A bicycle odometer (which meajno- hyfo3 00t.*f vkn+wh-7z4oofjp 0h /cpvsures distance traveled) is attached near the wheel hub an0t/0ko0f ojnfpv-o.h+-j4hp7nh cfw y3 zov*d is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point onxg7cp:w ar h3- the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifw: r-x73ghca pormly, 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 b36m cjbgl)/b2lyk it1e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever ex42b4 i s7vjaquert 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 has pwg/me-b .l9)8sqkm nds behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thip9)gms.-q b lwmks8/es.

A 21-speed bicycle has seven sprockets at the rear wheel ucx9 ulcm9tfud;,a7 3and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Wu9cmua,;c7 tfdxlu 93 hy? 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 lowerg6zup f;n44 op(i/qtel vq26r legs with flesh and muscle concentrated high, close to the bgq 4n; ieq(u6p6f2vtlo4/ rpz ody (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) 7 k4,x.fjshz)b o0jqgcarry a long, narrow beam?

If the net force on a system isu;rkpx)4f -a .l)-mod:gc0hu geexb; zero, is the net torque also zero? If the net torque on a system is zero, is the net hfkp- )4 eu0deac;u.xg ro-b m)gx;l:force zero?

Two inclines have the same height but make different angles with the horizontal.;q sadcue9 ell31 (+uo The same steel ball is rolled down each 9u+eulcqdo;1(3 sela incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneous*7p8agmo*hm m 0 pvy:tly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom ofgp y0:tamv mph*7*o8m 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 radiuyp(ku : 6g)j3ghyonay)r j72p s and the same 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 n ug2jyp(jahyp:)gr k67)oy3has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or roa cp:-*j vsbz6tating objects eventually slow down and stojvs pbc6a :*-zp. Explain.

If there were a great migration of people toward the Earth’s equator, xvmt:-( ko( 4gaqw)b vhow would this affect the length of q(:mwgvtak-4 )(bxvo the day?

Can the diver of Fig. 8–29o 5cyz3z 9cl:n do a somersault without having any initial rotation when :oyn 5zz3c9c lshe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centy wgfd+o:3xl *y9 pd )oudsmaty:- 2l:er o*:3 a uy pd92om::l wdyx- tfyos+l)gdf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdib(2zb)q;wzvt/k *f xb ng a 2-kg mass in each outstretched hand. If you suddenly drop the /btxzv )bk*z wb;q2(fmasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holle:p :.mpe sxk*ow and the other is solid. Describe an experime*s k. xmp:epe:nt to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points zj 14idhi/o7a 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 increasing or decrei7zi1h /od a4jasing?

Suppose you are standing on the edge of a large freely rotating turqmsm 3;c)cpr jx 7a0v t8-be:untable. What happens if you walk toward the cenv0c jb : cperma-) t7s8mxq;3uter?

A shortstop may leap into the air to catch a bz e4xd/nkpl, o*8a0dat:.hg9 u 2mq op)+fmlvall and throw it quickly. As he throws the ball, the upper part of his body rotates. If you lop 0tzmf/4aad*.2luq xpv+nlgd ehk):98om o ,ok quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuoqh+ b g0wnp60ss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep n 0bog0p+hq 6wthe helicopter stable.

  Express the following angles in rad7oh.:7xqe 1ls13uf+uxeltdh ians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because1drsu )cc6ypo/yxs (l59 -qoh of an amazing coincidence. Calculate, using the information inside the Front Cover, ycl )yp(dxq ho cus-sor/9561the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at t cku/pbcx --w2he Moon, 380,000 km from Earth. The beam diverges at an an ckwu-2/xpc -bgle $\theta$ (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 turni4zh 569x5bkmncfs 0med off during operation, the blades slow to rest in 3.0 s. What is the angular acce kmm4hxnfs 55b6 cz09ileration as the blades slow down?    $rad/s^2$

  A child rolls a ball wt,;v4jfm.d 0mpkdp 9 on a level floor 3.5 m to another child. If the ball makes 15.0 revolmtp jpdvw940,; f .kdmutions, what is its diameter?    m

  A bicycle with tires 68.b 8p tny/ opb5thj2:l cm in diameter travels 8.0 km. How many revolutions do th o82b .htnpb5tyl/:jpe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotata6/c ffupfl*,:)xaq/l ib qht-)jy+ ies at 2500 rpm. Calculate its angular vp t , b6fi:q+a*/h-qf)j/lluacfi x )yelocity 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.f:b ;m/f a9 3)m o80fk)v yqfeztdjcl8 8–38). (a) What is the lkl0ome)ctfqfy j 8: fv)93 8mbd z;/afinear 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 oe 0.otecs048zem1yk p rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speoxuvc3 +y8; s4xoob9a ed 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 c* -1kiz2jixdqlyen(i :u fy1t1i9.flentrifuge rotate if a particle 7.0 cm from the axis of ri.ezj2y :tkn1i-y 1i ifq *9fdl1u(xl otation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its c y zjnyv7 c sk2/3+c;5tejdxyf9s57ncenter from 130 rpm to 280 rpm intsc fc3 s7nz;yyvj +5j cn5dx/e29ky7 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 radiu+ sxlvo6 wp,5hs $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themseln :czm8f9n (im dt-h51cueue.ves into a slow rotation to distribute the Sun’s energy evenly. At thmetnd uh -e.:m5 zui1ccn9(f 8e 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 r5krt,w. .dgo.p 9yhm ccc:3zkest to 15,000 rpm in 220 s. Through how many revolutions did it turn .t9,3cko mk5w. : z gpccyh.rdin this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. ygwkexe8 5x8n e96vjb: v*+1 egpc0lnCalculate
(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 v 2zonup2(6ia highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complin vpoa2u(62zete 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 2400f* lp(w 7ypzpps 3arf,g ss3* rpm to 360 rpm in 6.5 s. How far will a point on the edge of thps7lps f,zfy3**pgrw3(spa0e wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/miq5iq24d t rw4n4m2jsd n It turns 1500 revolutions before it 4qjdimwrs24n2dq4 5t 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 makeenk,u ayq ) yeks;**.v 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires a ;ku )*,k*n eeqvyy.shave 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 its speed uniformlb9ql/gv(f1 zj l l94 id(o0adu w*5cpfy from 95j qlz c1f*ab(li9wpov0 9 gd l/df(u45km/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

  A 55-kg person riding a bike puts all h 0 g3wwyuwd zo; t1)0xsgu:f0ker weight on each pedal when climbing a hill. The pedals rotate w0g;wg)twdfuuz03yx o 0k 1:sin 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 yucr 9tcg,h ;)ks9ok2end of a door 74 cm wide. What is the magnitude of t9o h), ;2ckrutcgyks9he torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of t5n2ohum 3uaxw* fp-ga :r)4fche wheel shown in Fig. 8–39. Assume that a fr p g3u-5u xhacf*m)rnao2f: w4iction 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 endsxn/-i6,0vn7 anca w3 lekf.9n*;gfw ej4fi fd of a massless rod which pivots as sho0if.x6l,na;devf7 fgf*/ijk 9n-a3 4c ne wnw wn in Fig. 8–40. 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 head of an engivw1 n7u,e4q4tya d au8ne require tightening to a torque of 38 aw4uen qa,tu147v8y d $m \cdot N$ 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 0.648 m whel2gx( 9owk5hg 2:-ltpvn)i9 a gs0j xen the axis of rotation is through its center. 2kgx0plw g9hojg9 a)5in2(t esl-:vx   $kg \cdot m^2$

  Calculate the moment of i, k pc,ez7mn 4nmtck(av67 4ln ,o-phynertia 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 hub can be7,nko c mnank(vt6plm4c,,y p 4e7zh- ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the *oqk;(9)giooy uwu)oit gj*( end of a thin, light rod is rotated in a horizontal circle of r *gu o9q;(y)jwgi(iku t)ooo* adius 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 bowl on a potter’s wheel -) z)eo2zmtq(o 52s q4; l,ikbqzeqpb0bg bi6rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 ltbqz;,)pozq6bib-mo 0z 5b4e2kq q2e ( g) isN 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 pointc0*ab :s z7h/ fbdjof. objects shown in Fig. 8–43 abou0ofba .:b/d*7z hsfcjt
(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 whof3m6ilhml /m5 bp (le4se total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ratd. bhg 10 mv)qip4 fx1di)smp;e, 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 rx0v)h g14 dqm1p.fb;iidm)spequired steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rc5pdf6(- bk /*yu eqktadius of 8.50 cm and a mass of 0.580bq y*pt-6 /uedk k(cf5 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rl (48r )4lliueterhyrx4di:3pdk+ ;fest 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-d 5 uz-hzkw nev9a.s//33qzjee riven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 e-hvu z k9wnq3/a5/ z3eje.sz 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 produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating ab s9 9/+uew.zvtmkq-o :1w zbvt 10,300 rpm is shut off and is eventually brought uniformly1 :q99obwu/v -+mze.vtswbzk to 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 accelerates a 3.6 y;8qygn2hvv5fe 77jt3h-w p x-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, wa5nlajgub:tj32: l m ,hich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated3u5j a,t:2 jm:lagl nb 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 zj1w;sv wpw1+ d,pj: fin Fig. 8–46w1j 1vpws,;zj pf:wd +.
(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 mass)ky5qxja8 yxsr gk-r24wu; :bes, $m_1$ 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 acceleratejs ,t58jbi)mu0dx-zvs the hammer from rest within four full turns (revolutions) and releases it at a speed of 2805v uz8jsmjxtb ,i-)d $m/s$ 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 u mn,rte m.gzl8u 74fn*3pt,fof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque d/ kt4-yeq-hczi:vh 2ux(f e(of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rollst6yyz/isi:h+ ft 99et without slipping down a lane at 3.ii+ysz6et 9ty h9f/t:3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sb- tdl*) .vkcuun as the sum of two terms,dl)-b .*kct vu
(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 r vfd qpe6ut8xyl em/v*d1(l8ety,9/a mass of 1640 kg and a radius of 7.50 m. How much net work,lv9e1exf8e(p mt / lddvyuy 6q8t*r/ is required 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 2l q5p8mghq+mrm z9 r)(0.0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30qqrz mmrp gm)(h+85 l9.0 $^{\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 stamucz-2 m8s. n k hu(uwpx39bh.rts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before itsu-whz( .n3bku8u9px.2 c mm h hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a t kf/;pnucf 01yhin strp0;c1fkyuf /ning 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 cyli uw9yjw-3jf/ th1 z5kjndrical grinding wheel of radius 18 cm when rotating at 1500 rpm?k j5 wzf- hwtjj/9yu31    $kg \cdot m^2$
(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 rotatinhcmz4j8iz : k/ju7 m,jg at a rate of 1.3rev/sjm:c, / zu4ijk jh8mz7 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 y l1dubi888g4gk(qqt–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck pos 8ubyiqtdq gk4l 1g8(8ition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r8 8hfehd,lo d 7z/c6ofate of 1.0 rev every 2.0 s tozfdh8e8/l 6h7,d oo fc a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verticalolh w27uo w)c8hm)k4s 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 radiu2om 7ww)cu8 s4 hl)hkos 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skateru4dlev+ tj, 0c 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 )io/nk y:plo4momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an ident6 8-cahp0mjp*j5otueg x ) 0ntical disk r0 gpo ca65xjj th*8netp0um )-otating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns aqv kju)g9jd/w -j6mh. t 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-go-round po-1wisuc o65n)mg5+bpwowd t h b7 h;+latform of radius 3.0 m and o7 +wbn +gmishuhoct)o -61w5pd5bw; 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 angwhpy.o7qru tm r223,dc.t4t ad:n ss :ular velocity of 0.8 h3d74w.yt s:sq moat 2u ,. npcdrr2t:$rad/s$ 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 d2yt +z j3a 4aez8b0qupwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing rajb 0atyu3zz8 p4 eq+a2dius. 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 involve winds i9 4 osfz2t2ucfl/ .*q:ag hwvdn 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 4i -is - 2tegbyrui9 :4nkg0 khyuc+q1.k.mgt$ 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 we73bw-us/ihkmu 9)p uithout friction. The mo)meuuu9 -ikw7 p/3bhs ment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merr*a+t 9n4ec ir+xut at4y-go-round turntable that is mounted on frictionless bearings and hrn xca4 t +a4ti9tu+e*as 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 end r/q3b ur9t a/1d9l 5d)7l fiokam( jxsof the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–b/q59lk /j( li7fm 1 r 3sdaxt9)ruado50. 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 alwaysi u8m4moju q4npsi.d -wkeh:/3cvm -. 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 theu. -.ips/4dq v-h c:8e3n4 mmwkuimj o Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

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

  A 1.4-kg grindstone in the shape of a uniform cylinder of+qs ox7h;;3c 8ct 1kz a3oetxd radius 0.20 m acquires a rotastd7o qz8 eth1;+cxxc 3aok 3;tional rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylin6((s lnaelz cet.gbf xt9zg;(sq0 : ;ldrical disks, each of mass 0.050 kg and diameter 0.sbcsna(.eq;:z6l(ztx9(;efg 0l t gl 075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed (-gasg6xw/1ughw dr 3 ne *k3yof 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 si ,nyp:pulo)hdvnh6l:ug5arc -f 5 , q3ze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass: 6 ,hpa ln35yrpfh)log :u,cun5dv q- 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 automobile is for il:+c/e q4:;v;o ohat gvaw7 o1hkcrn5t to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass v agc nheko q 7 ow;rl+;::4vt1h/aco5of 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 illustra;2o5za3wl c pq /zdi8xmavp6z;o0 . pqtes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on n9lv*.e 9syy0.y ec4 ygt4jeaa 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.,8e4v,qemvyq 2w 2 ah+n -v1pdk7ut8 gxj1js:h 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 the2 va ,-jpk8eg x24m17vqjvud1 tey 8wqnshh +: 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. I; )t raphyg/x b5vae5 zltq9ew/ke57*t is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against whi5;eetyaz rwahv/ep 5 5 /kq*l 7tbg9)xch it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with sjiu4wu4pg14x peed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normap4w4j xuug i41l 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 1mf;kfuo ( bk (r7ied;kddf6/a sling of length 1.5 m and begins whirlinr (o b; kfd;emfi6dk(d1 /fku7g the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rodsce7t0 wtljv1yxs6mtb (qlj dzq6, 5l n2+63ws, making reasonable estimates for 0sqdwbw,2n673t tt + lvzj( xly66emj 1cslq 5the 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 twobxz mt n5i8a(1 cylindrical plates, of m 5zain( x1bt8mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loo:z0i )5ooc lbfped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble mustf iz5l0c )bo:o 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, butg7*cu6 7l p dk99fmdit do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces,v pe+j,oe 4b0xkqg uc++ sk4b its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wao c+0, +kuexbsqge44 ,+j bkpvs 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