A bicycle odometer (whicau 21gl/smnzt(s 3t y*l2gg6bs(i )o uh measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch w/)gsb(2u(m l21g ta*ztls yin 6g3suo heels?

Suppose a disk rotates at co*be y z: m)v-wgetyf2;nstant angular velocity. Does a point on the rim have radial awyg)y*-m:tf vee z;2bnd/or tangential acceleration? If the 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 single valuev yqc-eroon 82i cz6-l7o d.qpf61mv5 of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than axp i0 5f**fsxx 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 hajy.wita/ /)yaj8zkz +nds behind your head than when your arms are stretched out in front of you? A diagij.zt+a/wyjz8 / ay)k ram may help you to answer this.

A 21-speed bicycle has sevew0.thwsokh -)qbut bowok 2/ 5*.d5nen sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front s*2o ..odktbe)5noqtsw hwk/ -b0h uw5procket? Why?

Mammals that depend on being able to run fast have sc( lp ut1pa9h9v- :5) 9snw5qtickpnflender lower legs with flesh and muscle concentrated high, close to the body c-h(9pa pq)v kl 5n9i:us9fc 1twtp5n(Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) c1 6 7djdg ah*bss)vk 3w8dgkt0arry a long, narrow beam?

If the net force on a system is zero, is the bp d)e) sbz6e9net torque also zero? If the net torqu) 9 6bezbdse)pe on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hjd(2d* k u7eicorizontal. The sa7keu 2j(*dc dime 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 r.y 6gmoo-xdpz.t,h 7iest) down an incline. One sphere has twice the radius and twice-i.o.o z 6m 7pxyhgt,d the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same zo(2kg mjv:g 5 :u/jztradius 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 kinet(t kz:zjoj:5 ugv/g m2ic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserv h2t fnfn)t +s-j2a.caed. Yet most moving or rotating objects eventually slow don 2taactnjh 2f.-fs+) wn and stop. Explain.

If there were a great migration od18*8b y wgvb(ybi8aif people toward the Earth’s equator, how would this affect the length owybyavig81 88 bb*(d if the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatioq v7nee6aso* y9x05zon when she leaveseq9oz5so7e6 ayx* n0v the board?

The moment of inertia of a rotating solid disk about an axis through i2e:yns/- p 4;lwj-z q.jyiclo;synx +ts center ofy-p /o:;jjcs ;yz .q-nni xesly+2lw 4 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 sittirz2tq5syb/, e i)r d:g9znef9j 2alh *ng on a rotating stool holding a 2-kg mass in each outstretched hand. If you sudd5h2*e2ld,nz:jz9r eq/ )af y9rtsig benly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hownq1tmyb2 a5(kqvq1mxev7; 9 aever, one is hollow and the other is solie5yx1qv(a ammn2 k97qv tbq1;d. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poiu.nu :ksy75 obh;gbw+nts 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 angbbukuw h5o: yn+;g.s 7ular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable.5i:lo g 6d1rzp What happens if you walk towlg di:r z561opard the center?

A shortstop may leap into the air to catch a ball and tgbr2(nhjp*.(up f8z yhrow 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 rota(j 8yzg rh(p*f b.pn2ute in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conse73azu b 9pw6aorvation of angular momentum, discuss why a helicopter must havou6p9w ba3 a7ze more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followinb -he9/derno-4re-up ex+ xv3*r qa4ag angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Emw9i. ja51g7 qrc;vdl+ m gdk+arth because of an amazing coincidence. Calculate, using the information insv.7g5d jc kr1 +;lq i+dagwm9mide the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moongg7fg2vnu87 n, 380,000 km from Earth. The beam diverges at an anglnggf2n u87gv 7e $\theta$ (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 egs6zzt)4xl 23:w y,(xivc)d7bze: z nj;o bl the motor is turned off during operation, the blades slow to wi7xbeg3jl ot(v:zxcy6l);,:s b zdnz42ze ) 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 flo)5ra0unu sjq6 n7c6p5wuh a4 sor 3.5 m to another child. If the ball makes 15.0 revolun5usu60 7a 5hnp cwa4j)us6qrtions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter qeiih0r2c 3 8stravels 8.0 km. How many revolutions do ths3ce08h2 iirqe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diamv miep4vn9f88dakkvq*) l*a3jx 3v:t eter rotates at 2500 rpm. Calculate its angula :lxdk 49i8e vv 3 m*qtfk*jvapn)83avr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. j86b:ef 7y xpa .a4klm8–38). (a) What is the linear speed of a child seated 1.2 m 6xja 7fly:.48mb epakfrom 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 oix;rv ;6 gfkfhyw,vjaj* r m9:vtl5+ :rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear s fl 8,1x;zngcopeed 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 :pnz 9 s+ 0 twu/gpei-o9um/kxa centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acpg /e9 in os9+u:/m0w xpztu-kceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abodw9v b/4)ks jaut its center from 130 rpm to 280 rpm a)49b/ ksvdwj in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius0zhvto/ki hc5 (j89rgwl 0 fg;g s+ f9xr7xt3e $R_1$ 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 ;z9 qgew03 5od sgx3bxMoon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time ixs3d ;ewx b9q503zg gonterval. 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 unifouy+ *7hgt ms 46sgl-yxrmly from rest to 15,000 rpm in 220 s. Through how many revolutions did itgsg*ylx6 su -4+ym ht7 turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula7tg :la)f :jd vev2v.bte
(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 stressesii r n,n4apm6s (2a3(zl s fkulhw6v24 of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cos(3ur, l2(vsla44 fnk2wa6pmz6 ihi nmplete 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 ru(7p 1 omkq r/qtpfnf;g)1z,dgr z0d ;pm in 6.5 s. How far will a point on the edge of the wheel have traveled info1kzpr(p gzd;q )d7/1t; f,g0nuq rm this time?    m

  A cooling fan is turned off.ft7 jr4y,gxc when it is running at 850rev/min It turns 1500 revolutions before it , .47 gfrjytxccomes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the cw 5piq/qrz4r/q bfw /6hg;eq- ar reduces its speed uniformly from g 6qqw4 e 5h//ipb/wrqzrfq- ;95km/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 revolutionkumzql;5x )z;y8evg7s as the car reduces its speed uniformly from 95km/h to 45km/h The tire 8;kev)7;z5 zmq uxlygs 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 weight dfb9v:pcv 5 wvk :i481xlwn) mon each pedal when climbing a hill. The pedals rotate in a xw:1bp5)liwv 8fn:4 dv9mvkc 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 n dg6wgg cxg 7fj52:a8,a1f)kjjwmm. on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exercgg,j 6 j5djmawafkf)g:8.n xg71wm2ted
(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 putz8 vv:zc7 *Fig. 8–39. Assume that a fz8 tvz cv*7pu:riction 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 th rwllv*8k )v9;ez:r mte ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod 9vmt8rzkv:);l l ew*r 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 of6mp/7c nii-5 rlynb6 cm .ivio:6 7o1 m7dbacx an engine require tightening to a torque of 38 cc.m77-6v m:1ybip lc iox n76 a5rm/6bn oidi$m \cdot N$ 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 . rtzs:5;ixwsz97s r qof inertia of a 10.8-kg sphere of radius 0.648 m when the axis 9ws5x. ztr 7i:qrz;ssof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam7ao bx*q9wn 6deter. The rim and tire have a combined mass of 1.25 7bon6 w da9*xqkg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotate:l/73 mw yesayd in a horizontal circle ofey/lam7y s3w: radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angulry93 r 7yes 1joh*79ks ,wdejzar speedzd93 1 seho,jr* kswjr797e yy (Fig. 8–42). 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 Fic2 9dt, r. l-/icxiurjg. 8–43l j/itrix cd ,-r.u2c9 about
(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 oxygeao, .rb)xq 8acmoq;2/k7a t omn atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corre*kui8 ouhd5d;,w) .ldjq8xmact 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 j5 mo8xq hd.,auu;id *kwl8) dthe required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5pgg0mh4q 0zh -0 cm and a mass of 0.580 kg. Calculapgmz-04 g0qhhte
(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 rest to 3 6 p,h+2;gmczevju6a k $rev/s$ 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 ablewcyikzm8 (h) :a8k i:f 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 edkz8::)fc wyik 8ihm(a ge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broxqes 61 2cwx2eught uniformly to rest byx1q x2see2cw6 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 3lb 7 e9z6jjr,8 oom9ob.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of pxch+gn /wn7 zh s1qx/,oj*fr4i.jz* the forearm, which rotates about the elbow joi pf4c /rnij*7.n*j+s,zx 1z wgq/xohhnt 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 conside:md c)s2:e xhho46vid red a long thin rod, as shown in Fig. 8–46.mo:hsed cx:4hv 6d)i2
(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 masses, 60 vafzkp3n t.-hssl5$m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hamz6/d, gyttv n9mer from rest within four full turns (revolutions) and releases it aty dnt6/9, gtzv 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 jah1ddf s3i .i 1 g9(ssazj3ito*-wd 4$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28h7p :kmg:mgn00 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9fhli o*55a-u vv:ola 2.0 cm rolls without slipping down a lane aav55* :iuo hl -f2olvat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sue7utmbfn 6 w..u- p:nnm of two :tbe f. nn-n.m wpu6u7terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. Howggk2vzqgj8bc0*q lg4vqv30p-1r l,ju52jc t much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cyli,b gpgk qtv 2 jc1l5-q g2rgz v8j*4u3lq0cvj0nder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rollu :xmacv67b-o-:nzcx 0g hi2g j3u;z kr qcc2.s without slipping down am a u:6k0 cqg2jzb;.:ch-32zcuicgr xvon7 -x 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 sij5p dd xv0rnh*/ze(1 tarts to fall and its lower end does not slip. What will be the speed of tjei pn105xd/h*r(v z dhe 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 bgzw)paix wf.afr5137 9or/ktall rotating on the end of a thin string in a circle of radius 1.10 m at an angu7g1 i5wxft.p9 ofkaa /3wz)rr lar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mom)qs7ii9 /l;7i ,-+8eu slxtbkzi hyadentum of a 2.8-kg uniform cylindrical grinding wheel olq;xhd9) 7l-ki7 esysz bi,it 8/au +if radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ra) *i/v n9o*l ka8q mi)azft vof(9rvz,te of 1.3rev/s If he raises his arms to a horizontal positvo*nrzlifmo/,f)a8z vqa9ktv )i9 (*ion, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of insr9y zfz-gj( 7 )0xuqurl zow0po(0i+ertia by a factor of ab -gz ry0sz0f)or7upl (qozj9 ui0x+w (out 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate fryrgkob96 ghkft9o9 ) (om an initial rate of 1.0 rev every 2.0 s to a 9(ohfyt9o g )k9 rgbk6final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a f2p 6dxsi;pel1ciig828 v7gzz requency of 1.5rev/s The wheel6iz8 c8p;e7x p vg iz1dgls22i can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angulakcg0o fv mps .2i4:da*r momentum 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 momentfmtioa;1m2as4 dt dqt8gg5 fq,( jcmm240 s2lum 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 iden.67ol i bit*setical disk rb6s t e7*.oiilotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsfthq7ahg5c7kw z17-zbpk 1 kraa;*r/ at 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 kc y v)b.2,xoje8n ig4kg stands at the center of a rotating merry-go-round platform of radiusjy 8 bk,ic24). evnogx 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 velocity o jhn**lu7*jtl f 0.8 7 tl*hujlnj** $rad/s$ 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 dwar jhbj7-elxku2kcon.+ 518co qf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s 7k-oo clj. 5+b12 8ukqjxnhce 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 wt*0( l hjqwc jl2o2m3xinds 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 rw(g*libps 1; +8kzsz$ 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 r2hx -:4yewb irest, that can rotate freely without friction. The moment of inertia of the person plus thexhb 4r2 -:iywe 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 pers1nu z0k-e 9a kq.lx7w;*iv +eow+ qexvon stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings a l7w9-ukqove+0 eq*x v ax;winzk 1.e+nd has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyinlxnpx 4tlu e,(e4ph2hv d21c-g on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from thn-vc x2lu(ltexd4h1,p 4 p2e he spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thb2go, h mot8(d1d 5sfbe 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 ph(,18 gm ob25btds fodarticle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rm w :xwx lz(eivpkd ubk ;a,o)3/),8zeate 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 tr lg:ta,q( p7qid 80pfhe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acce:rlgqi87fd ( q,0 atppleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical kq 7zorf7,gmii ++5nvdisks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. + m57z,firnoq+g 7ikvUse 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 thmtz c * jj6 d fzkophj32gee23t697o;ke angular 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 ourn dp sj 0(dh;k:sfj98c Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lnskdf( hpj09dc: 8 j;sost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automom lnn/+ - h54flnbgcq)bile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, u/l n-lc4 mb)hgq5n+ nfses 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 illustraak:d 5q3o, jg2m5wtrrtes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at spe+ qgszp2ci;o0 k *7zrfed 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.,foxgp.a.9v +uh +yp54t ddx(z we ignore friction) about a hinge attached to a wall, as in Fig. 8–54.xf5ag.zpth u.pd o9y +dxv+4( 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. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on okgrn k7lid+ wm 9 :.k6jt,vo9the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55gko9j7nmvi:+ 6k k w.l9dtor, ). The step has height h, where h

  A bicyclist traveling with speeddf 25(pokonbk.s*a94qa .3 9 k jlm1v8bdkhws v=4.2m/s on a flat road is making a turn with a ra 2ksob v .3*8kdkn4a wbqhljs5op1ka9 (f d.m9dius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg ro(/wvidpov5 ,wck 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 this feat, and where does the wv/v,podi(w 5 torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as )b1fz ay4pib3l kvq5: thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wiy4bq)5la v1p3bzfi:kth arms held tightly against her body.   

  You are designing a clujj(.ra/l u /ycx (umk6tch assembly which consists of two cylindrical plates, of mamju//yl j( k6x .ucra(ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped rough tri*;/etv3du2nfvk do-q5(xjb;e fau -ack 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 wittbvvx;-fu2o5qk;i au*e /ddj3-e n( fhout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumeb zxxh6/i8r b5arl,k ; $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions e* mdyzf4 6 *om;nhi:vas the car reduces its speed uniformly from 9064;*f:y ih*vmo dzemnkm/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s