A bicycle odometer (which measures distance travelbb3:)nabm x 28/zjlg b3ytvd3ed) is attached near the wheel hub and is designed for 27-inch wheels. What b3323 xbz)b n8/y:lj matv gdbhappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on t2/nev6 *ahaey he 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 eitheraa26 e */hyevn component of linear acceleration change?

Could a nonrigid body be described by a single value of the almj6mvuec :oc;,q/a b0tvd13ngular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu* ;lpqi vmq3 vwtvi0;5e 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 han+y.f1mjuhtw2tnk7i 9 ds behind your head than when your arms aj2 u h+w.m9ty7tkfi1nre stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven spi fs)py7fnu-i) s k4x4wsh1a2 rockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or u4isxpsw )nf24-7ka1iyf hs )a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower le95b-w6 gjf qzz*u o8uhgs with flesh and muscle concentrated high, close to the body (Fig. 86 fg*o5-hwjub uqzz 89–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35.qpag;yu*q +, gfyy f5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the nb akd 0ba/adq;ujbg 7p 95+f6fet torque on 5 7/akaba6p bbfd9gq uj0d;f+a system is zero, is the net force zero?

Two inclines have the same height but/ 5maf4b nb+v0sp l, n-;fpbgy make different angles with the horizontal. The same steel ball is rolled down each incline. On which inclin-bpl4n m;, p5 sav yb/f+b0gnfe will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneoum6g ou;rj,uad /*jkfq 99m+y yaqp ;/asly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches thma, +fy6 p ;;k*mqaj y9u/j9qar/g udoe 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 a-oxm*mv jguuklor( k. 6n0 jj/+nb7y:nd the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which hk jmkl(y* gvj.bxu7 mu+n/ -0oor6 nj:as the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum e9col(,fdy7n are conserved. Yet most moving or rotating objects eventuall7 9ln(f ey,ocdy slow down and stop. Explain.

If there were a great migratioo 1k g pdb36oh.lxb8rj- ;xj4lx4ykx:n of people toward the Earth’s equator, how would this affect the length of the da1gp4x8 kbkrdlx yxh ojx: .3o- j;46bly?

Can the diver of Fig ,66xxldl0my ag89jllp/ h rc.. 8–29 do a somersault without having any initial rotation when she leaves th llldg m,6jxrp6xh8/y 9c0.l ae board?

The moment of inertia of a rotat oik09 1pdt 53vhd5cmiing solid disk about an axis through its center of mass khdmp9 vto55c0i i31dis $\frac{1}{2}WR^2$ (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 r6up gzb49w;ss otating stool holding a 2-kg mass in each outstretched hand. If you sudp9b4zu6sg; ws denly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hod/m3.v0/i d z ehk,+dd8ms hgswever, one is hollow and the other is solid. Describe an ex /s/ ddmm hv3s,gid8ez.k+hd0periment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axl*i8 f9/nypqr fe points west. In what direction is the linear velocity of a point on the top of the wheel? If the aqn8 i*yfrp9 f/ngular 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 freel((-yp /z j0y5q tddvisy rotating turntable. What happens if you wiqzs-(jydd0 p(yv/ t 5alk toward the center?

A shortstop may leap into the air todw0:87ne 5 o 9i(dlvawb s-h.or;imzn catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Ex8oni0 (9w5an7d;-l we:sm. borvz idhplain.

On the basis of the law of conservati48)zf 8vbt; qzvx3nvpon of angular momentum, discuss why a helicopter must have more than one rotor (or propellevfvv3bpx tq)48; zn z8r). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in ;zuk 6vcs c38zradians: (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 oft8v *5zioo0;ut5ao o y an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun auz8voo t yo *0;55taoind the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dzu/y; (ht9 *7bewn okgiverges at an angle7 w9e;k( */ynbthgozu $\theta$ (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 .xijk h/p1(qq9 ;hr+wz qki c1ac63g mmotor is turned off during operatio3ah(1q hx+wgpzrj 16.qi ;k q9cc/ikmn, the 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 baa(2 u aul/*gpjhy8a6ttg (w2kll makes 15.0 revolutions, what is its diamettt6yaa kw uaj(/pu*2g2 8 lg(her?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How masgvk8 oj - cosc n)2;pkt;j5au):w y0nny revolutions do tcycsj n;jgw)t8a;5-:k)op u n0s vk2ohe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in dia/blee2 i f63kameter rotates at 2500 rpm. Calculate its angular ve 3ek6l/ eabfi2locity 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 rew8ax.+bndi7ud 9u1uhxy se: +volution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child dyne+h buuw+ua.8:9 x7ix d1sseated 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 velocity of the Earth (a) in its orbit ar+8r,x dh* .g(dlxgc + gpvri4wound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o+ cnpo/ 4 de-t(1difdqf 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 al0 c 4hqhd9vz7p vuxuf9876ucm from the axis of rotation is to experience an acceleration vhu7u0 pza 4fd79xqv98lc6huof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to ;1em9.gz k3qgdcz5z-mkbt (n 280 rpm in 4.0 s5q-dk9g t(k mmezn3 z cg1b.;z. 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 radiuv; k:cnth)sj:q xs9e.mz 5q8es $R_1$ 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 Moo6sk 0qvjr,e 20))i;k lemcudsn, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy 2c0su0,imdk )e6)l;kevqrjs 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 res7bjm9( py m ad8css1h q54*fmvt to 15,000 rpm in 220 s. Through how many revolutions m y( 7qhf*psd9m5 j1vm8cas4bdid it turn in this time?    $rev$

  An automobile engine slows down from 45ez8 f/-h,y rnau+qbl-00 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the str:j b t3inby9;messes of flying highspeed jets in a whirling “human c n9yib3;mjb: tentrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fri)u iz -k49vsa zjux:0om 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have trav ) uvjax: kuz9i-0sz4ieled in this time?    m

  A cooling fan is turnnw7qbuzwp :no,q1y5x/l-v le4 *ed,ned off when it is running at 850rev/min It turns 1500 revolutions before it comes ton o, qnbelv51y*z:enp x u4d /-7w,qwl 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 reor asr3o0870thgt. :md1chfcduces its speed uniformly frosh :3cm f gt0078c. ooda1hrtrm 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car ma 3kfwx :m*np7 r.ukyn2ke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter o:nuxnky7fk wm2 * .3rpf 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(yo93xq ino *4w ujt;v all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiusnj* 4iv y3(qtxo9 uwo; 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 m3cwp0mn mj0ls +7-dl end of a door 74 cm wide. What is the magnitude of the torque imnm73 slpmc0 dw0-+jlf the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the whee+.h, x61ga jb(vcyil+mv :lahl shown in Fig. 8–39. Assume tha iy( j1v.lhl 6c+x+:b, mvgahat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless r/c0jz/ umql-2ar l(tk od which pivots as shown in Fig. 8–40. Initially -(lmcj//ruq0klt 2a z 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 tighte ny-v a+- h1nkccv1ay:ning to a torque of 38 v y+ah nc: v-ay1c-nk1$m \cdot N$ 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 inj4,redcrs ) q9,q )hawertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cente),ea)c,dhwq 9s jqr4r r.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheely4 8dznsj)srs)j,jfw 5+- sq t 66.7 cm in diameter. The rim and tire hav) rsq4j5y+ sn-)f8jd sstw,jze a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizo,0yle0/i+ o *h2oiw;awcrljzntal circle of radi+/o0wzc iw j r;,*2h0loliyea us 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular spee) xex )c)8+d x*sgjoced gjos+)c* )eex8xdxc) (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment 8d -k qqnpq2f2f 4g lc,p)unint;7ge2of inertia of the array of point objects shown in Fig. 8–43 abo fqd g 24ke-gl)2qp7i8tcn;,pnfnqu 2ut
(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 whose total qm8+f jpkf/99(fth hl 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 correckc b4i p3gg;p.h3g mo0rz) 1zbt 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 sr bg04g3 hg3zo.;b m1izkpp)catellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5w+dzp q-r3 qs98 l;xxa0 cm and a mass of 0.58qrp-x8dw;3 s q9a xzl+0 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, accelerati:j8 l-op 5 vrz/6 ssahh8cljx6ng it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangens;tc*2d/g mpmib:f-1ng4p g otially on a small hand-driven merry-go-round and is able 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, n; -1modg*m:g n s24tp f/cpbigeglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is 99qcso0t 5qn mo(3dtnshut off and is eventually brought uniformly to rest by a fr(qo9nm 9dt tsc035onq ictional 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. 8z).srtk (r 4 .uovie5h–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, wh;(urcn 0 2jv *h* wzjpa8:wnhhich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45nru;8:n wv 0 *hhc*h2z (jajwp. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as sxv 5oj-c:s3az hown in Fig. 8–46.:avc3s- ojx5 z
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of twoc(lpq034hfn g l49j ze 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 four fy(ozp4 n)hb bcl rx2zo:v3 c:3ull turns (revolutions) and hbozo :42ycxl(33 v)z pnrb:c 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 of ie+lc,b.q lqs, nertia 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 nac+b-rj06 tx4:7 lrx xrrec/ develops 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 7.3 kg and radius 9.0 ct*4r6b pf,fib 16mhhv*c8wgt m rolls without slipping down a lane at 3.it th* vmghr48b b6*c1pw, 6ff3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as .2 812i5v8 ,vjzoshp pmksry uthe sum of tormu 81 vp k5 ,pzj22iyvhss8.wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kgh9j/n-mj/r 9mar*h t p and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8. n*r 9ra/h-t9m hjmj/p00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg star2,z 3 jw3cjh:-u;fegsqy nbi1n0 bu(x ts from rest and rolls without slippingi,x-3z(2 b;u1qh 3gu en0y jbswfnc: j down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and ii7hpxi rb*09h ts lower end does not slip. What will be the speed of the upper end of the pole jh7i9 rxp0*ihbust 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 tbix(7bb- 2u kn 95uszjhin string in a circle of radius 1.10 m at an angulubsz7(2iub j- n5kx 9bar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular0jmjc. jm*31w0b leu mdlk;7i momentum of a 2.8-kg uniform cylindrical grinding wheel of radi;ecl1ml0jj3mwkj* .u bi0d 7mus 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, ond4 nxowq0 m:d y+u3l.2 ie8ya.vwwb/q a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Figa2q.ww / ymoub8 yx+d l3d0qve.n i4w:. 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 red f+c1zd.sct eg2vt r-;uce her moment of inertia by a factor of about 3.5 when changing from the straighf1+. ;c-rd vcgezt2stt position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase hq1uw k5:up: i +cmz98 bony0a3ccb3ther spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rank m5ua8:0zc1b 33h y co:iq+tuc9wb pte 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 verticco4dkv-4 d2),x x yodxal 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 3.1xd, xc2dyo-o vx4k )d4-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 momet 7c*cu8xha0;n+qc vqntum 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 angularaq+qlup*sg6 dpv6 3h5 momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylind0m6l mpjz h-m/c/e;usrical disk of moment of inertia I is dropped onto an identical disk rota/ej m-sphmz;0 lmc6 u/ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2ka6+lyio5i .d1k8.l h9nwor;:a8wjys6lhlk .4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-rx-- l/oujpg m;ound platform of radius 3.0 m and mome /m-gl;x-p jount 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 bmku 29h fu1qezon88ao:7 )li-go-round is rotating freely with an angular velocity of 0.7) fu881eno:o9 lkb m u2ihqaz8 $rad/s$ 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 a3dc1caljipfnp) s9 :5 bout 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 momentu1pipd )9sa jcfn: 3c5lm, 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 120 e0k4z0/j mvtgg4;t pw$km/h$ 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 pfhd+do:w/*xy ;s2cy $ 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 s9h 7d mnzjs-q1qy4dx l1y6a*rotate freely without friction. The-mlxaq ddss1qhy97z y46* 1 jn 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 dievnw xs549o6p,462 bwbqb c1 :tjd ecuameter merry-go-round turntable that is mounted on frictionless bearings and has a mo 5p6woswbecx6eu9 q4 t:vb2cn b1 4j,dment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of therdxt (.op* +wf rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L,wd .ot*(rfpx + 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 tht:ff3y xruly 61dsla3b- s6h 8 qqt-f*at the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its or*1y- ffxqf: qyd6 s8ul rtl a3ht3-sb6bital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate1j3z* fhknbd6 s from 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 0.20q919m61gpc -y7qaa4fu gau:ab7sx:qjaetc - m acquires a rotational rate of from rest over a 6.0-s ijm9-at a:a7ucafxs ueb9q76 gg yp11aqq :c-4nterval 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, easzj3v* mwjian-.t+wv 8 u z.:h whb+t0ch of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub t .*jw3v+ 0wuhtw m s.binz-zjhav+8 :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 an-wum ngw/* tbm :d18nlgular 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 masxle6s/n 96-gmsd6v *a( lcpkettei3;s 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost -3nep/(s msg6l xce;alvk t6d*6tei9 mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy szzy3.ckxv )t 7edi6s;tored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diadx)kzsvz;. 3y i76tecmeter 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 anm ibr76 vnmxk/ kdyvn0 3a/*9n $H_2O$ 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 rol 53nf/ez2spz:zduk9y ling 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., we ignore friction0mw.an lj )gim -t0*ie) 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 rod. Assume i0)na0 i-.t*ewm gjlmthat 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 i x3 0vt4lbh vg9j7,izss standing vertically on the floor, and we want to exert a horizontal force F at its axle so that itixl,g bv 3hj4790svt z will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2 9r)bqv-pjzly tkvvxl24m7e: .naq :8m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the n: lvn2-jq9yl p)8mv 7 .x akterv4:qbzormal 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 ;zkr tvv :dx5ymvy)hb50t( i-begins whirling the rock in a nearly horizontal circle above h0); m5kbyth-(rvd z: i5ytv xvis 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 qte)mti 1, c*fsolid cylinder and her arms as thin rods, making reasonable estimate1*,t qt ecmfi)s for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two 9duho/ /e-pyscylindrical plates, of masss-peo u/ 9dh/y $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough tran9ewfwk5akv78 4kqz .ck of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to k 4 vwfk87.9 wqnzae5kreach 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 as6 ndqhdcnbzs- .agg jv1( u:j11+/crlsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redun y-5edb9ki0ax+ n .zgtpgfzsh9 /, d6ces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wz65 ftiez.ypd g kn99/h+,b gd 0saxn-as 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