A bicycle odometer (which measures d8(hit nn;pw0jgkjx. 8)vzs)y 1 3z*xhk,o gza istance traveled) is attached near the wheel hub and is designed for 27-inch wheelsztxjozw,*gvnhzp gyij1 h.3 8x)n0k8ask); ( . What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocit3+k w9bdz3uuuy. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, doekub+ d93u3 zuws 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 61ws jh,0:way74fw4*lnjugf hcr2njthe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larhki)ehu6ea8c z k(.r1ger 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 diffict:0 rtbg ;wpgqpl 8;v/ult to do a sit-up with your hands behind your head than when your arms are stretched out in frp;qv0rtp:ltbg; / 8 gwont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and t f5ltuj d:pq/x 62r*smzk/9grh ,h.wghree at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear u6 g5/kqgwlp j:9zf./*s 2hm,drt xhr is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fl, 5v(rshx9fhea b-9mmesh and muscle c-hv9m ,(5 asmhrexb9foncentrated 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, oi;n3 5wnv 1v3z7,gn h)dadet (rfh+knarrow beam?

If the net force on a system is zero, .gn:p r4gmk v5sq;2fc is the net torque also zero? If the net torque on a system is zevq2sm5; gkfc.g4: pnr ro, is the net force zero?

Two inclines have the same height but make differ f;)fk 7+*36cvn w wq(py*ahclzn -bvuent angles with the horizontal. The same steel ball is rolled down each incline. On which incline w uf)palb n7(-f6*kcnc3 ;q wvzhyw* +vill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneoue-s o/*hq //zwwjog1ysly start rolling (from 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 ho/-syj egwq z*/1ow/ greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the0 uqfc w2a ); x8wah(ln:sfn0tf/ip e9 same radius and the same mass. They start from rest at the top of an incline.flx9ewh0 /f:)af n2u ta(0pswqcn 8; i 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 rotational KE?

We claim that momentum and angular mom pn8 b,mvo/ujpd k+x2,entum are conserved. Yet most moving or rotating objeu,nx+ 8mvoj2pb/kp d ,cts eventually slow down and stop. Explain.

If there were a great migegfzc; s0/d,t2 l *fcmration of people toward the Earth’s equator, how would thf2 ztcdeg *lc,0sm;/fis affect the length of the day?

Can the diver of Fig. 8* b(w0d:mfm2retbpe -–29 do a somersault without having any initial rotation whenmtr-p(wme :b2f0e b* d she leaves the board?

The moment of inertia of a rotating sg xm -vw+/ dqg)v,vv:jy.td1lolid disk about an axis through its center of mass iyd:) vgxgt/vml+v1 wqjv,.-d s $\frac{1}{2}WR^2$ (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 stofa92r7ym qma)m ,hl2 qol holding a 2-kg mass in each outstret7am)r22 ,qh9l mafy mqched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havngcr,d3i 4(rve the same mass. However, one is hollow and the other is solid. Describe an experiment to deter4c (dgr,rn3vimine which is which.

The angular velocity of a wheel rotating ond5pk ob :oex+k1.r u7p a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleratione r: 7u+x db.ook1p5kp 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 stany dm5 umtq3ho3rh5wr7;ava4r y(s38q ding on the edge of a large freely rotating turntable. What happens if you walk toward thesh;r5 m7ra53r4dm 3yuyhq83qa(wo tv center?

A shortstop may leap into the *z d b/xl df hwz.hugk0)9),azair to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in thelh0hw)u)9b,xk.f zz*d/a gdz opposite direction (Fig. 8–36). Explain.

On the basis of the law of con z;lab( a3w ntai-a5 g)b*o/trservation 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 keep the helicopter s b5z-tb*a3)raiow t la( nag/;table.

  Express the following angles in radiabkx7q9w5 zj9-gjxc9q ns: (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 am0wh-, u ,)jt-lasguig qi,un, azing coincidence. Calculate, using the information inside the Fu0wag,n-uuijhq i,s) -g, ,l tront Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00 el 7 krxt*)cved z0/jyce14/y0 km from Earth. The beam diverges at an angld 4kve1/rt)0 x7/le *yc zjycee $\theta$ (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. Whens5*h-h7 kx 3 ixg9qykavic 2+(9vxkqk the motor is turned off during operation, the blades slow to rest invy2kqs9+*hg q ka i973hx( 5k-xkv cxi 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 anotheykfo:-r u .ahkr6c(vux;/ h5nr child. If the ball makes 15.0 revolutions, what is its diauvh :o5u ;xr/- fr(n6ackyh.k meter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many re/4o m yeta61kbvolutions do the 1tme4kob/6 a ywheels make?    $rev$

  (a) A grinding wheel 0.35 m4, x:igcm dvb* in diameter rotates at 2500 rpm. Calculate its angular velb4g , cd:m*xviocity 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–r e4 xtt2pz 4.)6kr-gkaqrk9q38). (a) What is the linear speed of a chi k2qpkte-)rtar 4rg 64xqk9.zld 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 around the S5ibxd dyv); k)un    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ocs)z ecjg8 x -u33g*blf 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 rota/d,2a b6d semyyepcgj1l7l.x;u ,0 nate if a particle 7.0 cm from the axis of xugbl n,; c.dyajs7/ m02epdl1y, e6arotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratesw7f -i6;xxe ba uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determfx 6e7-ibwax;ine
(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 radiusj +gevva0n (/)hsv zik)0le -z $R_1$ 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 spacecrafe574tdkblst eof ** 3ut put themselves into a slow rotation to distribute the Sun’s energy evenl*lfk o3us et7b 5*dt4ey. 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 w1n,d a7cw l*xv)k/ed in 220 s. Through how mx* w/ na)7c wdvke1d,lany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1-cfdx.z9r . m7uq+n 05tdqdxc200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirlcuusl:;jl j xr2z*8t8 ing “human centrifuge,” wh8lt;jlu s:z*c j8rx2uich 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 6.5 sn/wpnx(+n )uiicms1;91 hr sv. How far will a poinm1;p 9h)ir1s c un/(ns xv+nwit on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when h b, us;r 3wo,yeg 74u10poot)awnqa8it is running at 850rev/min It turns 1500 revolutioe,a3qnuty;0 og,o boprhws7)81 a 4wuns before it 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 revo 3p7xdg 0 +slew0t4scllutions as the car reduces its speed uniformly from 95km/h tg p 7sceslx00d3+wt4 lo 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 9hcpcvm+j. (das the car reduces its speed uniformly from 95km/h to 45km/h The tires hapmhj.cd+(cv9 ve 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 bikj-:e h ap4 xub2ipo/j.e puts all her weight on each pedal when climbing a hill. The pe./j-:4p pi2ujobx a hedals 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 theag/r iyaj p 7g,y*rs9 ,k6h-ye end of a door 74 cm wide. What is the magnitude of the tok,g,p67 */e y yhsjirag-9y arrque 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 torqs t /e. 6tqajwqeu-e7.ue about the axle of the wheel shown in Fig. 8–39. Assume that a friction je e a6wttq.-.qes/u7 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 tol--)ow 2 v9z.tppcnna the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pos-o9 w)2-pancznpvl t.ition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine rehg;w vo up6 m9()aqp7pquire tightening to a torque of 387ha)6 p;9vp(mwg qo pu $m \cdot N$ 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 0.648 0zk,str) 3tsc eaxdgj 3-:(mvm when the axis of roja:3) z0r,g-dsmxvce(t3sk t tation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rd1)cw.vnhpf/2 q 9ub rim and tire have a combined mass of 1.c q.2w/1f vd)9rnbhup25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thiffl,)ml0ld s+ n, light rod is rotated in a horizontal circle of radius 1.2 m. Calculsdlm+ )llff,0 ate
(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 vlzczf5 .*otrbg6vi,e y:kchpx; 78 /bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tot8l.r5; zb f,evgzo :x*pki67hvctyc/al.
(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 poi1,rqaes (/altc*hc /3s(p.j /nql xxsnt objects shown in Fig. 8–43 laph *jsxs/nl/q. (x, tac /1cqs(e3rabout
(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 kt:s+u 9uqubzq7n : 2stwo 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 cylindricalsr*)taxzdot6 e-wc :5 satellite spinning at the correct rate, engineers fire four tangential rockets as showrs 5dze*wtx):a c6 t-on 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 cylind * v*e+njshri1er with a radius of 8.50 cm and a mass of 0.580 ki n+ *r*sehjv1g. 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 itqn(70ay h6 p4hogx kx8 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-go-round and is ab9f:.wd zw r+loo4caz( le 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 hrzf 4oz.:w a+dl(o9w cas 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 sfax;zpa)* ;( ,cdl jqpd, i6hphut off and is eventually brought uniformly to rest by a frictional torque ofpp,i*aaq;x); f6,ph (jdldzc 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 i n/v8vk prd1wk q)-*4uoep7s oj.xf;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 k5 3fdu58j qo9 vnk-gkkb2 py*thrown solely by the action of the forearm, which rotates about the elbow joint under the action kj g-kv* 23n op9qd8y k5ukfb5of 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 rocu*xwh x 030gjd, as shown in Fig. 8–46j30u wc 0hx*xg.
(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 consisx fk*qd5pw4 c2ts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (ry)db n6d e8cdx2lv j-+evolutions) and releases it at e6 dbly2d+v-cdnx 8j)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 inertihnuqu 05;)lu,hah:ona 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 torquesh1+hlc7/+r +ay+ ea w3ty2in6o aejr 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 r ugt,3 tlhje ./mhhjka5;3ah0olls without slipping down a lane at 3t u gl k.3;mha,ehj53thjh/a0.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth witsx 3q7g7+2df qwlo*ejh respect to the Sun as the sum of two termlo3fx +*qsgqwj772 des,
(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 16k1q4.bxxd (e h40 kg and a radius of 7.50 m. How much net work is required tbd 1h4.xxkqe( o accelerate it from rest 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 je doj :4jtnx1wtym b4 /;29 m-dwtf/qbf7r*ccm and mass 1.80 kg starts from rest and rolls without slipping dot /nfjm :9 mw7f2jd1 yqdr;bt4b4j*cwoxet/-wn 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 balanc179wr s+*gohm5 pnalved vertically on its tip. It starts to fall and its lower end does not slip. What will be the sp+ 5s o7*vplmn1grawh9eed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momd u(778a( xhxz9sptm bentum of a 0.210-kg ball rotating on the end of a thin string in (d x x9bs7h7zpta8(um a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mox6 c*n6 t ob1cibwq,r;mentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when cqxco, b16n;tir6 b* wrotating 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 si.tw:v2r) cblw de, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal pos)wv2rc.tw lb :ition, 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 p+ sdrltiw/,*) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position ,i+r ds/ltpw *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 frozaa+i:g f(xdc 4 )an5pm an initial rate of 1.0 rev every 2.0 s to a final ra )fiz(:ng +p 5x4adcaate 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 verticu6, pamb70zsyoqn4 h7 al 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 diam0usph mznoa b 6qy7,47eter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skagad3 db)cm+e7x+kbg1 8v2 qv wter 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 momen joet.ccbfvchp;,j kjc1qh7 j f+i2q8, 76kt0tum 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 cylindrical disk of momentq i8j9d7(dyth of inertia I is dropped onto an identicad7q(ty hj9di8 l disk 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 2.3w9s5ss vj o*f1 jdkuy9 r*q.h-rg7j -gywuf :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 cb;i7m 9fxhm-center of a rotating merry-go-round platform 9xihmc-mbf;7 of 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 merr8l kowji) dby48q4g)j y-go-round is rotating freely with an angular velocity of 08)ijgbl) dywq8kj 4o4 .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 aboutd/i z5l uplc7 +bnhwieh6 u+ 7kfe;-m0 half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what wohe;nw pkuuemi7 lfz-+0 d7 6bi +lh5/culd 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 ofx1*k e-t;lx :k:nvbk x 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 p*, 9(c nl +71lz ezptpm81mbw toko1z$ 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, inip35ajt q4p2hw tially at rest, that can rotate freely without friction. The moment of inertia of the person pa2jpph3w q t45lus 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 sp/t m6gt cv,z(i)b1-pz tv dp*tands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on fr1g*cd,(pvm6bzzp vtip-t/ t) ictionless 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 rolls on the ground with the end of the rope lyin bq:u +8,zhlhbg on the top edge of the spool. A person grabq+ bh8bu:l,hzs 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 the same side2uunj ,b 4sr1z always faces 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, z srun2 j4bu,1treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest acw (eg t:2lr7,6reigw t 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 un9 -jh lkvh;f+hiform cylinder 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 delivered by th-9lf+h hkjhv; e motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0 -ee6npcl)st-js k/r y/(ulw /.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is releasedenwt( )sesk/-/cpl ry/6l- uj from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the anguljg(0jv1 ewo:aar 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 8.0 times as grvph3t2ph gez. 4(slj )eat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational colthgpze).(hp23vjs4 llapse 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 angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-po79chgp/ tgpm.6:w- uqhn*yz illution automobile is for it to use energy stored in anw*ggu-ih:6p h.mcqy t7/9p z 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 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 illustratevp6t3: s.jyom sr2v)g s 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/jsg9.b ;dkyhn4: sf cing 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 /f s,+bxysf1,ur6rhgpnd u2 +pivot 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 acceleration of the rod, and (b) the linear accr x2yf,1+ nu s+supb,h/g6drfeleration 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 flof(3scgauxx k* eiu8cn3x c;,/xx x:o* or, and we want to exert a horizontal force F at its axle so that it wi,ocgxsxe8xck3uu*/xc:3 x; fx ina(*ll climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is ma3.vbw,i- bo sdking a turn with a radius The forces acting on the cycwsvdo bi-b .,3list 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 5 nghvo;ob 0b( d-)etpof length 1.5 m and begins whirling 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 featbdo e0t)-h og(p ;5vnb, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sohy1k) 4fezm er16cpl.lid 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 outstretch c.ek)p46ze11ryml f hed arms, and with arms held tightly against her body.   

  You are designing a clutp 7u kbf 5z75f yxdcy4d aiut1-ms .*-vfz,9wach assembly which consists of two cylindrical plates, of mvi7k -utxc , 9fmydb.y4fs*7f1 5az wdpa-5 zuass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rm /a :jek)ejo:adius r rolls along the looped rough track 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 of the loop wj )jmoke:ea /:ithout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do np;7q94 a7)isgmgyei5nrl i 7o ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 re wt9zk.wp6 j/;u d(rqfvolutions as the car reduces its speed uniformly from 90km/h to 60km/h Th.p( fd9u; /6jqz tkwrwe 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