A bicycle odometer (which measures distance traveled) is attached near the ,x ja ex(t+b)fwheel hub and is designed for 27-inch whexxe)at(,j +fb els. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doeok qeh2.ttm3ndv 71-cs a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component ond2tv-t.ke37 h m c1qof linear acceleration change?

Could a nonrigid body be described by a single vcg,i+b els;n,alue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque th zy2vaiyl-3 y1yabnb,( /cy *wan a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind yo7tqmfpp 5 fp-* utm)h/1hm+biur head than when your arms are stretched out in front of you? A diagram may help you to answer th1m *5f+-mb/h7fpi mp)thpqt uis.

A 21-speed bicycle has seven sprockets at the rdm)3sv 5q e8fvyd*mv (ear 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 spro sdqvfdvem)y5m*v8(3 cket? Why?

Mammals that depend on being able to run fast have slend+8o09nqbikf / s -bsy pqj-p0lr ,eq.aer lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this +spq9 -r0- s,b ki8b0 lnepyoq j/.fqadistribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow ww*f62q +adnlh/1 +: c5olv7sgoxfzbbeam?

If the net force on a system is zero, is the net torque also zero? 0gkfi.z ,h-+m ortb7qwm;s s6If the net torque on a system is zero, is the neb.rf ,itw+0szg mk ;6q-oms7ht force zero?

Two inclines have the same height butmmh1cv2hwe6 2qf7 n s- make different angles with the horizontal. The same steel ball is rolled down each incline. On which i-7 2nm fve s1cmqhh26wncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dox ;;x5gsjzx3k //vierwn an incline. One sphere has twice th v i5/ ;serx;/k3jxxgze radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the saoebf,rh ah(r8e3e696ql : qsume mass. They start from rest a6usae( e3 8bol e qq:6rfh9,rht the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum aredp7qxqj t,7f r 5kzd3b lsz.-0 conserved. Yet most moving or rotating objects eventuq-s qzdj0.fpk5773x zldbtr,ally slow down and stop. Explain.

If there were a great migration of people toward the Earth’s e,iusemh 7 45a*2.xwhoe h; (lk2xops k,uel/e quator, how would this affect the le2/ ep*lokxhex27keshhu m5u 4,(, e .aolwi;sngth of the day?

Can the diver of Fig. 8–29 do a somersault without havi/165pvtt8 os- t/qqf uyqgj b8ng any initial rotation when she leavo// fqbp vtgt68j-y 15tu qq8ses the board?

The moment of inertia of a rotating solid disk abo l:x q+lg:nqe+ut an axis through its center e:+nlxq:l +qg of 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 holding a 2-kg mass inr0 vp (5u75vqoc njxw, each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrcp75v5,qjonrvu w 0(xease, or stay the same? Explain.

Two spheres look identical and h w)5fd 4fo(kdgave the same mass. However, one is hollow and the other is solid. De 5o4f(dfwg) kdscribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In pai /8mt3io enxoro0 o5m;3+twhat direction is the linear velocity of a point on the top of the wheel? If the an0 o/o etp+n8m3aoi xm3ti5or ;gular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely roae6 2 zvtod*7htating turntable. What happens ifod76ha* z2ve t you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asd -txt 0nzf1hw,)s-vo he throws the ball, the uppetwh d,)zf-v -n0ox1 str part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angum;h4w,sa +0 z):k2yanm u wwntlar momentum, discuss why a helicopky0 uw nmhmas+ta2;4,nw: zw)ter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) svbn,dju /09t 5agbzpw/4of*30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincider b;b(yc6yj/ i/1 qghe6j,bldnce. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seeng; c, r(6jbeh/q /dyyji1bl b6 on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges 8,msbj 3pczr 2at an angle2r3s, cmb 8jpz $\theta$ (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. Wfu-8:s+e gkr lhen the motor is turned off during operation, th:lfrgk -+ue 8se blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball mksae,6o y /r).5dtsb(s 5(lhw;jegq yakes 15.0 revolutions, w5d (ye,gksb s)ws6e hl/ qjy5o.;( rathat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k4f9ms5raza - zm;g;lofl v59 km. How many revolutions do the wheels makek5mo g fm 599alfsav -rz4;zl;?    $rev$

  (a) A grinding wheel 0.35 m in diametczsr cf4+3oy3 (4 iyarer rotates at 2500 rpm. Calculate its angular yoar (isc z3y r4c3+4fvelocity 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.y a-sb7q c gqbhl)3 u5uuz,g(9 8–38). (a) What is the linear speed of a child seated 1.2 m from the centeb7 s zqg3) g,u(u-9lybhuaq c5r?    $m/s$
(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 orbit arou r6/wfobx 3x.t0qdf w4nd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p:qlig b;5w )lkoint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particlex0:4euoe8e z81 vo,rmt ks -rs 7.0 cm from the axis of rotation is tuxe :tz88osm- vs4ro1e,r k 0eo experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifoz- we w f9j7wbzii1/pc//ls s)rmly about its center from 130 rpm to 280 rpmwipz/f) -1 zibw sc97 lw/sej/ 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 radius 4z p 3pxi,hqxz:j2/ty $R_1$ 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 Apolloiia*l dda *2+o spacecraft put themselves into a slow rotation to disti +li*d*2aaodribute 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 cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through9n4uad. dyuykn9vju,/ 7tb0d how many revolutions did it turn in7 jnk/ avd4duu .yybnt99ud, 0 this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcx-yib rr ;,0f). u cnac;5iohhulate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets 04 zj/jk:6 b;qhswosr in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befojw /h:sko6q4sj; b r0zre 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 diam73mgx5 r5lzy heter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wil z3l57m5gyxhr l a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off w -28 daf6reg)m ,emw(vnjqkmh)sp *r -hen it is running at 850rev/min It turns 1500 revolutiom8k)n mraw,( -gjhd6 ef )v-*e2r mspqns before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduwz xul(bjw0:3ces its speed uniformly from 95km/h to 45km/h The tires have a ww0u(:jb3xl zdiameter 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 mqy8,azo x1w7cpoxy qo1b3 0o 8ake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 oowz,8xaobx3c7 yq18pqy1 o0 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 clo :h.p5 r/s xu/bhs h)yr,y u:((dasbjimbing a hill. The pedals rotate in a circle ofr 5(sy: )a(x:uushh/bsdbpr,jy / .oh 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 rfm2+ o7ph8y g0 soiu)on the end of a door 74 cm wide. What is the magnitude of +)0fhg 7i ru8oopysm2 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 torquers, dgxt; d+-aj0v*ub about the axle of the wheel shown in Fig. 8–39. Assume that a friction ad tjdx*,;u br0sv+g- torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod wndw0:b .re u+l2xe+r)w,lt xohich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and tho2lb)nerud +w .t+0rx, :xe lwen released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of 3vig eqqd5h,v-kms o09i6np(an engine require tightening to a torque of 38 kqpn( q m,givev50-6 is39odh$m \cdot N$ 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 i+i.bsqdqq/mj1 iul2b5/6 to jnertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is/.t 6u q2j+lm5/ bbjs1oiidq q through its center.    $kg \cdot m^2$

  Calculate the moment of inertia6+1cr ap uvkj8 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 be p+urckav6 1 j8ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in awbo)y z2h737ly3yay a 4kyi .i horizontal circle of radius 1.2 m. Cyzwiilk y4).ob 237yh7ya ay3 alculate
(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 r.ldkt s;,fip tyc5f*e9z7.ai otating at constant angular speed (Fig. 8–42). The friction k9* f ts;lp,te c.i5yzai7fd. 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 Fig. 8am o )key 14x51bw2tau–43 abo2mua yebk1o5a1t 4x )wut
(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 consadmnt+6g* tgil+bv3 5nu,v e 2m5x rf2ists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylq li5:w t5-s xxb(:cgzindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm ii5 : szlcxq5-b(wx t:gn 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 04m4ebt3f.; nwp(w mu kr3 ou3l.580 kg. Calcuoup4tn 3.4k;f um( ewm 3wrl3blate
(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 y 5jfdrw6j*)0n o:q62fwh qcjfrom 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 /. mq*h h-;cx pnxpl6ca small hand-driven merry-go-round and is able .;mx-6/p * hnpxl cqhcto accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to 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 broug; qtz l0y/siv- 3i5/5g0gvz/hccmbhl ht unicz iq/0c;v5gb -lz5v3h mihyt/l/s0 g formly 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-klfn;fu ( ng/v75.g+b1p rpp ezg 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,ah;tm1 uy7bzezv3w z bj.:x0 action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 1vwe: b;m hj.0zyu,a7zb t3x1z0 $m/s$ 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 co vgqk/gx9xpi ucf* *(+.x1x gsnsidered a long thin rod, as shown in Fig. 8–4 kcuspxi.v *g1x(+/9* qfx gxg6.
(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 consis7.syy aa j s2au5um:kh waq6.sgv: 4i+ts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accohk.oq. 6qw nxq m0)6qelerates the hammer from rest within four full turns (revolutions) and relea.m)kwo.0qo6h q6xnqq ses it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of xlq,anc sn:v 25*9jmbinertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2tq1z 0 xcn08yp80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 93lv0-knb 5:fjezq*uk .0 cm rolls without slipping down a lane at 3.33 0 -jvlnk f*5uqbk:ez $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy o -i5oyy a4 wvk9vqz)-mf the Earth with respect to the Sun as the sum of two termi v 9yv w-amy4o)-k5qzs,
(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. jv8 l .+d8jh(fbtpe b7How 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 soli.8 +j8efhvbj(pd7lt bd cylinder.    J

  A sphere of radius 20.0 cm an 3(jaif-8gctxh, j y9fd mass 1.80 kg starts from rest and rolls without slipping down a 30jx(, hy3fj af9 tcg-i8.0 $^{\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 tiqhmibe-* w i,nv..q,n p. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation ofii.nn qv*b.q,m w he,- energy.]    $m/s$

  What is the angular momentum of a 0.210-kg br03g: 9pk/rzfqc mnoz2tm),b all rotating on the end of a thin string in a circle of radius 1.1bopr,: zm g tcm3/qr)kn20 z9f0 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8c3zcliz- 7m-/p4ps csm p61px -kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500mc4z63-pp1 lxpismcs/pz- 7 c 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 rotatin8vicb*er(7;3 w /y c ch/chie*drs3 cog at a rate of 1.3rev/s If he raises his ayc* h* crdc; r7i3/(3wvcshe/ 8iboec rms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of ined24 rrjq -lm4nsp ;;t mlb1wt9rtia by a factor of about 3.5 when changing from the straight position to the tuck position. 2 144;n-mbwdtjt mrl;r 9slqp 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 +npcvll*w :dm.jd1- ufrom an initial rate of 1.0 revl+dnv1m: w- pd luc.*j every 2.0 s to 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 vertica ;i;soo3alt m;l axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the fis;l o; toa;3mrequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moment u3rj 17j6faq bp)8raaum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ea;ttcs5. mt ,zlrth
(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 d7j*9 m 61o:nuyo tfsww7cjjj,b 3hkw(isk of moment of inertia I is dropped onto an identical disk rotati7ucnj* sbtw7 9o6y1h3mk :,w jwjfo j(ng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4xxm,cfv;pf.:8b0ps j $rev/s$ 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 0no*l8q 6vits l2jigvv))tdov9x 0 )oof a rotating merry-go-round platform of radius 3.0 m and6lnsjo0vxv 9q )oo0ivg*2id)v t)l t8 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-e t zkdmvr3c2evkmg ;su:v126 oeak*af*(3f *round is rotating freely with an angular velocity of 0.8e e(svm;tucak6*2* *zm fo2regvkva k1 d:3 f3 $rad/s$ 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 az. 8;trwi 4o,xs-r 53v qnfvhsb3cyd 1 white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mao cts 3dhz- fxy1;vsv.i,bw5 qr3rn48ss 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 in excess of d*3 n3 *d ksgdu tcyk7ema)-k:dp/2u m120 $km/h$ 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 y9v;aa8 pvcz8urg /9n$ 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, init(oc6be2t 10)ak-z/w3lxbrs8 bpstljec 2 e s7ially at rest, that can rotate freely without fricti06o b(c w2at-ez1ceb 2pkj/lt78essx)lb3r son. The moment 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 merjk,q;3fkyfei( mx6 9 jry-go-round turntable that 39jkmij;fk(e 6 yqx,f is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the en/txyy* od r-w+d of the rope lying on the top edge of *-ydwx+/ o ytrthe 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 the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same zu+2 0cj kehq;side always faces the Earth. Determine the ratio of the Moon’s spin angularq0; 2ezujchk + 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 rest at a rate of joo9xs ;/1t+j+k ldor kjf) 8l1 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 radius 0.20 m acquireas/,.kav/ w ;u4k+tr1asba lqs a rotational rate of from rest ov sa k4la 1usr ;,aaw./+qv/bkter 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 cylindrical disks, each of mass 0.050 kg and 8.htlxa k.t,jdiameter 0.075 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 reache jtt,xha.8. kls 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 x,;bfk n9f(g3ti)z hu-zvtrjx* 4in )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 our Sun, but with mass 8.0 time6d2 7rc1dmirbs 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, 2ib 7c1rd6 mdrlosing 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 lo2xnt ,bfz; jx7w-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 3 xtf; x7,2zjbn50 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 illustrat3b5qdq 4f +aay1maf+ol0j3emp (qld- +ilc b 0es 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 suwvr ,0p -zwr )ln-w)smrface 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 caf/n:jh,og 31qgb xy*o./ pr3l uq6 odvn 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 ojfvo6 or/ gu13glq .p,3qxn/: b dyh*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 the floor, and wt 1j( efz9v*aj jen8l(e want to exert a horizontal force F at its axle so that it will climb a step againsveza(l 9(j fn1j8tj*et which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2q5k. iiy5.samw5dgr) m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cwg .5 ys55i mri).adkqycle 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 rock into a sling of length 1.5 m and bea yxez91l 69zugins 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 torque come from?19xyezu 6az9l    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rho:z;7vy p x5 ossghoopy89 2.ods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms;: 8 yhvhsx2szoy9op7oop.g5 , and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of t8, zyf 26m:p9q bsgrc9jqkf/ unyb 65two cylindrical plates, of ma6:m9qg9yp6f nckb /,tbrj 2y uz5qs8f 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 rr/ (v n5t 8rdi3v3vrse rolls along the looped rough track of Fig. 8–58. What is the minimum in5(s3 v evdtr8r3/ rvvalue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume 2/d c*2 zb10dd9weh)yobuvi kuei v+/$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions afhgb7*pmxc72;ro(h as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the;fxpgah(m*7rho b27 c 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