A bicycle odometer (which measures dista26gj32.iu x5x tpxcknnce traveled) is attached near the wheel hub and is designed for 27-ixxjtic p6gx. n u2k325nch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at con 1vc/k: tklle h0l,7gqstant angular 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/or tangential acceleration? For which cases would the magnitude of eithqlt1,0 h: cev gl/lkk7er component of linear acceleration change?

Could a nonrigid bodyobl+)v lf 6bobs/3r :s be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a niy97b;87bx e)*d tn:x s)mcwlgk) wmz g,-cqgreater 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:vja g8,dp:sr w :3zfn with your hands behind your head than when your arms are stretched out in front of yo 8zw,pvjas:fr 3d:g:nu? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three atvg)zstabu5+k,0xj )y 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 hardersjta 0gb,y5uzxv))k+ 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 wit;-l zr4zkrxu0 h flesh0z ruzk4;xr l- and muscle concentrated high, close to the body (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, nah5mx- *i-f,tm6c:ox -gl rkmxrrow beam?

If the net force on a g kh0qvfx**rj(5agd 8system is zero, is the net torque also zero? If the net torque on a system is zero, is the a *8(k0grdv 5hfqjxg* net force zero?

Two inclines have the s / (sw.(xacjyzw5r dci6,p1ptame height but make different angles with the horizontal. The same steel ball is rolled dowj wrzt(da /ciwpx5y (, 6p1.scn each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollin9q(xl dcm *9seg (from rest) down an incline. One sphere has twiceem 9 l(qd9xs*c 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 kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. T vhz ):7klyse6u k*.sfc -w)eohey 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 to l. 7-)owfy*ezsc:kse)v6h uktal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are consedzrs.x5h5s ci10q cw.9kdv j2rved. Yet most moving or rotating objects eventually w.dvx . csr jd 2z1cqi95hs5k0slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how 7txo*m * * :gykk2urbqwould t2kgm o:*ykx7qb*r *ut his affect the length of the day?

Can the diver of Fig. 8–29 do a somersault with ipf2 dzy svt/5xys7:st)o2 m)out having any initial rotation when she lets5isyxs2o y2/zp)7v fd)tm:aves the board?

The moment of inertia of a r0 lxsy) 70 frjk p3 jml*i)opv*kpo54dotating solid disk about an axis through its center o3i yrsvlj0x okj7m*k fdpol*p p40)5) f mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each oug9fb6 vvic a*g4vv/g lnf6/d2tstretched ha4*d6acvnv9if/ lbgf gv26/ v gnd. 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 ma1 3r:p- qszsvess. However, one is hollow and the other is 3 pv ssrz1qe:-solid. Describe an experiment to determine which is which.

The angular velocity-owh wch;n9yegkz5 6n :zp)5h rz6w9b of a wheel rotating on 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 tangenti5w-9hoh bn;5nkewz6rz) 9cyh6zg p: wal linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the egq;h /lhs lj0-dge of a large freely rotating turntable. What happens if you walk towar lhqs0g /-lj;hd the center?

A shortstop may leap into the air s2uix00y4 ;9htz 6: -w ittqozu ;oqjtto 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 9 s 6q 0qtu-2iohtt;4; twu:zxzjio0y legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of awy0sk ldp015 s k464udke rhm3ngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pmpkw0 41rsh 4 u03edl 6dsk5ykropeller can operate to keep the helicopter stable.

  Express the following angles in *sf.kbmrq)gfne 2 2c4f6m*jjradians: (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 ofh 3moa0 s:4rhsa+e1xp an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamet1o0as:xea hhp3s+r 4 mers (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam3x (5siw 3q/i yn)ixno diverges at n3o3)i/(yi5ix wx nsqan angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is4k.exttl5 x, c turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as th, cxt5x.4t kele blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anfh2sjk-qqs98 s:8 vf j/fq2lnother child. If the ball makes f jjqf 9vq82sskh-fnq8:2s /l15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 smos i0-99 hz j68hey00 jlpj8myom4 rkm. How many revolutions do th4-ip9js0mm 8o y h800esmjjhyz6orl9e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in j/ j2ozzv9tf4* 9kshcji52e idiameter rotates at 2500 rpm. Calculate its angular velo czoi 995j2/kztf2e sjiv4 j*hcity 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 rbk b-z/y)/otb o 7vv7kz *3vclevolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chi/z3 z7okbt bvb okl)*yv7-v/ cld seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arou2)v drvshe d r7s75ju)ty,dmz4z -g1and the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed cgv/,-q/k bjjay; u 1kof 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 t9j 5ena 4b ;96e-nkbbtfx+ubxhe axis of rotation is to experience an ac ab- xxtkf 9bb4nee9;ujb5+ n6celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its7t 7kfmn -2i/ftfn q2a center from 130 rpm to 280 rpm in 4.0 s. fi7n k7aq t-2f/mt2nfDetermine
(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 roiy (8;( a:,lhqaaorbpsu:0$R_1$ 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 s6wdfyo 5 ),lejpacecraft put themselves i, jedy56 )olfwnto 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 of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thro vcvpjl7 m5)t3ugh how many revolumcv7tp 5 vl3)jtions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm i1zwf g;kqqz 45u /anoeh-sz:8n 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 l9; sgnu ezks+n1a;m3yi) 3eoa whirling “human centrn3loeym; z; n3k u ag91+ssi)eifuge,” 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 uniform6;i842eef ubj+lrj 24xq humily from 240 rpm to 360 rpm in 6.5 s. How far will a point on the j6ihxijl;442 8q re +fueb2mu edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It acc*0x(e,0kjca w9 mgturns 1500 revolutions before it comes to a stopc *wajcek9x,mga (c00.
(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 lm 7/0il1/fxqnogrh l+j()mw reduces its speed uniformly from 95km/h to 4jm7w l1rqfholgn )xi0 ml /+(/5km/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 unis)mip sfo4(n+formly from 95km/h to 45km/h The tires have a diameter of 0.80 m.ss)fni(o4mp +
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each per 9m; jqpk,m(udal when climbing a hill. The pedm (9,ruqk;pj mals 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 thfjln8 oey ko:n93garyf f/e ln5w/ophy.+;-6e end of a door 74 cm wide. What is the magnitude of to3:6o/yrfj 8n 5h-k;yw /ag n ff.lpy+nolee9he torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown inb3p lw eyjz..4 Fig. 8–39. Assume that. z34yw jpleb. 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 15f/8sncgx -9wff-bspd) f/y kjkyj8a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this c /k-jy ys8 fd g5k fxb8spnf)j/f19w-system.

  The bolts on the cylinder head of an/yqz* kkhpu-c: ,cl8f engine require tightening to a torque of 38 c/k-y8q ufp*zhc,l:k $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere o,h.4vfz1cj0x b j ellzz 6xv:i+qos56f radius 0.648 m when the axis of rotation is th q zjs0vvz6b5hj1cl,ofe x+4zx.: l6i rough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dzs588huh f( rf3 xmfo)3+s pdqiameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ign zpu(+shrmd88 o)xff3f5s 3h qored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rowu -c)v83 uxzgd is rotated in a horizontal circle of radius 1.2-c)z8guvxw 3 u 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 bowlymt/nq twk8p bfgbxb4,6/j 5jt+: )kp on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction f/b/)byfp wpxt,8j+k btjm4g :k5 t6nqorce between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the a24djxonw(d5 iglv,sk2 eo c,w-utt); rray of point objects shown in Fig. 8–43 abo-xwoc 2ve tgw;2(t ik,5d)j,o4nuld sut
(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 atom ie2d758,gvyr1) .rwk qhwatj s 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 rate, enj(xrrw p n0id(1 (jyy mul)*,wgineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a ry,1(r0uj)dw(p jiny (mx*lr w adius 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 wit5-4ks o9 bfq(-sg fgggh a radius of 8.50 cm and a mass of 0.580 s5-gg -( 49bsfogkqgf 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 from rest to 37e tc5b gosf;2.y.xui zj*m0 f $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a smax7-vj bo +v* fu0vm+szll 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. Calculatxs0*m7 + vjf+vobzvu-e 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 10,3 rp96lz- mann.q mwxsfz2;/0w00 rpm is shut off and is eventually brought uniformly to rest by a pmrs9 fn6 nq0 2/;azz-mxw.lw 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. 8–45yio7enz qqj9+9 2s 3rabdc8 ;a z9d0kw 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, w+3oq+ew ,syr7d f0- zarr7)rrttr pw:hich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated unif7ra+tzr0 ,+r-r)w3ws7r yfptqor:deormly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–4qiw y6/o mbr3j1gny. 4m/rq64wgj y3.iny1b o6.
(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 twql1+*51av/5xlvat b2vyi j 30zwzv qb o 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 full h01k.5x: .hmvuu7p .c bykbxz turns (revolutions)czu x1.bkk 5ub .:0vpxy.mh7h and 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 momen4.f*24s j)snp b zwicwt of 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 develops a torq7ti r0oxwem57ro: ah :ue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 705b,(rn9ynw b u7b dfi;:teo,lr g c6j.3 kg and radius 9.0 cm rolls without slipping down a lanbbf 0 ,lyj9;edot,n5cgn u :w6(i7rbre at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun aswz+skd 9 mn;i4 the sum of two termswm94 i+z;ndsk ,
(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 m;nl1spdk 10c xass of 1640 kg 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 8.00 s? Assuxkd;n1 1c0p lsme it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest m; qtz*woikl( 3:/1 a9flazsoand rolls without slipping down a 30.0/;9z(3 at:m1wll zoq*s kiof a $^{\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 a6eaf; .i* wbxcnd 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 o;xw bai6 e*fc.f energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba:1xx- z,oq ebfll rotating on the end of a thin string in a circle of radius 1.10 m at an angular spexb- x,qoz :1feed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whee80f (nsbprsq :l of radius 18 cm when rotatis0 f:q8p(s nrbng at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotaluvb1 6iw g zhmmh6-w7l,0slba0v /j8ting at a rate of 1.3rev/ ,-v0 mz6bwu0h gaslblh16lw7 8j i/mvs 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 her moment of iner6j,dxwf0b 0 cz(s a;:oogs sp4tia by a factor of about 3.5 when changing from the stra bf ;po0xawj (4s,s gczsod06:ight 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 her spin rotation rate from an initial rate ofq : e5kavrl,nm x 403th323aijayq wq6 1.0 rev every 2.0 s to a final ra:,e6qmj xr i vaqn320ta45y3qkw3h late of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis througsb l440s *5 ivrhth5hoh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the s *5iv5o s4h hhltb04rwheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinninxh (, dnqtxpol: h1g58g 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 tkj,;j fcoaxa*o+ep o+tw -dhjg:r2 29 he 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 droppe8 jobg8s7:av md onto an identical d a7gbvo:8mjs8 isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns0jhta i6n:1xq)e j;q7 +jxji x at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotjpdj+h( y;ag(ating merry-go-round platform of radius 3j j(p(gd y+;ah.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 witc04 m)z2 4a;c o.j4k 1f hpmhmkrts/udh an angular velocity of 0.8 hp0fm1)4j/r ;mdcmu hcos 4 kta2 z.k4$rad/s$ 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 yer( yc6jegb8 /judb)g k, k 1rxx,38qabout half its mass in the process, and winding upy1rb 3rb dec),(8j g6yx/ekg 8qkxj u, 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 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 involv kihxc0y5yyi.l o *(c8e 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 -xk )qzn9)r(*mre izon4 wzo )$ 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, inia lfpj3l- s4 ,sx, zf/xqk .rk*t i:bssx+-dr*tially at rest, that can rotate freely without friction. The moment of inertia of the person plus the plat --qj/:sx 3. ks*l,xxsb flp ar 4izrfk*tsd+,form 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 ac6- 8 29u,h.z+h5 an9rcnuz sslsmn azt the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700s5r n,us-8.zs9h6uzhn z9c a+mcla n2 $kg \cdot m^2$ 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 gro0xrqa h08jf+n1mz xo8und with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwihq 1m88ozxnr0ja+ fx 0nds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always fsbt3hm unw)dtztlp 6:(u- 1-races 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, treat mz6 tth-nu)sp3ldu: ( wrb-1 tthe Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ra,kp174kx)-sj wnuqc rte 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 jvfa4(y 5h;q hm acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliveredv;af(h5hqjy 4 by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mau870cij6 wo2wj1x/pk gc 5eyo7w mgd6ss 0.050 kg and diameter 0.075 m, joined by a (concentric) wj 6 5wgiej77c1/6oukd0c 2ymo xwg8pthin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed offgp z.ep w2q1: 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 si0i;*npzp6 ltqn /r 5hoze of our Sun, but 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 be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no apri/pqn 0 lht6;n *o5zngular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it (0mg(s uvyx wn6b7cwzbb)-r6 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 km without needin)z6vc7 bwu0 xg(ybrw-nm6( sb g 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 illustray0eu+a6(:( lx(m aw7gxs 7pg.tdirnttes 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 roll xq5zwnnec7 /8 fw*yb3ing 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 lufgubzf0rt so)8zdvk *1t) q s0b.-0 freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceler d k-08rb) s0 gtutvf zz.*1loqf0)subation 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 sgbb5v 0 p3::0 b b6bi7knfz5zsmhltf:tanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which itkvs00lbfhzpm5 5 :3n 6bbfb:gzti :7b rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wphz vvdyp0 h2a2u,:. zith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycv hy d, .zvp022:ahuzple are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins w folh2 *z*;xquhirling the rock in a ne*fxu z*hq l2;oarly 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 solo*i:u wn :0vruid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of theinrw:0o: uuv* angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which csx(*tqm hw ;v-vj8n k83bt jz;onsists of two cylindrical plates, of kzbmhvxj;;vq*t3 8( w sjt8n- mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tr z7,w g9lhq6zr3gv hj*ack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point ofgzqzj7*6l9hvwgrh 3, the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not kzxha/0 9onxf /:o:np assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fr temu) .yjbwx(urp/+/q rz 1c6om 90km/h to 6jwxurc(+y b/6e.)t/rpq z1mu0km/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