A bicycle odometer (which measures distancel1s:18bw tr. fgumio/ traveled) is attached near the wheel hub and is designed for 27-inch wheels. What ha:o mu1.bw s1r/ lf8igtppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular y2dnz-qmrx6-) vdr; o velocity. Does 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 m26- y);voqrr-dxnzd/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a swc8huh c 3z8u4g akv413juj -cingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert ads0m vu99j; k r6ake*x greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head x0gan9;j tl4 kpn gmfl79i5,m than when your armsil 0kx;famg jm4t5lg9n,9n7p are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has4wn ekn k/ )am0hr+b9t seven sprockets at the rear wheel and three at the pedal cranks. In which g)e+nkbr4ham09ktw /n ear 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 sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with(:*rz ; ldocja flesh and muscle concentrated high, c(cz;alj d:*or lose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkcedu p5 ey93ymk rowlli7 36(:ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a zh,r8 v8o0 sgyihr.5dsystem is zero, is the net torque also zero? If the net torque o rgoi8yzv,80 rhh5.sdn a system is zero, is the net force zero?

Two inclines have the same height but make different 1fc f(pzr* q5y.bbr3f angles with the horizontal. The same steel ball is rolled down each infbc3 *pyz.(f5rq r1fbcline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (i99 xm,5gnw mhoh p6p+from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total ki69,+hph o 5gpiwmm9nxnetic energy at the bottom?

A sphere and a cylinder have the same radius and thv-zftd fkcer7ljv:9wv6.l 0 ze7p4g 1e same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which -v z7c7elwl1 6v4v:rzkd eg jftf9.p0 has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mosg4 + np*hoscof739d bjt moving or rotating objects evcf9+n7gjosd4 h b* p3oentually slow down and stop. Explain.

If there were a great migration of people toward the u5iu 2y35ay jghxdf r+o-2cj4Earth’s equator, how would this afff xyjh35yru+o-4uj g d25ia2 cect the length of the day?

Can the diver of Fig. 8–29 do 9of.fqf peygq:5 ; ikn5+lf)5 vu, lola somersault without having any initial rotation when she leaves thf 9uqp,gfq e +yinl fol 5.vof)55;lk:e board?

The moment of inertia of a rotating solid dif)xjci5 ) l+ n.h2 mhdsk+ur8esk about an axis through its center of mass ksi h2 )5)erdl+hu .nj+fcm8xis $\frac{1}{2}WR^2$ (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 in each outstnyvq9x9 bme;( s:s-bxretc x(qby9 -ssbnmv9;x:ehed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.hk ukayqo6s9z yo0djd490 0e/hl26 g n However, one is hollow and the other is solid. Describe an experiment to determine whic4 dko uqz6d/y02ehnjgo9yak09l h 60sh is which.

The angular velocity of a wheel rotating ongmo -r7t r-,woe.ffn8 a horizontal axle points 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 linr. rftm-e,g -wo8 nfo7ear 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 freel+ni gb fr 5;tum,k,lc uj h0y0z5ful;0y rotating turntable. What happens if you walk,izhuf+ul m0r;b;j yf5 0 0lkg5 c,tnu toward the center?

A shortstop may leap into the air to catch a nm :bys2;y zq+mszw*; ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly 2mqz+nw;m b;:z yys*syou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mo5fph 92pc(ufom0 hq6c mentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to fhp0om5 9hq26fp cc(ukeep the helicopter stable.

  Express the following angles in radians: (a) g hvjf6n( ;oocm/4m7eq ;-urh30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calc7 di71ll6d 0qhjfmpy2ulate, using the information inside the Front Cover, the angular diameters (in radl 2jq0i7 f7 1pl6dymdhians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed e9;vv y 2p5ehto*w6vk;hq /my at the Moon, 380,000 km from Earth. The beam diverges at an angle pyw;hkvvq;o y6ve h*e29/t5 m $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnl9lo; zq r/. fuloi8i6ed off during operation, the bladesfo;l/li9 uq.i 6zr8 lo 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 mw /v9f -7sq7gtvnf ;hfloor 3.5 m to another child. If the ball makes 15.0 revolutions,h /wmv t79s ;vf-nqf7g what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How mqz fkx 6s6ax8yu.tbeh;- k,z f1.vk (iany revolutions do the wheel,.ziakx sxq;ybt e(668k. -f1hfkuvzs make?    $rev$

  (a) A grinding wheel 0.35 m in diameoruv(7*a ;q /ueue) alter rotates at 2500 rpm. Calculate its angular velocveu rq o/auea);*ul(7ity 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 revolutiouqb4ywcc . w; uf718 l)e:t/k/izi ucbn in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seat7w8 t/ib. /k:1wfl c ;4 zq)uecibcyuued 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 velocityk/kdx8 q2 *bpb 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 speed of a poin, 0uqhn0(m.a 1 r0vhkxv;wz+h2 ysnto v eq(y1t
(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 particle 7.0 cmn+x 9 n:j difosl6q+f, from the axis of rotation is to ex+ +sj9x 6ofnldinf q:,perience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center f 4*fhn8sjb1sz rom 130 rpm to 280 rpm in 4h 8 s4*nzj1bsf.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 .sl9eb p l-h joz/y 1(;jjk,xr$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themseln- (vfdj4*zztves 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 cylinder with a diameter f4(nz-zd*v j tof 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 dt hv-(y(9puo 220 s. Through how many revolutions did it turn ( t-d hyvo(pu9in this time?    $rev$

  An automobile engine sl msnbxep9ii,,zn.n a6. f0pb7 ows down from 4500 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 of flying highspeed jets in a wht 33*uff p2ikdirling “human centrifuge,” which takes 1.0 min to turn through 20 complete refk2i3f p*ud3t volutions 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 v3,n.:qbt jy 2q 5x)mtop0 urnrpm to 360 rpm in 6.5 s. How far will a point on the edge of the who pbr53q: nu.yvt0 2q)m,n xjteel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1gnqt)gj: jk 0fi: a;gi5/o8ub500 revolutions aqg8igk/b);jjuf5ng :ot0:i 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 9c3g93f i p3vw6a prmqthe car reduces its speed uniformly from 95km/h to 45km3 gcpf r9q3mi396vp aw/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 reo1a,)l.*gkhu -aoewf volutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires havw .ghlae)uk,-o *oa f1e 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 u;2pvtj x/s7vher weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 1jxp72t vu/v;s7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcp ,hmrju )f v3:)o)utwe of 55 N on the end of a door 74 cm wide. What is the magn vr) htuj3p:uo )f),wmitude of 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 thekbn.,epj ; h6y wheel shown in Fig. 8–39. Assume ph.y6kne, j;b that a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass h1yjonc98d ,lm, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. ,9lc1yj8d nhoInitially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening toxqs5t :3) sg6mc cvey5 a torque of 38cq6vs)tx5c5sy :e 3 mg $m \cdot N$ 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 axi*2s0ihb) 93cvesc10.8-kg sphere of radius 0.648 m when the axis of rotation is throu)i 30v csa29bexhc*sigh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim y6( i4bnu6vveand tire have a combined mass of 1.25 kg. The mass of the hu6unvbive (y64 b can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balxuz7xs ryey b2+ 47e;i zk)ac4l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cacy7;ekub 7z xes+ )xaz424iyrlculate
(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 angular sp eio-74 dlq1v(pm-b fseed (Fig. 8–42). The friction force between her hands ae m7d-qfb-sv41( liopnd 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 arraylk25sees74*knk:c 7 vwht +qchjtl03 of point objects shown in Fig. 8–43 ae*5jv 2hct:t lnqc4hk+ls kk 7w7 30esbout
(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 whcfc/u0 v451rf0o gy:mmc+dy p ose 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 spinnir9e u:ov0vm(g ub)6 bing at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite ha vmgv(6 e:9u rbboui0)s 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 in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radiusz(8p , v7wjluk rq*t8a of 8.50 cm and a mass of 0.580kaw j8r*p,q(uvl78t z kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fr1t ;hw) ;5u0qb jshm7a)ouwxtom 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 onjgeycfckn 0 pf1 ++1y2ypn s.0 a small hand-driven merry-go-round and is able1egypc 0yk1+ sj2 y+nffcn.p0 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 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 o /mfcffhuuf9-v+ x *7pff and is eventually brought uniformly t9 mpv-uh/u7f cf*ff+xo 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 acck; gxj t7, :svq0li0x/8owu coelerates 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 c23ih3vze7bj.x1kdk action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45i hkedkx712zc j bv33.. 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 b(kobpy)w 6g-hwqx)z / obi ;r8lade can be considered a long thin rod, as shown in Fig. 8–46. rzb kphq6ow- ( gwox;8i/by))
(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 cons b):w;7 ;u ur(vdiuwwlists 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 acceler)w+3zdjm effvf)ob 3+ates the hammer from rest within four full turns (revolutions) anz)+f3mewdfjf+) o3bv d releases 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 o 2gv3h8 dgqru(kp /crx s1kl2/p51uzmf inertia 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 dev7(t qat 1q/n2yij/ckeelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.35v-v vs+;ux5uk ja/yo kg and radius 9.0 cm rolls without slipping down a lane at uavyu 5ks;+x/vovj 5-3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to ths+83bgeehb:hc h.m( de Sun as the sum of two hg8 h3:dhe+bb. cme(s 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 13 :r 4bq1cxzup640 kg and a radius of 7.50 m. How much net work is required to accelerate it from zqb31rxuc:4prest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st2 s r 0zyqu79s6juxlj0;x/gul ,mjl1carts from rest and rolls without slipping dorx9yjlucl0s mu 0su1 / l,x6;j2qgzj 7wn 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 to fall yl ,tq l4+yt/kand 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 yq4 +/t tkl,lyof energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a th i- i; 4eltyt -:gt:nm7v7jafy 5rts3din string in a circle of radius 1.10 m at an angular speed of 10.4 -t syv4d7: fayimn :5tij-7 grtlte3;$rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o.k+zzt vu-vq 7f a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whe7vu zztv+kq-.n 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 pla-pa/+zybb51 , xi ynbgbry6h,tform that is rotating at a rate of ybnr+,zbp b6 x/a1-,yibyh 5g 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce hes4d92 ;.jb) )y+m4yenbe rgi pkyeb; er moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she m+. yb;)be ngkj)ei 4ybyr;e2smdp9 e4akes 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 fromk r*h; af,3kcn an initial rate of 1.0 rev every 2.0 s tc3rkn;a,h* f ko 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 vertical axq q sh4 z rhm8/p*nff*l:83kfois through its center at a frequency of 1.5rev/s The wheel can be *fl m4k f8s3r*q8fnzqp:hh o/ 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 angular momentum of a figu :y p hz(aqzv.i0omc(5re 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 .u) si z0*oqkc;2a+rmezb ez+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 idnew(q.gsb08 b79+ itq29adnekn )cgb entical disk rotating at angular bei gg9 bnk.bt8 70a(2n+nesq wcqd 9)speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4v8c0 2c :unwgz $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merrmz hdp ed)9 , /oc a0p3vutygz9idn-ds*he702 y-go-round platform of radius 3.0 m and moment of ic ddda0gopv 7zd3zt0e n*m,9/isey 9uh) 2ph-nertia 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 withl9 7(n vdzse;/9kpjq y an angular velocity of 0.87n/sj 9y(dpk lq9v;ez $rad/s$ 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, losing+ol.1 lgfh r 3k*9 20uzs,g+emcarxalouh* ( j 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 momenl efgchkaosl 9l o0(u.h+*2r +jmr3zx,a1g*u tum, 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 involkbgr/bn tw0+ 5ve winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass m +dp+vuaaneg7 o4a+zt0 /nu 7$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely without)gvezg; .7j qx friction. The moment g7xg z ).;qjveof 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 merzq-8v5 jfb q-bry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia jbbqz-58-vfq 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 w)d rat55z 6bpwith 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 spool rolls behind the person r)wd5p bt z6a5without 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 thkmr+se *tue(; elh*9s at the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abh*ls* r emut+ek9;( esout 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 fyzl.z u2t -mx2rom rest 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 uniform cylinder of radius 0.20l cenlumcu37 (5xc e lyp)fgt:m h7703 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor. 7 g(77c3:mt mcylune)fuxp5 el0 3lhc   $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kgb)j d4r)p7dbt and diameter 0.075 m, joined by a (concentric) thin solid cylindricd7jbdr) ptb )4al hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whe vb:wtr7v)-*agwz ) .oef(nddel ($\omega_R$) 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 o ,rm3 b*dvofsty:2l+mf our Sun, but with mass 8.0 times as great, were rotating at a f* 3m l,sm+2tb: oyrdvspeed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is wp d/ 3s+vxa:efor 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 350 kp w+v:/sx ae3dm 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 illustral np,0tmh1v7v u:iv2ytes 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 *o-q7diq/ eq cxz1j tal0a((n rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., w;w:gc ts 3ohk,e ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the ro wtgs;c :okh3,d. 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 fc; x y(7+3ya5xrsfzcstanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will y7sfx3;z5f+a(cy rx c climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling 2+1kwit gmects1gh) ,with speed v=4.2m/s on a flat road is making a turn with a radius The forcesg1tei+2kg1wth)c , ms 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 )fmu t(-/xrifgy8s8hp n z6r00.50-kg rock 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 ts y8 h(r8m/n-pfg6uztxr i0f )his feat, and where does the torque come from?    $m \cdot N$

  Model a figure skaterneatg)w8c3k:)i36 bn bbf jlw0j- x3s’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretc3t8k3))blgxbnwfna3ji6 be-:ws0 j ched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical e6p2l so yo.oau/fyeq7c0qr .m4* uz8 plates, of mass p.oc.s6om* zflq 7 /ae4uyeqyu0o82r $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rad* z(ep3m)r nwoius r rolls along the looped rough track of Fig. 8–58. What ( ow)* pme3nzris the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bue 7uee e,,o+int do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as t4:qmirq4, zmhl63msful: np 7he car reduces its speed uniformly from 90km/h to4 :,m:f3 6msq ziun lql4mprh7 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