A bicycle odometer (which measures distance traveled) 0x hkxp4, tzk5is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with,pxk45x zhk0t 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a p01 f(em0)mpetie au2mf llv46oint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity ine4v6i(epmml ffeau0 2 tl1)m0creases 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 singlh h .b 3tp ,z8a1d4k9gl1bvv japa+d/we value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exp1w)v46ef bb)tn+3dn6nfslnz +l :siu lain.

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 gsqxvw 7+ .yy2with your hands behind your head than when your arms are stretched.72+wxsy vgqy out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and throvt4)li44+ad w;m(yvxs9 xk oee at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gea)md+stvwx xyo k4 ia(4;lov94r is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs withbtno g g1rd p(( j-y*-l;3ilqi flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis oyn1-3j g;-i*r(o(pti dlqlgbf rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowa+at x4dbyfts1g8e8x21. wu y02f fqj beam?

If the net force on a system is zero, is thetn7pky 6:: ivp net torque also zero? If the net torque on a system it::py n ki6vp7s zero, is the net force zero?

Two inclines have the same heighi:do 0x;.ow d9n ppvu)5qmra/t but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom ud/ oq.m0)5 xi9wvpd;anr:pobe greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down 1nam kkf 0 hjo /w1(zsn9eyxe:)nf,:gan incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inc: e n 01xsff:yo9a/jk1z w(ghe,nkm )nline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the sy 6q/c/6wqic inp/ou9fpb:) hame mass. They start from rest at the top of an incline. Which reaches the bottom fw 9bip: piyq6/q 6ccof) u//hnirst? 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 conservefkoe2mw :m*; i m2ewkabg,( q;d. Yet most moving or rotating objects eventuallymw f;ea,k;gw2e 2 ik ombm(*q: slow down and stop. Explain.

If there were a great migration of people toward tk6v m;v9tlw 8u.o-h dghe Earth’s equator, how would this affect the length of thel 6 w vh.g-9;omkd8utv day?

Can the diver of Fig. 8–29 do sqh7x2 8gz:kqj ,7gk gu4(laua somersault without having any initial rotation wh, 7uk h: gls7aqkz2qxj8( 4gugen she leaves the board?

The moment of inertia of a rotating solid disk about an axis th.ckcd,o4bp 7f rough its center of massbcpd .,c o47fk 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 rota6orw,/txhmbc g. ;/veting stool holding a 2-kg mass in each outstretched hand. If you s /6h mxvc/g; tre,bw.ouddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have thxxdw)g5 q2s luyjh.9; e same mass. However, one is hollow and the other is solid. Describejuhwdxy ;9l)2sq5x. g an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Inj5zrk.pf98p i 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 ip 5 f8k9.pzjrincreasing or decreasing?

Suppose you are standiw5oxc6pik2 tjc q(k-0kjq* a 3ng on the edge of a large freely rotating turntable. What happens if you walk toward the center?a*ct2qxo qk w ck-5p(k0 j6i3j

A shortstop may leap into the fs3rn/o- /rm( ):5xscmjp xar air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite dis frx:p r)-(3c ra5//jmxomn srection (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dif42 kmepj+1 b .uuala-x1-(owj9ivbq g7h c4 dscuss why a helicopter must have more than oa+.9xiuhpb71w4 bj2kjo-( 1uv 4 qldgc efma- ne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: e9 jb3,ueeswd gm39p0pxn x:+(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 Earb3g1kuy3d ao9 th because of an amazing coincidence. Calculate, using the information inside the Front Cover, tgk3d 3bau1y o9he angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km f ,2jd/difodqa2bwy+ ;rom Earth. The beam diverges at a+bq2dfd , 2 ijo;adw/yn angle $\theta$ (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 zupf1khg, q)0 zn/yf )at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?z) f)gpkuq/ n0z,y 1fh    $rad/s^2$

  A child rolls a ball on a level floor 3z*4 adplc 5oir+iq;2b.5 m to another child. If the ball makes 15.0 revolutions, what is its di *i;z2oicq4+dp5ra lbameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions doti37:o-g3yf dol :bo5 jnxivm-l9 qr7 the wheell7i3oi3qtr l:gx oo7 5 -bn:m- j9fvdys make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angulg:1t,q* t c(hzh9fg3wclfsm- arh ,ms: *9th(z1cft w cfg-q3lg 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 (m0h);s6vyccko:vw x2Fig. 8–38). (a)osk):;y6 0xc2cm vvhw What is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocir xsvm64l9 2tm /6fmtoty of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe) e7pp r ,te*ryvw(jt-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 centrifuge rota-h0+ mot m4lmite if a particle 7.0 cm from the axis of oi m+4 h0mmtl-rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly /b)rwfszjrou z lj4v 0 :qymg84b/6o:about its center from 130 rpm to 280 rpm in 4.0 s. Determif)j6yul sb/0rj: :o 4goq48v wb/mzr zne
(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:-ss qggm itn1f/r s(8s $R_1$ 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(gk 0qzhd, kw/(ul xh2kxhq e-7n,qu6lm si9( 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 interval. The spacecraft can be thought of as a cylindeeqhnw -2ukkug7 q09qlx ,l (hksdm,(/ z(x6hir with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 2mm0y6 43*( j xya ugpuwkuu:1n20 s. Through how many revolutions did it turn in tkayjw* p(nu um4xmu6 u:0gy13his time?    $rev$

  An automobile engine slows down from 45)3thwuxie+-j 00 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses ;wv,1 ycak b s*y9u(mn)aes:dof flying highspeed jets in a whirling “human centrifuge:n; yy(scu1dsva* kma,e9 bw) ,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 l. oz rn5gqhu:e)gc82*+kerxrpm in 6.5 s. How far will a point oucghz kxr ) nleo+: q8*5e2gr.n the edge of the wheel have traveled in this time?    m

  A cooling fan is turned:lghz;)c mr g j(0ma6m off when it is running at 850rev/min It turns 1500 revolutions before itm:zl)(jmm ra6;gh cg0 comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed-ofr8a/,i-9u -ql9lu(c edvmffelh: uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 mvfr q --l,lh:uoe8m f-d/i(9 9ucafl e.
(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 unifozj0e.pv,+ 8n clh qs7y (9damyrmly from 95km/h to 45km/h The tires have a diameter of zd+ve a0ylqn9smp. (h8jy c 7,0.80 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 on each pedal when climbing 6:/ku , f1d*i yzayfzlu9dk2 aa hill. The uzdf2 lk1ua/zd 9,iky6:f a*y pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a d. co zbpdhao 7t(i5/.door 74 cm wide. What is the magnitude of dp. t7 ch.bd/oizoa(5 the 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 the wheel shown in Fig. 8–jtn 4siyr(7negc7m qi5+0yv0 +dmk z:39. Assume t:g s0jqde0+7 4yyvnik+m7nm tr(z 5ichat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a m0v 0fjd7bxfrrx(-r* f4 l(wf uassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the hoxv0(*ffrr 07rx- w jf(lubfd 4rizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head -t ,0 wvpwvde/of an engine require tightening to a torque of 380/v -wtev p,dw $m \cdot N$ 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 0.648t hp1iv-+cyor6 o,yz8 m when the axis of 8 i o+, vzoyy-r61phtcrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a 4(ap5ekef/j bfxk 81obicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of fj ak(x8e4/pbo ke1f5the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on thx94 4:umz 1uskzy/ xuvip)3wae end of a thin, light rod is rotated in a horizontal circle of radiuss9m xz41wz/yuax ) 4k3uup:vi 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 rotati z355znfl0t eyq2 jo)4nszd1jng at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5y qsjtlj3n2ne 1zz o45d f)0z5 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 arraa9j3h 4vp/y o e2op0qqy of point objects shown in Fig. 8–43 aboutv2poyaq4pj 0/ oh9e3q
(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 whojnp6bo+m .3ss ,3eqnbse 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 rat7zglc694i:ckkd mf - ve, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg6zg4v cfikkl-cd: 79m and a radius of 4.0 m, what is the 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 +nojz8ot*h:jh pf:8 tlt 3p8+lis6og with a radius of 8.50 cm and a mass of 0.580 kg. Calculate6*t l zh ot8p8+l8s3:jhgpnfji+:too
(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 oi4g)au( 8wajt;f zb: a:k+ieit from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round anqc) j z(f7sp4/ l zxk gx2s(zl1o*xsl;d is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a sz2x k 4*sqjlp(s(7g/zfx1xcz;ll)o 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 rotpdzw-8(b;y vm ating at 10,300 rpm is shut off and is eventually brought uniformly dmbw- yp;z(8 vto 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 acceledt b7 (j(br4herates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, woh;0pot *cz+b hich rotates about zob*p;o+h ct0the elbow joint 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 im.ggvymo)kwv+0n5l)t2m a/ k n Fig. 8–46. kmaw n.y/mt+ )ggkl2 )0vom 5v
(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 m0jvr-zrka*82: as9,kmkd pp ( 6az ,psjlm;e jasses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (revolzncawc7, 3bb(b*b a27 qdqfvwkga5v6c;:b 1lutions) and releases it at a speed of 5vk b62a1v a*a7dc cc 3q :w7lbwgb(; b,fbnzq28 $m/s$ 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 ofx3kpcv-8m3 r x $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque 2pnu5nl 0r evb 0u4ud23y :hkfof 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 radiuskur9t(k3s+ qa*6a jbz 9.0 cm rolls without slipping down a lane at 3(6kuz9j+a3qtr sb*ak .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of ( wwir-hfme0d0,+ tw ibst/*+m ,mkn nthe Earth with respect to the Sun as the sum of tw,wi *knnebdmst-wm (0wf,h +0rmt i+/o 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 and a radius of 7.50 m. How much net 9r+7 ghkh kh3hwork is required to accelerate it from rest to a3kh9 +7rhh hkg 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 starta0)sy:h1w iiss from rest and rolls without slippins1:0 iyaiw) shg down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts tol 15c(xke/5 wb yedkx6 fall and its lower end does not slip. What will be the speed ok 6ckx1x5 y5 d/blew(ef the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momews+p32wmbnk9 cqxv ./ ntum of a 0.210-kg ball rotating on the end of a thin string ixcw93bkwn2 mq sp.+/v n a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulam6ycyihuy f:true d s/,)4k;73kaa,jr momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wh,yucs t:k3)ae r f ,aki/u74y;yj6hmden 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 allg q 150 o6i6.e-dgooth ha.wt a rate of 1.3r. qliagtl5oo1 d6hw0 - .hgeo6ev/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 swj ) ab7 8o r/co45jorsl3)bojhown in Fig. 8–29) 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 position, what is her angular spr8 r 3ow7l)o o4csbj)5/ajjbo eed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can incre/(ai/+ pqiid isd e6*kqb z2d2ase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final(kzd eb iii/dp/2a2 +d6*qsiq 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 throujg /z;b .fsjao0;1 hvizm i(,ogh its center at a frequency of 1.5rev/s The wheel can be considered a uniform difoviob/ashi z zg,.1 ;(j m;0jsk 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 angular momentum o b5n*d6-isqqpdr8j h; .*siy ,hh5rqc f 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 angularz4e -:opa abz;o .ec us-j6s:v momentum 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 cylindrikex,03o jq+jz i-bz ivkqh.+ 4cal disk of moment of inertia I is dropped onto an identical dq ioj,jkb3zz.+eik+- 04qxhv isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 0fd.3cj40 ip(ttx2ohih;8j6a vjsd f 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 ce8ne(- m4:.p mho,ep zt eao0rqnter of a rotating merry-go-round platform of radius 3.0 m and moment of inertinqt0po ho.p,:z8mma- (eee 4ra 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 veloci d8e 3trlbn15tty of 0nle81tb 3 5rtd.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losbvomby2ol 9b9n.z 4z1ing about half its mass in the process, and winding up with a2 l4mnz9b.1bo bzv9o y radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve wia h0yne i-.rpi)dz t8:nds 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 : 6,maar3g(p/ iirvgd $ 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 platd zyz4d*fs)(zform, initially at rest, that can rotate freely without friction. The moment of inertia of t4ddzf(s)y *zzhe 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 9cqf8 ) j p23i51r3etkjt9z)umxka o p8.jplu diameter merry-go-round turntable that is mounted on frictionless bearings and has a 3pmuue88 lx2qjpkrokc)j3t)p z 9 1ti95 .jfamoment 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 grt8n2i3wri9 ,gxek.ca gaj . fnax36/w ound 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, holding onto it, Fig. 8–50. The sp8i rtk,. cejan39xiwag2 aw/3x fng.6ool 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 yqssc7-3u 7hib3yigaqm* zcm7l7 ;j5 such that the same side always faces the Earth. Determine the ratio of the Mi q753ysbqc7 y ;m3jsgl7i m*z-7ac uhoon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from ru9 (b upra:rw6) mivadu:6l- iest at 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 ur/lc v*9cu: tzniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interzuc vl /rct:9*val at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindr5t oeq ,-vc9 (5+kvclsu/jxeu ical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentris59 e cl/uxo+,-ktve5ju(c qv c) 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 ang 4b)o g4e. dtdfw,o+ngular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gr :mszjk/ ai u+dxxt(:-eat, 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: stuz-x:i( djmka/x+ km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobilet. xmemv .cq+ o0,6+2q nmqmxb is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical fl.mqenv+x.6qm 20mxt q c+,mboywheel 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 illustrates1 5 dnnoew et+n-)la7c 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 horizo0a rxh1qn 1a1 ukkbsnk8: ef2*ntal 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 caj8 :em.t(xmu zzsp+mte:-ccowh* z 7hdg3:n5n pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine nht8mxz tj eoez 7pwdszu:c:(gcm:-53*hm.+ (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 the floor, and we wq hl /3h*( qb1qlg zn;(1;pavhbcql*i ant to exert a horizontal force F at its axle so that it will clbbha3zq*(lc h;1q(pvgliq/ * h;1lnqimb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making))x.* 5plgfd 2gbvzqt1vxx 2t a turn with a radius The forces acting on the cyclist and cycle are the normal force1glx)x tbzdv 2gq)2. fxtv *5p $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of 0vzsil nzq01c:1n sk 5length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceler:5szsnl01zvc1 qi 0knating 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 rods, *z,98cqiyuy r;i3 s2u lq;lr rmaking reasonable estimates for the dimensions. Then calculate the r ;;,q9qy2lilycr8z*ursur 3 iatio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch as f0:qmk/ ofp:/*ecr hylc 9t-zsembly which consists of two cylindrical plates, of massfo ft0/h::lrk *9-zecyc /pmq $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 me :)kg2en:w01 jtg ktlooped rough track of Fig. 8–58. What is the minimum value of the vertigkw2 ) tgke:j1n:e tm0cal height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ;kqhvb9e6u1wk+m9hyk 62x wh assume $r\ll R$ h =    (R-r)

  The tires of a car make oyg yv6:pgd3/isl vw i xue506i+9 ;wp85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What waiyg+ :e6 /pd6wvvgyiui w90ox5s;3lps 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