A bicycle odometer (which measures distance traveled) is attacxaakz0fp 6nup318 v:)-z zifihed near the wheel hub and is designed for 27-inch 8n6zaf 1u-)i z0pvxaik f3p: zwheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant (w9am7bbdy,u y/rwo /angular velocity. Does a point on the rim have radial and/or t9y7d b/uw(myrwba , o/angential 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 either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloc:sl)uf5 yh j9c,0lsruity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explaivp(a nguuf0g 4c 4,wb8n.

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 8mwe.q2 u9nvrmgn-s+ to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? usvqn+2-8 mw 9n m.regA diagram may help you to answer this.

A 21-speed bicycle has seven sproclzk3t1p8 9yh zkets at the rear wheel and three at the pedal cranks. In which gear is it harder3 z9lkhz1p8ty to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to iv 4v-i. vbuogb6 fj6p, xle/3run fast have slender lower legs with flesh vfo6vxil/j g6bb,viu.4e 3p-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) -f(,c9 tb.xk-cdvmoq carry a long, narrow beam?

If the net force on a system is zero, is the net torque also ze,c-na/ g mvow,wgxvtv *7 o.p,r 18wibro? If the net torque on a system is,,r8i. /w- vc t,vb1 v *g7noamwwxopg zero, is the net force zero?

Two inclines have the same height but make dif; r2c.bz xq(fxl4fb a9ferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explax9br;la2 f( .qxzfcb4in.

Two solid spheres simultaneously start rolling (from rest) down rma)* kcfj,h euho5-(fam/9q an incline. One sphere has twice the radius and twice the mass of the ot/ejf-qa,c9(hko h fu*ma5rm) her. 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 ma4ldhb0 tw .4cd4ara3 8e3tsqhss. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Wdhlsc t4qe4h83.4wb a a3dr0thich has the greater rotational KE?

We claim that momentum and angular momentum are *q:fhyb4 fca- conserved. Yet most moving or rotating objects eventuaq:acy-fh *b f4lly slow down and stop. Explain.

If there were a great migration : e6q:w -fihdoof people toward the Earth’s equator, how would this affect the length of d6ef:oi:-qhw the day?

Can the diver of Fig. etwssj;z l uv+i 8 1xe1djzg1q,f (6)n8–29 do a somersault without having any initial rotation when she leaves v;)qgdun t(si61 1z x+sj fw,8e 1ejzlthe board?

The moment of inertia of a rotating solid disk about an axis through eev6ju:x(k : eits center of uk: xe:ev 6je(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 yr4x7 wgfe,;vl p4q, xobd+p7rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, wil,; xvqb+74e p,4xylw fog7rdpl your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have :p p z x+sch)3:4h+e km*czphuthe same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is:zsz px m3puh ehc h+k:4)*p+c which.

The angular velocity of a wheel rotating on a horizontal axle points west. Ih,b hsv(gsxhs 3.k.u7 n what direction is the linear velocity o sghbs,3hk(vh7 .sx. uf 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 the wheel. Is the angular speed increasing or decreasing?

Suppose you are standins y7*shm,-o; lp7y fok/peez -g on the edge of a large freely rotating turntable. What happens if you walk toward they z-/k*-y,pep ho sol7s;mf7e center?

A shortstop may leap into the air to catch a ball and throw it quicjby4m*m/ri1- l.e,cge use -fkly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hiyj4 * rcg.im/e ,l-fe-s b1emups and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of con.us5usaj cs: ,3yl(ac4+ yfvqservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to 4ssl,vyy3c+5a:cfq.suj ( au keep the helicopter stable.

  Express the following angles in radians: (a) 30ggb0 gmz9ax 1) $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using+e b6/ trco4 fmco3pj0 the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Ecopfb4/3rmt +e0o 6jcarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth.n poqx h:ihgc53: fc5a8 7.fdg The beam diverges at an apqcff oa cd837 h.h:gn5gxi5:ngle $\theta$ (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:bxgo1 6p og,kvb 0n8*jw;n5pkwo )ld 6500 rpm. When the motor is turned off during operation, the v61pn );jw ,bxwo b0lo:onk8g*g5dkpblades 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 ball makes (mqto j yj;i135ucul)15.0 revolutions, what is its diamet1cmtujli) 35 ou;qy( jer?    m

  A bicycle with tires 68 cm in diamek32zjij3a )aac5q at 9ter travels 8.0 km. How many revolutions do the wheels makea5jq39j ia zk2)3 aatc?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculateg:hc(fi *,w 0n(8wwsag 3q-c5tnpltnm ) -gq e its angula) fwqn8gtt:ewiacs q lp3h 0( m n(,5-ngw-c*gr 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 comple739 jv:sty9u ;huuquonzd9 (y te revolution in 4.0 s (Fig. 8–38). (a) What iy;nd 9uv3 uosz uy 9:9htu7jq(s the linear speed of a 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 Earth (a) in its orbit arounjpvb mvxn)ch0,of(x 0v5 1p9pd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed/.*ho uiw2 t(qo;ccbso; ao8q9cmad2 of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fr jx upt yp-a7mdfz)(32om the axis of rotation is to experiencep2ztxm a-7u(dy )3p fj an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from nc6; kuv5pyx c1 ij65j130 rpm to 280 rpm in 4.0 s. Dexypkjn6; c15 vi5uj c6termine
(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 v 1 rv61y(o4j;tx efq+jbh.zo$R_1$ 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 spz: *nvvie2er57u 8;8k sy ji3lu+w gczacecraft put themselves into a slow rotation to distribute the Sun’s enezckjvu*852 + 7y sinw3: zieevg8; rlurgy 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. c 7ss a76a0uaaThrough how many revolutions did it turn in 7 cua0s7a sa6athis time?    $rev$

  An automobile engine saesi; jl d.ll)+oz6 h5lows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspe8e5 nzm9wxb2j1ek:gd ed jets in a whirling “human centrifuge,” which taj9edm2 :xz8weg1k n b5kes 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 w1 ux3wss0v b-rfp2b0 rpm to 360 rpm in 6.5 s. How far will a point on the edgsf0w s3p1u w2r0b-bvxe of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running awg xxtg r(3 1r/u.iqv wdx660rt 850rev/min It turns 1500 revolutions before it comes to a sx.6r1(xdxuqw /vr 6 rgwit30g top.
(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 reduce6h:bpef c3ht :8:nygzs its speed uniformly from 95km/h to 45km/h The tires have a diameter ozh e:3bncpyh:fg86 t:f 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 uniforb2 ukeeqt6,pmiz 68* fmly from tfee6 bumkpi,8zq2* 695km/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

  A 55-kg person riding a bike puts all hqqa(v9)yj,d. +qhxbgqb 6n4 her weight on each pedal when climbing a hill. The pedals rotate in a circly q,4.hn bd g96qxbq+j( aqvh)e 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 o1ebbg3jib*rdpv61 1 gf 55 N on the end of a door 74 cm wide. What is the magn1 gb11*dr v6jib3 ebpgitude 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. Assumeslauv(ym, 9g - that a frl9u gsm(v- a,yiction 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 rod which pm7ap dv65s o7 :jzycv+ivots as shown in Fig. 8–40. Initially the rod is zmy+cvj7p :o a675dsv 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 4nnvw1 b( v,xqan engine require tightening to a torque of 38qnb ,(w4v1 xvn $m \cdot N$ 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 apn4fl i1;pic /9qz1sv 10.8-kg sphere of radius 0.648 m when the axis of rotatl9 ipv cnq14ip;f/1zsion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thzuloa es/5- 1fibtqck1 fc14 4+ jdag(e rim and tire have a combine11+u4s5b gal4 jzci takq(ofdcf1-e/d mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on;txkgs+ iu. a: zqrx8,ypks s n::oz02 the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calc:8xizkqspyt k. ;:2gu,xsz0 :aors+ nulate
(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 aw1-o n 0t:jjcc j8yuoos0 ;(rl bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between 0-1suloyt noj jrcoc:(jw0 8;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 :wus3ulrx .k 5inertia of the array of point objects shown in Fig. 8–433xlku 5srw .u: about
(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 b*5xriz d jv:m3 il6b3pj 95yxtwo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sat abk+ys*(m:t yellite spinning at the correct rate, engineers fire four tangential roct(+ kyma:by* skets 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 c -idg04 )ysxhfylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate0g hd yfsx4)i-
(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, accelerativ .4/h6 kntz/tk 7rtfeng 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 merry-g 5m/qis7omh4 ho-round and is able to accelerate hih5q7s m mo4/it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off a-vw7;m4zrf5bpnn .mbc (r r.knd is eventually brought uniformly to rest by a frictional torque of4rzrrbncmm 7w .p5f;b (kv.n- 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg 6 .r0pm5gt 7jxx-fa *d4flnflball 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-kgz m owyt5 v3+g:1a:wiy ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The vywiy 3ta5g: mw:o1z+ 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 beko. pg-*.jrmb considered a long thin rod, as shown in Fig. 8–4m.jrbp.gk o-* 6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consistsu8p8eyu6au7)z,/g o.egfy kh of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower acce irr::b:;ak8-.yr n4 zy9,kjldmid islerates the hammer from rest within four full turns (revolutions) and releases it jazy;y,-r:.kblr9diir i8dm :s: 4 kn 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 inertia 2 ic6mmcehu3ngfa2kb jv, :0-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 tol( 7lr) ghcz b(iz4m/rrque 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 cm rolls without slipping down a lazu gq;vj)pdjgz2 +0j6 2bt) kone at 3.3qj0)g) vuj+ g 2b2oztp6;jzdk $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thlfuj/p6* )r6 n(ykd+ cmnz2eve Earth with respect to the Sun as the sum of twokz 6rc* +)un( v 26jyed/nfmpl 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 kg and a radius pr+whx rgxm- d8d7ha r8f;;k5i9yn*dof 7.50 m. How much net work is required to accelerate it from rest;98n-dhx;*+ ia rd rdp fm5rxy8wk7gh to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls h d1ey cazj.25o9w .nuwithout slipping down a hz.d.w nj y5c2e1oua9 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 viz qv -jbq:1kew3t)y -ertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the)v-3yqw:qiez1-j tbk pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bal+/x)y8rdt,3)q a swtk4 usqdt l rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1a8s3/yrdxqquw4+tdttk, ) ) s0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform hzjr u9ni8o;0 cylindrical grinding wheel of radius 18 cm when rotating at;0ozh89rj ni u 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 aarmy /m /29v 2m 6ioehxmao,2pn g8ni/t his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, v p/grahoem 6/9i,2/mma m28y2in ox nFig. 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 her8 bd nyr r7v9qlj- :lbeky.2r,pwf(w7 moment of inertia by a factor of about 3.5 when changing from the straight posb: -w yq9ew,(7d7f.krp2n yr8rl jblvition 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*reo9+ k0 8mddma y p ncu9wi977)plpe of 1.0 revn7de r99omky)u7 c0p+lamwid 8 9 *ppe every 2.0 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 vert6ct8/i , 77/fl6vvx oz rx gi-dgo8ucx74ejshical 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 0.4/-lf6 t,i7 s84c vvxxgid gzoj 67oc7rehu 8/x0 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure zd w/ 5yzqu2zhp k;hmu/gq::. skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum fwzyfg1x 15 b8:,wk5y:ga 9.ugrxhpm 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 cylindrix3 40g,vdf dvocal disk of moment of inertia I is dropped onto an identical did4d,fovx3 g0v sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 7z2t p tzy0 yrntp0k 2u+(c7oc z25ahu2.4 $rev/s$ 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-round plat 1nxm z0 mg/rx :gyjrc.:,9ldvform omx xjyl:n9 ,rgz.v01c m/g:rdf radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular veloci20r jp1ap 4rigt. n 87q;jl;0jqg i3ezv*sf rhty of 0.er1i3*gl;i0 sq0jan; 8pv fzh72pjrj r .t4q g8 $rad/s$ 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 6zke6 6ret (;f3eb01yerjenywhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost m ;6bky (0fe t6nryr61je z3eeeass 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 involve winds in excef5zje rz)8dpq s 1d8s7gka;n 5ss 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 x .46un0uoxpk dt:0c -td 7lct$ 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 c 9 ;+9vr+lvfbs0ht gv z:nrq+ian rotate freely without friction. The moment of inertia of the person plus th 9rg+s 0i blvhzv+f9;n+rq: vte platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-i g6d3,an o9uhround turntable that is mounted on frictionless uh9 d a3iog,n6bearings 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 rolls t.+ d: rldw1hxtbbx: 5c68/nf(s yldj on the ground with the end of the rope lying on the top edge of the spool. At(xyd 6ltn.:b/dl81fd rx+j:bchw 5s 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 unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that -gl s8obwn5l lf1* rk(the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abou 8o bf*gkr-l n5l1wsl(t its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates 7.yj*f5+ xiym gj cd8afrom 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 cylinder5hwtkx .zefm3 ws 8-x( of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deltmze.(k8hx-f3 5x w wsivered by the motor.    $m \cdot N$

  (a) A yo-yo is made offub4njv 0e az9,gk -g) two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by agagn9)0v,kf4b jz ue - (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long 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 cg,t2q 5by fn*4ga*hmangular 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 mass 8po1z hxq bh-2t9py9c.h7 0tvs.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 los1v7qp9p.hhts0y zh- 9o t xbc2t mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autom+wn(q k e+ l a(i3/ar9brvf4ahobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywhef+9( ne/k ra a+(awl34iqbvhrel 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 illustra .o p,o:y2oiewtes 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 rvt ope 8e,aj)b)jo, l6olling 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 frictionct43-6ft9 u )ydtsivn ) 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 that the fod c 4fs96-)t3uvny titrce 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 th8iamu7.ypc(z h .2qpje floor, and we want to exert a horizontal force F at its axle so that it will climb a step against 7ai pu2m .j.zpqyh(c8which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s jk) w2ywuxjn-9nlr ( 16hbi:zon a flat road is making a turn with a radius The forces acting on the c(rnwkh )z6 ynjjub-9:wlxi 2 1yclist 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 0.50-kg rock into a sling of length 1.5 m and begins whi5yjjfkcq 55tzp 29ow1rling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come fr91j 5 5zp5cf2j kotwqyom?    $m \cdot N$

  Model a figure skater’s body as a yjk gr3e7;d2rvj 1r7e solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with ojv ry;72 r7ejgder1k3utstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of twv a1 j;xp **kys+c,qcro cylindrical plates, of mass s+ka*rjxccqyp,1; * v$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rads9 h b;5r7oimiius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of thei;br 7 59hsoim vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assunt.5u9f4k z 2 rnnrb/m:k myv7me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revo 6il9ii 0f:vb 2wtu9tflutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wh i wu9btvlfi f26it90:at 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