A bicycle odometer (which measures distance traveled) is attache3oqd/nw4k k)4lyvvs ,n3u4)xmid 3gl d near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inc3 dul3odixl/n) gnv3)ms44kw4y v kq,h wheels?

Suppose a disk rotates at consk3m8n r 4u5 .zvn+vxdztant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity 4k z+vxmz5n3uv.d8 nr increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloha4uivn bbr53d(m.v86 wx -ancity $\omega$ Explain.

Can a small force ever exert a greater torque than a fo8 h33x ns62mzvw- p g8q17rf xnga.elarger 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 sk +*ec+fc 4:blojtx :c1z. adpit-up with your hands behind your head than when your arms are stretched ou a.jt4k*fboxcze 1 dcc +p:+:lt in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sptyhor;:x0 xgc;r3e+.iblx,x v2 s rr6rockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small.6;rrrtv or;y2,igx xhbx +l03xs ce: 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 fn d,)v+cmqme- o,9dgbda 5 b0plesh and muscle concentrated high, close to the bodmbn5 dbqe m-+ap0cd v,,9go)dy (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, h/1q 3xthb,q evs*- e1cqwou *narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net k wq:vpurdj c-d+0:z1torque on a system is zero, is the net forcwcd u1qrzpj0 d -v:+:ke zero?

Two inclines have the same height bc.mze oj;injt7iw4)d4cc m +q5om5y ,ut make different angles with the horizontal. The same stj4c 5iq+5ode ,tzy. icmom4)n7cw ;jmeel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulkvgav ehcb;c1n7hppe..5p ( . taneously start rolling (from rest) down an incline. One sphere has twice the radi kh.vp. vepb51cc.agph7(e; n us 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 twbpttxi1;*yyzbu r 2 vf;0+9whe 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 thu+*2 z b;;witr09b fx1ypvytwe bottom? Which has the greater rotational KE?

We claim that momentum a2yx( aiii+owy.a;rj *nd angular momentum are conserved. Yet most moving or rotating oaa.iyxy *2oj;(+ii wr bjects eventually slow down and stop. Explain.

If there were a great migration of peopw jgv00mb3410 y*nf (8 wssljh oeyu3 rn+.laple toward the Earth’s equator, how would tuo3 gslp(be81f00*wjm njav+w. 4ynhsly3 0r his affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any .yvzx , 53a/+7c:swswxz eoq8xmrzyxe nu( :2initial rotation when she leaves thews,3+2/sy:xax(.qoex x rcnyw8ze:57uzvzm board?

The moment of inertia of a rotating solid disk about an axis thbzqvx(o e,g ,nz,mlvd h+ 15rz 27ra4vrough its center of mass ih4mre2vg,+(v,zon517r ql v d,ax zzbs $\frac{1}{2}WR^2$ (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 am39tq 0m5 hwxy 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same5ty0w9qmxm 3 h? Explain.

Two spheres look identical and have the same mass. However, one ir 3w athezhv2.h 39xetjxp5)0s hollow and the other is solid. Describehhph3 t 90xrw3e5eva.xz2tj ) an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In whuvy9(+. w7 m lzbc -qoopmvf/ sz5e9j)at direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangentil5j)fzzm. e vwou9y+cqp/m-b7 9o vs(al 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 rotatiw3on)f c q5re())xxazng turntable. What happens ifxw eq)c)z )3r an5ox(f you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As mm bk3 ;+l p*mg:gka,nhe throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legsk ga+nlm;,k: g *pbmm3 rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mo b5b.c a.ta.pw+ yw,tmmentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propec b.. t.aam5y,wbw+tpller can operate to keep the helicopter stable.

  Express the following anglv sb/y)kl(xx6 es in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usicnab -5 l/ib4eng the information inside the Front Cover, the angular diameters (in radians 54a/ceb-lni b) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at thkow .ix*xl- 8ulxbozm ,1. z-te Moon, 380,000 km from Earth. The beam diverges at an angletxm-*.z,1bz ollo- xxk iwu8. $\theta$ (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 c:zw)5 hiy:kv2d s* zfih0ux4the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow ihifdw4 k2u :*:cvyx5z s0 z)hdown?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to q-08(0ut y vdlakmr32ar ux6s another child. If the ball makes 15.0 revolutions, what is its diameter? xq(l-u 60 30 ry8katur2dvsma   m

  A bicycle with tires 68 cm in diameter cm v1:*jt0/qwgs1gkz travels 8.0 km. How many revolutions do the wheels q g0/t vk cjmgs1z1:*wmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itspg r: *h8o.zf0lxts+m angular velocithf.rsomx :t8 z0 +lpg*y 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. uk qa/3nxh nfpz7)w208–38). (a) What is the linear speed of a child seated 1.2 m fr 0x3qna khpun2w/f)7z om the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Ea.)mncmhmnx*67hsb /3/g zxy vrth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedg ,fsr5kcc nl0ac13fl5r 2)e t 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 rotate if a particle 7.0 cm from the axis of n72 oyz6dk cil3rz.q, rotation is to experience an acceleration of 10diz. cl kr y2zn37,q6o0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acci52vauhxi01r n1njq 1 elerates uniformly about its center from 130 rpm to 280x0i qhn v1 ra51nj2u1i rpm 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 radiu5)nz l :t,/qv lnc5fhgs $R_1$ 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 Apollatdk. ykd-5 x6o 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 revolut5ay.kt d xd6k-ion 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 15i,m386f5vnm wk 1k )b3whacjm,000 rpm in 220 s. Through how many revoluhmwwki56c 1jmf,nk m a33 b8v)tions did it turn in this time?    $rev$

  An automobile engine slows down v+ jq2u4oujt0 ;;gob /hhfv:w 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 whirjir04bya n2 ,mkvi46j 37f ybpling “human centrifuge,” which takes 1.0 min to turn through 20 complet4v yb ,p3b4i0yia 2jkrnfm j67e 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 unifooc c0j*8rbab +rmly from 240 rpm to 360 rpm in 6.5 s. How far will o0b+c* cb8jraa point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running /ntr*i;4mol nat 850rev/min It turns 1500 revolutions before it comes t;n4/l*mr otin o 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 b r gs7o,xh3w; cge2 rj*4b*easpeed uniformly from 95km/x rra e7js ebb3g*h4,; oc2wg*h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform:: 7avneni *(cp jq0ghly from 95km/h to 45km/h The tires have a diameter of 0.80hej qcv :i 0:pa7ng(*n 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 weidaza *v3dz mg: w7 .zrvm4)m5wght on each pedal when climbing a hill. The pmg :wz5a. zz3 r7madd)* vwv4medals 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 door 74 cm wide. What isk7 e53h,jep+ m gq+pyl*zjb4l the magnlk73yelbe , *p 4z5gqp j+mhj+itude 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 ax d8eq)r0zc e:yle of the wheel shown in Fig. 8–39. Assume that a fric)d0e8 yzecq r:tion 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 ma,fm 4mff(s2 6u;rx z)zjg1 xx/tqqui3ssless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizonta41q,/t zrjms ;zf iu g)x3x2m(f6qufxl position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of ajh9/n/w 0l mptn engine require tightening to a torque of 38 whj0 t/pl/m n9$m \cdot N$ 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 in7. kz ; or)ipke+idcd4ertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is thrk c.+kero)i47 ; piddzough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dia nc8 9ky)u v,q0gsbu5kmeter. The rim and tire have a combined mass of 1.25 0 589ukkbngqsuvc ,)ykg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, uei1u)u(oj8g u2 ez0xlight rod is rotated in a horizontal ciugz) ej2ui e( 8o01xuurcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potteqn9 bczk3 8(o/0g2 vqw4 ecimar’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5ok0cw 3 gzieq bnm42c9 (aqv/8 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 arrayb /8op+aqajee,3 (ors of point objects shown in Fig. 8–43 aboe3 (r+eo ,j8/bs qaapout
(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 j- kel fi..yf7two 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 cylindrical satellite spinning at the correct rf(fz ;bsb +o1kij(- dmate, engineers fire four tangential rockets as sh mf-bbj1zd; fsi(k (o+own 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 in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylin ,mup w94fvn9dme:0cpder with a radius of 8.50 cm and a mass of 0.58f0d n:mp 9m,ucp ev9w40 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 frpq0s*p:r llr2om 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 tangenti oxm/f 1z998i-amx fbbz8xm2 lally on a small hand-driven merry-go-round and is able to acceleratex2-89bz8f 9fxm aomz1x /milb 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 rphxbu,*m z n+ta ;82amym is shut off and is eventually brought uniforzax*mbt y8a;+mnu h 2,mly 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 3.6-00 5c nfm1duhq,uvip4o c(vy-) u,hybkg 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 bada 4xgb9-co/ m2e/7svi m4jewll is thrown solely by the action of the forearm, which rotates/o ea2cgbx dsmj9 w7 /i4me4v- about the 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 cw88+yci ).ksb5uusa ban be considered a long thin rod, as shown in Fig. 8–4u8b5)auk8. yc ib+ssw 6.
(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 consiia2evdw0k fv 2n86-bists 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 accelerates the hammer from rest within-) iwsibx:f tw ah5xpe rq).74 four full turns (revolutions) and releashqw iei4-) s7r.w) x b:5xftpaes 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 -a g43hd* gemzinertia 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 u*:px;0kxe 8u ocp6phzy z12f -z5uak 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls 8g-.qil8zkns zr5uxo4hd+e;5h zx5 qwithout slipping down a lane at 3.3-d 5olq; ie nuhsz5hr 5g.8+z x4qx8zk $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of b2wj2jhdfp y-*un1e062(ew uys2naothe Earth with respect to the Sun as the sum of two ter0jw 2sw y 6n2d*ey 2uuh2fnepoa-1 (bjms,
(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 x1/gm6jfyag.h8(cw i work is required to accelerate it from rest to a rotation rate of 1.00 revolutg x1cwy 6f(h. ga/8imjion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm a,9dnyd g/kyzkg +m-m*nd mass 1.80 kg starts from rest and rolls without slippinm/gzky d -*y+kd gm9,ng 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 onq3 1n)fguf jgh64sl(m(b. mht its tip. It starts to fall and its lower end does not slip. What will be the speed of th.gtguf6ln ( 3fqmhb)(s4mj h1e upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin s1 r2o0lagvlq , r9ejl.nc3pz6 tring in a circle of radius 1.10 m at an angular speed of 1veacq0op3., jg9n 6z l1l2lr r0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a,l) xerfx qjv0h d86z( 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500x d)jlf e 8(qzvx60h,r rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate b5i.dbxbm bhu7:/0 -sq2f ifhof 1.3rev//m. u07xdbbfbh:iq5sfih b- 2 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 7/q bn rrzpv)2f2 c;kuher moment of inertia by a factor of about 3.5 when changing from the straight position to tqkb fu2c/z7r ;2 vnpr)he tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate3m (h8y*4qxx 5kr pwif from an initial rate of 1.0 rev every 2.0 s to a final m*hx8x pkr f54(y3iwq 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 rotatinj7 u *ax8,xexug around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg * xau78exj,x uand 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 figure skatvqmk:(6mjik * er 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 Eua)0s rrgg9t+4,f z3h o )soc 5uiyh(xarth
(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 mooeqygpxj87.3 3hi2 c-qh2 psyment of inertia I is dropped onto an identical disk rotating at angular spepqs82q3hpeg .- xoc3h2jyi7 yed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2. nbpz 72ovp*1m4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-r 6qax7swjxpa+z 3r+)eound platform of rx)q3ja+xerw s+6paz 7 adius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of/8b mgpo w :na.:/ikc-a:vdw h 0.8gm :w:-p h/:wn ai.c8b/akovd $rad/s$ 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, losixgvlmg 0)u8mqm ;o0v3q+fyzwlw *ge ho:7.t4ng about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round toeuy8 mvwg .0m zt h+7gfglq 4o)w*0:vlq3xm;o 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 involv:1 chlcwx im /q0zcpcr4t: l30e 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 h. 1 m7etep5ja$ 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,-w3zekg (edu(5 e l2 zihb:on( that can rotate freely without friction. The moment of ine: o2( gzb3 (e5-ewn(eiu hdklzrtia 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;0+wv5yitn ge.c7ox,5:pw figkr e6l the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings +x 6gk7inw e:g0cfly vr,. ;5pot5weiand 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 witbnxl7 x(yfpa z5ecgp5 .5;m/oh the end of the rope lying on the top edge of the spool. A person grabs the end of the rof5xgnzx(m 5l.e;yco pa5 bp/7pe 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 side always faclgh0f0b y9- qdes the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, t fly0g0-b 9qdhreat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fr0kfgn,x:vpup f8:)bg4c k 0ywom 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 0.20 m acqu 7xs0ys yb8:/hdy,v ouires a rotational rate of from rest oveo:7su/yvh 0bsxd 8yy,r 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 oqojuh 5( e,ey3iuhxd8r* *8yqf mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long stri,o(*yu5j 3xueqq *eirh 8dhy8ng, 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*-gu k .,9fo5iw/*kmhjv . /vbfi: iexopvwp- 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, b *62 o(vrnskbbrhre91ut with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed behr2*b9(eorkrsb6 v 1n? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ener;)o gvibwf:f 4gy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindric fob:i;4w)f gval flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates and pwco0x- o66q f/d9n ylpo*a: $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 e-k- u q m(yv a7jj-ezahfi:.;8h1l lg$m/s$ 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., we ignoqd8t h(m3f7jpp )dmj(re friction) about a hinge attached to )3d djtph p(f(7m8qjma 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 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 t ;wa6vd:epuw rk hh;9ctdes9x+ ,*y5floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step ha,h r :5hys9k6d;p*xudt +weve 9 twac;s height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flppn9/ce v 6cesjr8-4n at road is making a turn with a radi9r6v 8s/4p-cnj ecpn eus The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.b hjsd*f3hi 6750-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearlj 7*bhsd6 ifh3y horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a 8-okueetp ,r*;a1mtt 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 skae,*e tp a -ru1omt8t;kter with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists obs(+oe; ;y58otwtq ydf two cylindrical plates, of mass 85 bo;q;sdt+ owyt(ye$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t4 v 1:xpy.pazv,g2uurwe/a +qve g6x)he looped rough track of Fig. 8–58. What is the a.r:qax)/6wv42u+ uy pg,z gv exv1peminimum 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, but do; lezd9, ug p1p(nzc(g not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its opu )52b.c4az o;dk ylspeed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What waao4k. z5ud2oc)lbp;ys 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