A bicycle odometer (which measures distance traveled) is attached near the whewf pp(:h)bd/uys5 t.ael hub and is designed for 27-inch wheels. What hpby. f/p)dswhu t: a(5appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cet / ds;oq 33air:)v.vnnsmg 63zu6iqonstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If th:m ;.3znqr q/ati idgo)e3vv66uns s3e disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a ofxf3tlb p89+ 2bb bw/:i2kkcsingle value of the angular velocity $\omega$ Explain.

Can a small force ever exf;* th4/f qvqsert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your han l4edxam7 r /b ny.ae4xht6m8(tajw40ds behind your head than when your arms are stretched out in front of you? A diagrayt4na0b6m4 th.la w mr(d/ 4e87eaxj xm may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whe) kr e73jfh/zv l 5:1oihaned6el and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a larl)h3 :h6zaj5ikrevond1f 7/e ge rear sprocket? Why? 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 fas0 :f x b; vorfixn5w9qo41)hmft have slender lower legs with flesh and muscle concenqwv x;1r)0n hf:xbfi5foo 4m9trated 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 (F*m9wbq;o ys40-pk i koig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero?fe*x 8 fo .2vje4yhci. If the net torque on a system is zero, is the net forc*4yehvc f2 ifo.8 exj.e zero?

Two inclines have the sa7 9g zxr5f;ro3taff,w me height but make different angles with the horizontal. The same steel ball i 57r f, a93;zfxgrwftos rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulfli/jh3zr83mioj, a)taneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic ene 8 r/3oif,ajl)m zhj3irgy at the bottom?

A sphere and a cylinder htrr pc y;8*nik(yowav6zi 9(3ave the same radius and the same mass. They start from rest atr8 a3 y iwkoc(tp6nirz9v*y;( 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 greater rotational KE?

We claim that momentum and angular momentum are consera+ ob4x0 4v0.kr*ydhzyw)y toved. Yet most moving or rotating objects eventually slow down and stop. Explzxy dy440o*kobt.r0vwa + yh)ain.

If there were a great migration of ped0 snaw7/u zd z r8sd+-e(di.xople toward the Earth’s equator, how would e(z7 zxs+sdrd/ 08uw.id dan -this affect the length of the day?

Can the diver of Fig. 8–29 ;un7j+*lhi nhgeoo8n8vn1+ jdo a somersault without having any initial rotation w*v81+nnu7ojonn h+j8l; he ighen she leaves the board?

The moment of inertia of a rotati4 hmrp(uly+q- ng solid disk about an axis through its center of massuh (lm4y+q pr- 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-kgzw974n.lfw jhxn4p;i mass in each outstretched hand. If you suddenly drop the masses4j;7 lx9nhw 4n.wzifp , will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look ideni e (sd mkk2z5szo-p01tical and have the same mass. However, one is hollow and the other is solid. Describe an experimek mpe2(osdk50iz1zs- nt to determine which is which.

The angular velocity of a wheel rotatsmia p wz :/um4qc5ic2zx*:+jing 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 acjc4/2:auimmizz+pxw q:s5c *celeration 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 rotatinngb9n; +.gy yeg turntable. What happe ;bg.+g9ney nyns if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As hg;iaj4 4k cskm zgq/67xrrp,8e thprzg4akmki8g/;7 rc q,4s x 6jrows 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 (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentumoqzm k9+q31p 0:nc ncs, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or c 9qn0qp3+n szk1m:co more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles inn s6* rebu(2ga,te5i6xkq oc6 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 of an amazing coincidence. Calculate, us8gum.bs:kzm;5u8zxoc.gg n i.q k6b *ing the information inside the Fr gm:xgogimb 8.u*quk kzc;n8.s bz5 .6ont 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, 380,00) 7 wxzip6n 28ep:cvyx0 km from Earth. The beam diverges at an ap7n8:iy vxcez 2w)p6xngle $\theta$ (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 turnedtav+o cz/m t-psl+ 7 60yly fi4oay6o/ off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades yl4vlmsp+6 c7o oi /z60+/yaa yftto-slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball m98ptdm3.m ewb)gvc x7 akes 15.0 revolutions, what impv8w3bd c79t e mg).xs its diameter?    m

  A bicycle with tires 68 cm in diameter travelv rhf )j:)v,jvs 8.0 km. How many revolutions do the wheels make? v )v,: jjf)vhr   $rev$

  (a) A grinding wheel 0.35 m in diameter rot9xys ai;d.4k: s4ze14+lpwcmd (jom z, 8jfakates at 2500 rpm. Calculate its angular veloc jp9 4i8 ow:yf .m1(;emdlz ka,d4a sjx4cksz+ity 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 inth rg8xij .c42 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the rx8 jthc4g.i2 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) ins d-bag -bq)nkmhlt r 9a+3;3f its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin y2tyjb h9gq*7x ykig:0c(im; t
(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 rotat)makdu9g,e*bv;o2zd8,n ue ie if a particle 7.0 cm from the axis of rotation is to experience nzuoig;8,2 9edmbvk* ued),aan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ce : b;6ko5delxhnw+axk3v )/nz nter from 130 rpm to 280 rpm in 4.0 s. Determinexza;bk3ndkve:o6w)+5/x n lh
(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 2 2;v u(s be:9gap1q2jyrmbg nie:q,n $R_1$ 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 space;ox6cuh 9 1zrq9b-c:j1tasi jcraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At 96cs;1zxi:9qjbu orja1t-chthe start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 18a/(cwirep, 0 vazz e95,000 rpm in 220 s. Through how many revolutions did it(cz 8ipa r90 z/avw,ee turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to ( krqlr0m.:j) vicq u,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 wcimj- b m u8r5+-lh 2hlhd:y:djk33sjets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching itclw5 b3i 2hrlk-mh8+:myud dh:j-sj3 s 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 o48qo-39k im7tqh )i1 munshy240 rpm to 360 rpm in 6.5 s. How f o4imh hqu 8tn-yi)m1 7kso39qar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 83;jxjs 1 p2eoj3ffptn(xj )a/50rev/min It turns 1500 revolutions beforfn;12 3eo()j3/jfx apjjxs tp e 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,/kqoie c q-3ugk4/rl the car reduces its speed uniformly from 95km/h to 45km/h The tires hq4l e/r- kqik,3/gucoave 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 , hupnpj772fu;oev2cg3 hxf*its speed uniformly from 95km/h to 45km/h jf3c,fp2h uu7pg 72;vexh *onThe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her wewm9vwrj,))u -g yhgdhr 52 4kyight on each pedal when climbing a hill. The pedals rotate in a )g)5wuwy k 4jvr, hr2gyhd -9mcircle 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 oj2 ieudfe-3j+ep9qae9 cn( k+bws 3/gf a door 74 cm wide. What is the magnitude of the torque if the fondfq(j 3ug j2epe/+k+3 -wb9a i 9ecserce 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 F8kv)gmy3t;)i bw6od / , ihmruig. 8–39. Assume that a frictb; im86/kvg,wi) m y)uohrt3dion 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 tdy ;78zm-c cadhe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calcu7zcdd- ;ym8ca late the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening t0 gg15kmyras.+wu1(rwbcaz q6*hv0 w o a torque of 38 s v.r aw bwzu1h10c5 gkm6qwy*0ag+r ($m \cdot N$ 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 radiu1yuw2n5i mh r:ybt+x x03+cves 0.648 m when the aw5h0xtni y3m u+1:rvc 2+bexyxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in pfnmt c qe6(ju4/ne1ri*ee 35diameter. The rim and tire have a combined mass of 1.25 kg. The mass of t/ ret epie*qfm(3 1n65cju4enhe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on 1pqmbo 0 9zde4kpk remv-fpu) da6, kuti :*)-the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcu4-,feir q t)zp mvddekp mu*1 kka-0:9b)6poulate
(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 wheel rotating at constant animo6:rp d a: 0uy wn5ni,ixs9-p58gvwgular speed (Fig. 8–42) 6,uoy0:58imv nd: -i apiwpgw5s nr9x. The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in rnsqto:/.x-od gdq9 4 Fig. 8–43 abo t.4n gq/os :9drqx-dout
(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 twseag -; kin :ni.st1 uigs(-u.o 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, uniforms xr+vou/- mn; v:vo,l cylindrical 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 t-nm: + ;s l/ovurvv,oxhe 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 c2w + -f0;y w 8 x82ltssixajx.2iclhzbylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate 8fxtlchsaj yl;x-i2+z 2w.wbi802 xs
(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 t tk1 owrhymn(8.bthm )az8s4/:n/psa 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-dpfud0)zf:qg x02 r(zqwy+ xk.riven 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 .x dp:0wzq ( fguk0zrfy+x)q2mass 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 xjiq 3i.nnkv,n;iy2n(c2x j,;g. y wtat 10,300 rpm is shut off and is eventually brought uniformly to rest by a fricxn. .yjcv,wy3, t ijngiiqx2n; (2nk ;tional 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 acceleratx9tdstc882 ei es 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 thrf.aja *v)d(d xown solely by the action of the forearm, which rotates about the elbow joax.v dj)d *(afint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in n q(rgn.;c)eaq2s7 mx Fig. 8–46.m arq.en7 nxg sc);(2q
(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 twof3-pbkh 3/ydj 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 acc 8 t3-7+;jm*a5txh mclireq agelerates the hammer from rest within four full turns (revolutions) and releasese;+ gx8mq a rt-j5lim73 cht*a 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 inertia of l(,b5 za,q afc wp2zz($3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deveq4 6;fp0kq 1)cxwye d6iyj d9qlops 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 massn 5+nqp4nypj - 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3. n4qj5 p+p-nyn3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with reskbz1/ eas k7:wpect to the Sun as the sum of two7 /askw:bk 1ez 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 7s3gkv t ut)p+ j3bbzia7o7l:and a radius of 7.50 m. How much net work is reb73ja+utzvo:l3bp i 7) g7kstquired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withox322 pe4vksfgd./ wemo )u)7p, jajhi ut slipping down v j) x g3he.47aw2opmj u2/p i,ek)fsda 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and rfnklok i swkso )(rt3b)bmz *9z89ury./*8w its lower end does not slip. What will be the speed of the upper end of the pole just beforeko nr9.i 8sb3y)( l/swf8kwz)zbruo 9k rtm** it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momnsow9 s-vl7 0h k.z7uumb )bp.entum of a 0.210-kg ball rotating on the end of a thin string ns s7b0mbp-.).hwo9l ku7v zu in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding iat8,tzn2-a 1r vcd1p wheel of radius 18i8,d 2ac1tvzra1pt -n cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rj85pncg ruh.g ja )3s. 1mqs2votating at a rate of 1.3)82h qvjuss .pjca3.5 n gmrg1rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can pmv/by;(0z;(2 t xwymi,sslpreduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the ppb(s,xi(/s2lz;ymtm w;0 vy 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 izvkom*hys +92 t :yas (/hvua7ncrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finalhvvssty* amy+:a k/hz9(o 27u 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 throu 1+oo1cjf2o aagh 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 rotfooa+ao2 j1 c1ating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular 4 wf0ro0j,( yeg zhz0ymomentum 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 momentum of the Earz r fgez w81(g *pwtm8c62wqsm3*b:coth
(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 mome0 afiijb n- cip2/jhy1l9k 2/vnt of inertia I is dropped onto an identical disk rotatc 20lfjki ab yjpv/2n/i1ih9-ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 1 -aeuyn* *vdyv ) .ji)mod4ycat 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 mekf lar83 4w69wh :uy a3th;zkva:oud8rry-go-round platform of radiul;:y r6ofau k: udzt3 8a4ah8 3w9wkvhs 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular vezen775 *dnn/u pb6oa ylocity of 0.yd5/ nep*o6 z77n unba8 $rad/s$ 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 whiqwcfodjn jg4)lyqx/w+ p 7:*/te dwarf, losing about half its mass in the process, and winding up with a radius 1.0lpg*j c fd 4+q/jyq7)wxonw/:% 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 involpuc fe76j bdan 8t89.ave 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 bxniiji(sm:x( f(+ 1x$ 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 rotacpg8y pd49b k9f3 pj8:kpfq)oby p;5vte freely without friction. The moment of inertia of the persy f pj:b f p9pp8o;3bgy58v4kpqd 9k)con 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 dip)ocf: iooedz,oig7 a9;( m b 6zqx6h8ameter merry-go-round turntable that is mounted on frictionless bearings and has a m,z8 o; d9hipg maz obf7:qoo66)xe (icoment 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 rolj utw(,-/wcr jmt1.,2edjzt ols on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holotr-j 2,ej.w,tj cz wt1(/um dding 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 s.dhx.)rmnhv 0+ vyao hto,0j (uch 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(.m)hnv yt ,jao. 0hh+rv0xo d, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratex 2mpu7j.9/qe8grxo m of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the hpj;h 0r-:xm wshape of a uniform cylinder of radius 0.20 m acquires a rotational rate ofhwxp :0;mj hr- from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylid0)a aido,.syq6v f yyb 23mn.ndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cyli6q.0m v2onibdy f)yd. s3y,aandrical 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 an:r 0e otg;: ki;)mxtikqz l wjdv)-27hgular 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 yx;b 62f:jw: mr cfd1jSun, 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 amb12j6 fx jcyr;::wfdngular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy storedo)brpzo.s.n kk+q 4- p in a heavy rotating flywheel. Szbq.o + sk okppnr-)4.uppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrat 21js*scxz*hcb2se5 zes 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 ta3)b0pexw :/ur7nrprolling 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.eo 25eddf,p96s zw9ehctfi 6a2., 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, 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. 8–2if6od2he9 9zafe, 5wdtp26s c1g.]

A wheel of mass M has radius R. It is standing vertically on the fr64 t(3r d(cz se7er6yb -hsn q;zz0pkt 7jng:loor, and we want to exert a horizontal ( e7djs6z zcy(br et hsgn;n 4rk0z3-6tprq7: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 travelinxqisymk+ g .u8fp 5 )(f3/mkpng with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting o/yns ()kfx. pmup5kq3 g8+ fimn 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 r v;;5f(frz gj (2zq(s inwvy4gock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve th( z ;(f2iggrn(svq4v zwy5f;j is 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 rods,bd.loyh b) q+1 making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skqb o+lb d1y.)hater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consish3mm6-pr0g bq ts of two cylindrical plates, of massg r6 -m0bqpm3h $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loop*o7rrbbr, 1jn a( htd)ed 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? Assuhjrr)( * b7r 1ondab,tme $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do ndfzz:k68 y c-uot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions asvrk: 0q20 aso 1pbx8xih :4jbc the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelcrvix0 0 o xh2:kb4q8spb :ja1eration 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