s canof a massless rod which pivots as shown in Fig. 8–40iw*ncss g lg 9au:/*1h. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder heagg 2m:*fm .x) *bzl 6p* wt*uffaum0sqd of an engine require tightening to a torque of 38mum:bt 0f6m lg 2a**.qfuxf*pw*gz)s $m \cdot N$ 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 of radius t9 4qdkh jt)s 2)j;yhe0.648 m when the axis of rotatioyqhh;9dj sj t2)tek4) n is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thevu .02j kv*svj rim and tire havkv*u2j0vj . sve a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rot+h3d me2t zmth9zgv 5p93b:ef ated in a horizontal circh9tfm9 he: 2t3 zgdbmpve+3z5le of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowopjn wjk.p6/rhtqsuf.5q sr36 9w -2nl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her h spun/-r9ws.3. jrjno 65f2 thqpq6wkands 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 invmc skd 5k2;0n6sgn5oertia of the array of point objects shown in Fig. 8–43 k nvndc6g20 os55 k;smabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms wh*nv:mfwg jlt m:of q6l9z gwjjft(yw3+(1 /6kose 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 cylindrifggqox:ys2bb yhkpu) ) d+i9h64m,h+cal satellite spinning at the correct rate, 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 tg,:fyobk +y9imhu) x b)p qs2g6+4hh dhe 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 rh5x kb/ l6uua,nq3am/26 vmh fadius of 8.50 cm and a mass of 0.580 kg. Calculat /x2huq,um h5l3kmfa/n6a 6vbe
(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, acceleratixncyx, ;m8h(png it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is a lk+2 ijurcg;: 9sugy7e 3z4mvble 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 kgivgc y7:9rzs43m ;eul kj+2gu, 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 is shut off and is evet 529 o2zbi lz+rposczu5 xj4h+ 1xw:zntually brought uniformly to5h iwt5b z o:9x+zu o4zj+p2s12lrzcx rest 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 acceleroh rjnr5 i2 fwji;:5h8ates 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, whic f4o96k4 ek 79ds)o:frzaqqm ah rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uf e z qqor)ka:697d4o4sma9 kfniformly 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 coz x0:k 52 ks9dfp2w2zlz* wgmonsidered a long thin rod, as shown in Fig. 8–4 p2lx*5mz9 ks2 g:kzf0w2ozwd6.
(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 ma86hh7)uyt s qwsses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withi i3 dtkh+vqk*9zs /-gfn four full turns (revolutions) and*qdh9/kk+tgi v-z f3s releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of int2hz: kuvr l1o77- 7fi8ngc.stfb8zv ertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 sg:lgye v /svu/(47b j)f/zvt $m \cdot N$ 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 5x;wmu 6b lh1 bgp*0qsdown a lane at 3qlxwgu6 m; b1*0pbs5h .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic eg ks/g0whgc/9 adk7b,nergy of the Earth with respect to the Sun as the sum of two teg/hs,wdk9ggc0 ak / 7brms,
(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 3e45w hmvvnh;b ur gb .2)t,hxand a radius of 7.50 m. How much net work is required to accelerate it from restthbm4wb rhv5nh2e, u3; .g vx) 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 rollsv uxw+yq: ;w.s+zc wp3 fmnbz+ffm,n- f/5dw7 without slipping down muq/y-.w pm+73,5wd w +wxsz +;bvf:fncffnza 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 fals9*0szl)nui tl and its lower end does not slip. What will be the speed of the upp)nul 9tz*s 0sier end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mocgi+n0q(*r+ep yq o:umentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angu0q nye+i+c q rpo(*gu:lar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mo rqp/ bswo6,7jaq/a i.mentum of a 2.8-kg uniform cylindrical grinding wheel of radiuqaa p6qjr /s.7ob, w/is 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 hi ptlpmh4 tuvs49x ;47ss side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positi;l94 msuv4 ts4ptp7x hon, 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 of iner// recz74x a: 6xdpadrtia by a factor of about 3.5 when changing frrcp:drd 47x6aez/a x/ om 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 rah4 m6svo) (xsdte from an initial rate of 1.0 rev every 2.0 s to a final rate x6so(4smh d)vof 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 aw+w2oa0 lbx)mt 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 after the clay sticks to it?aowxm)bw0 l+ 2    $rev/s$

  (a) What is the angular momentum of a figure skat fg sm8-1c*-uc :blgjper 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 Ea20cxxc,d:qif 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 cylindh, 5zgi*6ylt6.n a inurical disk of moment of inertia I is dropped onto an identical disk rotating at ang. *ti65ug a6h,n linzyular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 q fo 7 q 65ptq6i(etio+wh)tj($rev/s$ 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-gzz:.bz-1,c5ig1 jjfk bc i) uio-round platform of radii-, c11 k:)5gf.bjzujzibczius 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 rotatingub e: pt8e )4kzjjm6k; freely with an angular velocity of 0.8 p8jmkk 6b: ;)tueezj4$rad/s$ 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, li6 jqw/cxn 4(eosing 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 /wqxjc 4n6i(e 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 in excess oeu)e2c i q6q/yf 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 /7;hl /w7*nox/ ud0x* btc8nkk+yialaxvr;k $ 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 witho 0vkx(: xoldp9x+xvb 6ut friction. The moment of inerbx90x+(:x6v l dxkpo vtia 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 pers85 wjodz7b 5y1w1pdgron stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearingjdryd18b wp5w o15 7gzs and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the endxbhr sc*zz)wm 6p(w;e3-h gp( of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a w3;c*wp(hm(g- xz zb)h 6rpsedistance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces thq+8/p9hvli j /zf y2oze Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbi/ ljy9vo+f2z8p/zh qital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate :d :cbfs jb*1nq2u+ihof 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 ilom i1de2z;(pn the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torqueiz(1 e;2dlo mp delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 krur+p c;gpcf,jqc/p8 x 5;2cvg and diameter 0.075 m,rcpr p2jxgpc5,q ; +c8;f v/uc 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 is the angular speed of the rear wheel (5qp c( tf0:7bjix 9hqq$\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as greazn 4izrl6iu-+ oi e,-tt, 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 u4iiteo, -i+z -zulnr6niform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fl8x di e;lb,s8or 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 35le, 8sb;8xl id0 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 illustrate3knza;m/w u3 es 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 rol p0/og )owh7q i-qoa2eling 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 txe o:bh.s+tlh.2s0r/v z 0t tfriction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the mome .x/b:0z e2s ht.+0v ortlsthtnt 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 r4qg)2t d)blq vadius R. It is standing vertically on the floor, and we want tvt ) bldq4qg)2o exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with swy:-xe yt hq+3peed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclistxe+ yy:wth 3q- 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 s9jtk . ;mp:*lokbuy z;qi)z ,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 rl.t9uz y:bkk;oi*mqj;z,ps ) pm 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 ase: m:qq0vby 80 orlhh-u9l q5l thin rods, making reasonable estimates for the dhve lu mq059ll8 r-h:qy :bqo0imensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pl xn.(41fhqff+en 0qdt ates, of h 1+ nfff4 xq.0de(tqnmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alon0,zq7c(madu d43a n 2g. 4rxudaernj 6g the looped rough track of Fig. 8–58. What is the minimum valudqar0n7am3dne2.4,d4 zcj guu6r(xa e 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 assume a8c 01dlga trdj4r+t9$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformp:bp q/*6yy ztly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the az*q :6ppy/bytngular 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