A bicycle odometer (which measures distance traveled) is att,sg8zna(ia2 p0f8 k5ergjec; ached near the wheel hub and is designed for 27-inch wheels. Whatk8gga j (08np;sec 2fai5 ezr, happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constan 2u22u ee 3) 8qasolicere1y.jt angular velocity. 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 accelera2u j8yes2ue3)r le1io2e.qcation? 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 angulx*d6f7z -uz kffp8+owar velocity $\omega$ Explain.

Can a small force ever exert a grean5txv*6lfc fq(c 5 rk*ter torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behin7sm3bvw/j/q yd your head than when your arms are stretched out in front of you? A diagram may help you to answer t7b w3/ys /mjvqhis.

A 21-speed bicycle has seven sprockets at the rearncxa68,w p-*hncelw7 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 fr *l8 6,pw7ecwxahncn -ont sprocket? Why?

Mammals that depend on being able 7yuj q4e8koc8u .m/rz+cp y6dto run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, uc j8k6ce om.u87yr4py z/+qdexplain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cagr mzimb6. y:+ +tzch6rry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the ney.b93hq9m kjcm0( f ott torque on a systemkm9oqf(c hj.y0tm b3 9 is zero, is the net force zero?

Two inclines have the same height but mak1z*e5lfknxhl8uw bekmq9 4.z ;g a71pe different angles with the horizontal. The same.ekwmn5qpg u x *b7lz4k e1;9lz1f8 ha steel ball 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 (from restl ckfi(9w fk;7 r)v2mz) down an incline. One sphere has twice the radius and twice the mass of the other. Which reac(f;ficw k2 v)mkl9 7zrhes 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 sa ek y 54b9wb6vs**bj1wvs(dv nme mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the b(* e 16snvk ybj5vs9d w*v4wbgreater rotational KE?

We claim that momentum and angular momentum are conserved. Yet z eni,l - 70d3ho)ovjdc,3ksrqwh3x8 most moving or rotating objects eventuallh)se,ki h3zc-3o,x lq 0vr8o3wd d7jny slow down and stop. Explain.

If there were a great migration of people toward the Ea5xkf(3hr tdlo 4q5u0 krth’s equator, how would thur3o5(0k ft5d k4x lhqis affect the length of the day?

Can the diver of Fig. 8–29 d0 8cxr-8o nigco a somersault without having any initial rotation when she leavn0ixr 8c8c-g oes the board?

The moment of inertia of a rotating solid disk about cl: dvj58qatp2ui5 6 8i-d/o4eumro dan axis through its center of mass is c8d5r24-/:ta8iji pum od5ueldvq6o$\frac{1}{2}WR^2$ (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 irr +7bf2zzo4 a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay t7o2 fizzrr4+b he same? Explain.

Two spheres look identical anppi1n :y d:,) ixy5e 8ua6ycrbgja8 6xd have the same mass. However, one is hollow and the other is solid. Describe an experiment to ynirpa )uj6ic e day5gb8,6y8x ::1pxdetermine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In 114mw af4ehijpk7m 6q what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, descri7jew 4q mh4i1 1pam6kfbe 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 laratpd6z9m6 vp4nka;u1 ge freely rotating turntable. What happens if you walk towvnm;9za4u1 k pt a66pdard the center?

A shortstop may leap into6x7sv lv-rk8(jeo 8fu the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fi6fuje vx olk7r( v8-8sg. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wh*q0rr npfzap5es :6 ) itju.d9y a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propel*5 0a)su.fn jr6ie:r pzpdtq9ler can operate to keep the helicopter stable.

  Express the following angles in rlvuu7qh-5vn 3adians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an am . 0c9a3lidtxofoht(,azing coincidence. Calculate, using the information inside the Front Cov xt.foc a,hl3d(oi9 0ter, 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, uc;h:d k,igh km from Earth. The beam diverges at an angl ,,ikchhd; g:ue $\theta$ (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 turnre*.9g f6(x3ynrz nz ved off during operation, 396vf( * .rryn zegznxthe blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If theb:rbtll-5 5 ge ball makes 15.0 revolutions, what is its di5gebt5 r: lb-lameter?    m

  A bicycle with tires 68 cm in 1m ptyc6 id/q6.u*k vbdiameter travels 8.0 km. How many revolutions do thk u/6 tc*qpmyid1.bv6 e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculz6lw an g.q+yb bw77e0* gv;dcate its angular velocity in .0vl7+ wy;cwz*nbad e qg76b g$rad/s$ $\omega$ =    $rad/sec$
(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.anun+ck4jij-3yj21 m0 s (Fig. 8–38). (a) What is the linear sp1nmj 4ja+j 3 kcui2y-need 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 vel6fxcra1bcjhc603lq25e m 6vt ocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp:ev j3nm28 o)st 8sp( d(fcumreed of a 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 particlpvyorq2 z0 :lxx3o.k9 e 7.0 cm from the axis of rotation is to experil39 zqyp:r0k .x2o xovence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about it+84z c. epqchd lb+n3ps center from 130 rpm to 280 rpm inbc+3p.p d8h+qc4 ne lz 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 radiu6js fa2hdt+9ss $R_1$ 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 sa,b:y5k m v-wk zs9d .5o.eyr s,g yxmt.ud,e9pacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rog,m , ysbx :kdm.5eday y9ro-u ek5,w.zt9v s.tation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through hk3-zdl0e;(qlhqx jq 4 ow many revo0 l4-hel;qxz(qd qkj3 lutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 sdxf+8o vzek-16fig;ip3o o /k. 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 +u,g hth8ez0 gjets in a whirling “human centrifuge,” which takes 1.0g,80zghhe t+u min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rp pbg,q+z-oa r8m in 6.5 s. How far will a point on the edge of the wheel hav8z qa,rbo-+g pe traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/minfzin c 44-p,(7e)gbn6iobtt.v de 5wl It turns 1500 revolutions before it comes to a stinz,e.4 v7e6w nfgd-)(o 4 tpli5bbt cop.
(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 unk*eddq -unn1j( )7yp eiformly from 9u qk n* jpd1ed(en-7y)5km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces it okbej;+prqkni3 h6*y3tp:5 w s speed uniformly from 95km/h to 45km/h The tires have a diameter of 0 tk5p6*ph q:3ro kn;wby+3 eji.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 biri0+ vh1q869y bzzulxke puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiu hyvxz rz8b1 q+u9l0i6s 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 o(1xbvjrw pix/z4c ug 18dn1n-e/h ri1f 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exnrr 8w1(iu4x 1pvb - /cje1xd gz1/hnierted
(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 a y051 kuuyu euek-gtv8q79mj 0bout the axle of the wheel shown in Fig. 8–39. Assume vqgu105 j7ytku yue09m ke8- uthat a 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 ends of a massless rod which pm /bq4mx2garf+1(i m8- qbnslivots as shown in Fig. 8–40. I/q mmr(8+i-1sqmx24 abbfn lg nitially 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 head of an engine require tighteninp*: a. f:znnu+e/mek94hlp i ox :-khpg to a torque of 38 9oxpp e*-mf :knapni .:/uh: +e4h zkl$m \cdot N$ 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 o g4lcft wtsqs4,1yw::u -:2icfmf27ks:dr l of inertia of a 10.8-kg sphere of radius 0.648 m when the altko-: g 4wur2dm qs7sc:tcf,ff4y swil1::2xis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertiaasg 0ormu6z mkwi/l5q 5- 03uyg+za)m of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the h -zm3)gzqg5 mor0aa50ium6 +yls/w ukub 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 hkaj8 d8s s(vhqs ,k2:0 .jwqz.9v rnh;ad.dxxorizontal circle of radius 1.2 m. Calc8xkw(s 8j zvs.,k9 v jhda0qsdqh.nd;.rxa:2ulate
(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 wh vs9 v( jcf6v2a9t*zg 4+gku,rd ip/nrhzdv:)eel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and therdzzt +vs u 9 pdcj k:g(hn afgr/ivv*,v64)29 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.y6gdn3b7+ q zkbt; ck of inertia of the array of point objects shown in Fig. 8–43 abo b.;36d+ckzyq knbt7gut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose tml 43zb31baz-or r,xr otal 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 spin myv4wf+;vrakys.3/rqx:ammr61 zg0 ning 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 the satellit+va yrmzfxqs6r:v 301mm/w.y g;kr 4ae is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a ua8gge;: q tnni.e/9q -i-e;ikcx cs7qniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. - 7-9at n/xnqii s ;q;gqg.k:8eceiceCalculate
(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 ddqa nk3 w)1z5/5 smzrit 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 uphjtqe +g.hjo2x:5k 11 gs(lable to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produc512 u1jqtjxeoglhkp.:s gh (+e 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 ecz.)yw.5+ rf(nzs;mlbb om,gventually brought uniformly to rest c(.bm.z f,+y5nl;ogzsr) bwm 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 accelerates a 3.6-kg ball at jh 8 uf,feux8)xo7a1z7 $m/s^2$ 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, whichc9r 1.qefkqj w579-p8ben yih rotates about the elbow joint under the action of the triceps muscle, Fir1jfnq. kpqwch7 9 eiy 8b5e-9g. 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 beyb/pdh nq/1 k 6eqd0m53ha.buhqkg(8ps1 f9u considered a long thin rod, as shown in Fig. 8–4b9gk6bhfu(0ha15 /en3mq q.1d p/qdhy8psuk 6.
(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 5zmjq72 nk13xd :nq;1gqtd9wa3rsqe two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates tbz2bz )r u1x6 m1 )jut1ahpqo.he hammer from rest within four full turns (revolutions) and releases it at a speed of )1.tx qmohuu2a zrbj6 1)1zbp28 $m/s$ 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 mok8 kue*q:0aup ment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of jp)yjz )g8(dbh : zpv9280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down y3)lbc(y(p jaa lane at 3)c (yjlp by(3a.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect: j an,jhjm(,m to the Sun as the sum of two t mjj jm(:,a,nherms,
(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 has0 ev 8xl;p7arm a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate itrx 7ea;v pm0l8 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-- gm9uah py1fv byd*: skox-dy0d 6)m rolls without slippin*1oy yg0 hb:)-y kpdm 6-fsaud-m9xdvg 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 vertically495nem*7jsts n9 audh on its tip. It starts to fall and its lower end does not slip 4nh9n95 *ej7tau ssmd. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on th;y*-fogt0 ybne end of a thin string in a circle of radius 1.10 m at by0o-tygf;* n an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angul e vhzl7).s g;mr.ppd-ar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpep .)-lvgsm;7d. hr zpm?    $kg \cdot m^2$
(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 rotatingnckm3*cdnrr 9 h/bz*/ at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–9/knbncr*m*dcr 3 /hz48, 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 re unzo5ru( w*54yp)j mp3c(m krduce her moment of inertia by a factor of about 3.5 when changing from the straight position to thypuk o (3*cw4z5(mrn) 5 urjmpe tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate fj k3 hze ws9r*u6l8i5srom an initial rate of 1.0 rev every 2.0 s to a final r hsu9586rk 3 *ewjzislate 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 rotatinj9r*9j 2qjklw x p5tv8g around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and 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 thqjp8v2tjk9j w9 rxl* 5e frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moment p+( 3 dgf.edoov0rzlqw08xf7 um 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 xr w9rvt9iobh 9d0c)fie9l),um 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 inertiklh+/t/w5pzag du 9 h: 1grp.va I is dropped onto an identical disk rotating at a t:91 hzp5wph.+gl vr/audgk/ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns f7dyija1f y sc.h0)q8at 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 merry-*bj6ygs563 pwqa y h9w)t1p3d)de xungo-round platform of rpdn3 s3a6u h)jdy yge5q9 1*tw6 )bxpwadius 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 ro vu xdnnc+73y6/soy- ktating freely with an angular velocity of 0.8 dsy-3yc v7/ nk6u nox+$rad/s$ 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, l2 klp4lx4iv gk1y;qb.r k5;ybo bs*w-osing about half its ma*lok2 4i 5-psyl; qg.b1vx yb4w;bkrk ss 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 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 involveo; qvc j,dzq+* winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass uw b5gxjxbhan 32;l88$ 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, t 6mn g9v3bqsq/4 x*jlzhat can rotate freely without friction. The moment xm 94l* qvs6j3gq/bz nof 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 edge of a 6.5-m diameter merry-go-roundx8qezz5yn r)o)9na l4 turnt4z5lno aqy x))9en8rz able that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the j vgp,ouhqyk)y; 0s26 jiv1e0:ac;to ground with the end of the rope lying on the top edge of the spool. A person grabs the end of tht;sj gu2 jhv;o y,poq cay:ke060 vi1)e rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth sudr*)ye5aumba14 dok b91 lfx b4 a3i,rch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particl1kadl a5rob9raf, 4u4e 3 d b)x*byi1me orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/rlkibndo/.4.8m / j1- dcl nzgli1lg2 $s^2$ 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 +befe;*sqszy /oq6r mvl4a:x n ;.g+ yof a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constantfze/sgyx qre v*;;a+6sqblo.n:4 +my angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diszjesh (x h r,,v3rp)/fks, 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 0.010s3(x jvfp, h,hez/rr) 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 angula, t(xb: .jq7zt kgcue55gmuu/:x nwz4r speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rxwp -+0rzw1, u,n kjqvotating at a speed of 1.0 revolution every 12 days. If i,vr +w,pk0- 1wuqzjxnt 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 angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobiy ld9m(bq7oq5 xuk 5na ryuz/y5,13vble 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 d15ulzr /y yx3(b957 dnuy5qqm vok, abiameter 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 illustratstmm yiriq(:9b*ey 14es 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 rolli1i0 umy6szt)w( ,, pny+ ntj9cdmb jx4ng 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( v) h p:i*elkgh:zq.7n(k wer freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determi:(7 .r:keiqlwe(zh*)gnph vk ne (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 isd;ompyuc , 1(f standing vertically on the floor, and we want to exert a horizontal force F at its axu m1;f ,ypdc(ole 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 speed v=4.2m/s on a flat road is m+5vm,k/pkvp 1 u tu3ftaking a turn with a radius The forces acting on the cyclist and cycle are the nok3v tt,5up/u1 +km fvprmal 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 length 1.5 m and begjdaj+a ex.x3 3ins whirling the rock in a nearly horizontal circle above his head, accel3x 3ja .d+ajexerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rodsla(tg87ze0vngvc( ( g.ep.s+pnaw h 3 , making reasonable esth0 +gtsgg n v (3p.cnev7.ael8 (pa(zwimates for the dimensions. 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 platesbkg ass7 8u( *5ptede), of mass (ses 7ke8a5u)pbgtd *$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a92vb,-t e 7 fy1gcrnhqlong 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 point of the loop without leaving the track? Ass1,br yh ngvc-927eqtf ume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nolu8l8c. 6b ubst assume $r\ll R$ h =    (R-r)

  The tires of a car mzq-ln .9nwmi 5ake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires zq. -liwm9n5n 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