A bicycle odometer (whiep ;d sd,lgk;:ch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-in dd;ekp,;slg :ch wheels?

Suppose a disk rotates at co v,tz2j;n(nll34bdl5h6xnk knstant 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 accelell,n2n bk d 564txkn3;jzh(vl ration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singld sm:6cq/ (qc5k .x:ya c+gs*4c wsfohe value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lar5 jjhqv mnlxe*44ko)5dr40 ku ger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your hpv (u.vff zqzpq*sy;8s n;k-:ead than when your arms are stretched out in front of you? A diagram may help you to answer thisq.z:-n ;8 v pfu*;(szpksfqyv.

A 21-speed bicycle has seven sprockets c6o g2frky;gy41ss 2yat the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or2c2y; fg so gyr1ys46k 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 le/ na lzbu7) feza/zp;::i:pjegs with flesh and muscle concentrated highefu;az/:zn:7 j aie/bz p) pl:, 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, narrow be) q7c2v2k3zskr5mr wmam?

If the net force on a system is zero, is the net torque also zero? Ifho)*, so gu nbi-8q*og the net torque on a system is zero, ,-ohn* *o)bsquoig8 gis the net force zero?

Two inclines have the same height but make different angles with the h8dqs. a p rvla0g6ao:+6j +m2hyq ;s6vhep)fmorizontal. The same steel ball is rolled down each incline. On which incline will the sp0vsahpd ) ja;g.p6sqrme+fvhoa 6 :+68qyml 2eed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from,v(nf- 3 jl obnhpa,m9 rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the p,(n nj-vh omf9b,3 lagreater 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. They star;z vlq ,b* qjp /xlr;+z/lr-ktt from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater pxr *jlt-k r;+ v/q,/b;l qlzzrotational KE?

We claim that momentum and an;iwgrx,4 1qw fgular momentum are conserved. Yet most moving or rotating objects eventually slow down and stowq ;,gx4rfwi 1p. Explain.

If there were a great migration of people toward the Earth’s equah7 ho(-mcso:k, lwx2xtor, how would thik ,o : lsxh27-wh(mcoxs affect the length of the day?

Can the diver of Fig. 8–29 do a somersaulte t,q mf2y t5b6(i.irpz8: bgi-yt3fa without having any initial rotation wyt 3 -bit68 5:(,yzfqifbitepr2m.ag hen she leaves the board?

The moment of inertia of a rotating solid disk aboutnmx- ig 2a5t5n an axis through its center of mass isinn5g2 x5-mta $\frac{1}{2}WR^2$ (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 st q tzke5cg3eos,8 yzt-o0-b/i ool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay /qse k5zz-0, e-cgo y3titob8the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow0v6n26ez 5vj2 dalch n and the other is solid. Describe an experiment to det62 ad0h n2elvvn6jzc5 ermine which is which.

The angular velocity of a whm;*22tto2zie 0pj kvyjdb oju +6o);v s/ar,oeel 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 tangential linear acceleration of this point at the top of tkvaeo ;btty mj) s ;ju*oo dr2+26/ voi,z2j0phe wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntab,mj,7m04+knll x ncwht v8 )lile. What happ) lwljk8xmn, t ln047mivc ,+hens if you walk toward the center?

A shortstop may leap into the air to catch a ball and v.n 80k 8 .lhpps0ohputhrow 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 ps 8pk h8hlp00u.v.onthe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helic2 ma18efss7ovy,3 w mt(1rb njopter must have more than one rotor (or propeller). Discuss one or more ways the second propebsw2 t m( 3erj7mo1,ysvn1fa8 ller can operate to keep the helicopter stable.

  Express the following angles in radb8e*nwz;lsz(h- 3o7 gdunbh t;3js ;dians: (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. 6l+xti/w j.v putc,.m Calculate, using the information inside the Front Cover, the angular diameters (in / mw.cjtpl +v6. it,uxradians) 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. y.z ; i 139rygtnpv(sha j)vk8The beam diverges at an ang k)j1n yh93( ;ygrs a8.ivpvtzle $\theta$ (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 rpdy:z ;c 59a yk qmfzlcf;oirq3/f7/p0 m. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is tyfrdoc3cp f: z;0q afk 5/my;7/zlqi9 he 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 th.f kqvnls3*/be ball makes 15.0 rev3qf/v kl .*bsnolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.yxh(l. oqy+6t;w,b9fek b6 pk0 km. How many revolutions do the y ;q6 6(bpyfx, lkkh9+two.bewheels make?    $rev$

  (a) A grinding wheel 0.35 wk3(fhdh d*rkgz/w ca u9s96l 3rl/4dm in diameter rotates at 2500 rpm. Calculate its angul3uk9ddclfs6z/9(w l/a rrh hgdw*3 4k ar velocity 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. 8–38). (a6hmiy;+ oalp*) What is the linear speed of a ao mh+i6;lpy*child 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,95uc na/ww vv Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o0*r+r jw(imksf 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 o3k+- aub;t qhef rotation is to experience an acck3 e;-hqb+ut aeleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 xah3ka.9 gv; trpm to 280 rpm in 4.0 s. Determin hv;9tx.a3gkae
(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 z*shri3yh8 +cjoq6l , $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put h9:ov 1)htcaythemselves into a slow rotation to ta)ohhv1 :c 9ydistribute 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 i.zs4riz1g 7z en 220 s. Through how many revolutions did it turer 14gzz.s7zin in this time?    $rev$

  An automobile engine slows down fro -kew27:bns.b;d rwhkws;2 rpm 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 z3 tipch +ktv; b4bj8/stresses of flying highspeed jets in a whirling “human centrifuge,” whb3it4kh8;vpc b+ tz/jich 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 from 240 rpm to 360 rpm in 6.rhm x vu2(4+yyx9a o3w5 s. How far will ay 9 w+xu2v rom(3yxha4 point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is run). gci:gzkg0b ning at 850rev/min It turns 1500 revolutions before g:0c ik.z)gbgit comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the at xet (a8x3 w4hfc*r4mw+m 0 h.t: zszj,upu/car reduces its speed uniformly from 95k rh4 fx4et,su*w:3/hum8c wx(.zp+z mtjaat 0m/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 spx zlwff yv,p 8z 8*v az06/dbja1ead51eed uniformly from 95km/h to 45km/h The tires ha vwf/y zfaa610*8 v1 azlzjxd5pd, b8eve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbingjey:6vnti4 - o a hill. The pedals rotate n:vi-6 4oe yjtin 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 pdys0m0xqgf h,,o ; 4/qwe0p ha door 74 cm wide. What is the magnitude/ps w0o0qe,fq 0pmdhxg h,;4y of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. sc,dr,gr,cv j uuq11 (Assume that a friction torque of 0.4d(gs 1qru,, r,j cv1cu $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to tcqvcjr582jj: 1zp z2xhe ends of a 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 direj:8czzc125 pj vxj q2rction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to ayh jbe(51f3h t torque of 38bf531(yhj teh $m \cdot N$ 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 of radius 0.6i6uaclmy6y w+ vc4.2e48 m when the axis of rotation is through its c uc ylc64+2v.aiwemy6 enter.    $kg \cdot m^2$

  Calculate the moment of + *-hwoanz68q70 p etjausv/kinertia of a bicycle wheel 66.7 cm in diameter. The rim and tire t8keqa wun h/pv 7o6j0sza-* +have 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 onm4 hauzd)nb 47u,x; tf the end of a thin, light rod is rotated in a horizontal circle of rfz )7ath;db,mu x4 n4uadius 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 cocw iy/r*91qct o 9gvokff6.b*nstant angular speed (Fig. 8–42). The friction force between her q 1 *yvcocwgt6.fk9 r9 io/b*fhands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig.6a/y. myynl; z 8–43 abo/z .myyy6a;nl ut
(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 zhp6* :br*1wg.h ebhetotal 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 satelliqd/;*0xv(u8a 0i,hg dvpg bi 7xakdw, (fx 5vote spinning at the correct rate, engineersgqd ;(a bu ,,ihx8dxvvvxga0 k0w5*p 7oid/f ( fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a magdw)s*9yb;pj qkk3 *c rf3i7sss of 0.580 g*3d kq; bjw3yrskcs9* )fip7kg. 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 swingsr9c4m*sgn e8.sq(q n e a bat, accelerating 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 tangentially on a small hand-driven mf *n wxbuo5, e5-1 /tc:sryvd8*ezhdaerry-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 hdexw n sv5yt*r/z ou1-dc8b5:e,a *fa mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventupti 3 jr.zp497xgop e9ally brought uniformly to rest by a 7p4 poite9r. 3g jz9pxfrictional 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 ach n0 z3pc7eu3qcelerates 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 e5dymcma rytzq),g9x+ 2u.a2 n*bb c; ball is thrown solely by the action of the forearm, which rotates about the elbow joinaxgbm c;)q5 +ab.,2du 9rn*ec2z yymtt 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 can be considered a long thin rod, as shown inq /+ tdzsut r1p*ka5x1 Fig. 8–46xtar5 s1zk pud*t /+1q.
(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 two9m ze5)t jamh;fh64.ongk (gf 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 g,mj3 klye;+bwithin four full turns (revolutions) and releases it m g byj+;l3k,eat 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 inerj spf2eva8*lr +gx 9p:tia 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 tdg uiw 4)+/ 7evnasf )4gnut+corque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass i8rfrb(x 8r cp+kw0 /m7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3cw f8/r 8rik0pr+x(m b.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of *pq ,bj ygswd-1d7ms))* m(gn mjtd6 ctwo ter(d *s, gtdn)-wq6pb cmj7j1m )gs *ydmms,
(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 fzfzx7uoj 2w(c 3b,*nof 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 (xz3juzco, n wff*b27 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kwje(3tv54h3. 9pmh vbd3uk wi g starts from rest and rolls without sl tm4 j33vuwvh3kehdw5. (ip9bipping 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 tip07i60o:o-qn nwi+ u z/xkqc zn. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservationqk z+0in7- i:o0 / zxo6unncwq of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the ci3; mt(3js jiend of a thin string in a circle of radius 1.10 m atiits3 j3m(cj; an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentuy650ly il sz1rm of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating l 1r5y0lzsyi 6at 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, oncnl pn-, abvh()5cuw ds/4 l(m a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotaspncl 4a lb h,c wvum(/-(dn5)tion 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– vy4lxjv*nucki/l po2)9 vz5729) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what 2 luio7 59xlcj4 / vyzv*)pknvis her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin r3j5-hphvi pvt-u:fwv+c m3cd 2* 0 vxmotation rate from an initial rate of 1.0 rev every 2.0 -fp2xv0v35pmdmch 3 vwu*vcjh ti : +-s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throug ;,adgsw d4) 6x zlj2-q6fl-vkgreqo9h 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 frexqz-ja k)f6loe-2s;w6,rq ggld 9v 4 dquency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning*k.i)2/artrsq zui9 )k2rmu 3uzr ls) 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 Eag4n 1 4etl)xj, mah:plrth
(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 drop1c unega0xmuibv 6z/xh2*y-b.c) os 4 ped onto an identical disk rotating au.2za4ox u bi/c6bcmg s)-1x he*vny0t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns .s-hes 5zhnlzyg b*m u;o,rxbz0- +.p 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 c-lq2o sd5 hk/9/uydpat the center of a rotating merry-go-round platform of radius 3.0 m and moment of-yk/q d9us2olh / dcp5 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 tg*6:n)drx 7vpu, wft angular velocity of 0xg)r6pn ud,v ttf:w7 *.8 $rad/s$ 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 about hamqfa-vb /uf om)bs+dr ,1n:j26vqp2ny.f0u ylf 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 mo )2b:rymo2-0pfsv/ 1n 6 yja.mnuubdfq fq+v,mentum, 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 r,3u 8t-6) .gwpbsxecxzl9sj$km/h$ 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 ;8: +vgh+gw( :df p6c+kpniv oaa lc.m$ 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 restsu 4(2d9da.w v tr:x/irj7jc :p 6hvp-iojdr5, that can rotate freely without friction. The jj(dj 2xs:v9 pr/vua r otpw 65id:74.h-dcri 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 s,5qv 7skfy bv-v;p w,dtands at the edge of a 6.5-m diameter merry-go-round turntabld-v,ys vf,; vkbwp57q e that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope tdrbh/t l 9m k9fu5:r4*( rnknrolls on the ground 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 withouh*4l kk n: 9/(fr5b9trndumtr t 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 th*+notp kkc 0(up+poly3/dj 5y e same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axi 03dn jyupo/kk+co+ l5tp*py(s) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates hoi lzw6dy4sbs+ 8)p+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 0.20 m acqro* vekexn,py1:6 9iguires a rotational rate of from rest over a 6.0-ex1vyk*gp6rnoie 9:, 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,+ pl ufxp8qn;. each of 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 conse 8xqu ;pfp+.nlrvation 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 ofw)ejvwo5+0( wlgoog 7 the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times+0xxd mbjv89be skm+ 3 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 ijs+ve x 3d+kbm89x0bmts rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for np7 pe k;dvyn.8/hi-ca low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, up;- pdh/n 8yn.7vciekses a uniform cylindrical 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 illustrah*a sg;5t oq0htes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface atn 8i jw(eh6j i9-oo+az 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( 7 /j/w69a nm+xrvqbh h;xjbn ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) thrbn6jbh x+ (aqhn7wm;9/j/xv e linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we1jsn6r+ivu(rvv5 y; i.danb + want to exert a horizontal force F at its axle so that it will climb a step against which it resri5v.(v b; n in16su+jav+r dyts (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed 2*sf9fmn,yk b)u6 avuv=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclistvmk f2 syf)n*6u ua9,b 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 0.5o :f0ei,s 5ud7c/gfs f0-kg rock into a sling of length 1.5 m and begins whirling the :fcsu ,o7sfd/g0 fi5 erock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bod).vakj1 cgyvs/8 16b t9 x5wpzlib)x uy as a solid cylinder and her arms as thin rods, makin/b1j.vu 8gx 5p)bx1)atv9wzyi6lk scg reasonable estimates 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 assembl8 ywj8uqdg9sq* kun2j2 k1fg2y which consists of two cylindrical plates, ofj 282wqs*fk9y1gg juu qk82nd 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 rf qi:u:n )t96g/tfelji6 ih+b ough track of Fig. 8–58. What is the minimum value of the vertical height h that the mi9qfl:6j6b/ifeg h t :ti )+unarble 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 n 2q8wox/ u sol-h/e0isot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionptx2-3 o, wsdkdq8:ens as the car reduces its speed uniformly from 90km8 3exdp-tqn,sw2:ok d/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s