A bicycle odometer (which measures distance traveled) is 92eedlc p6v.rcwp ;7cattached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wit9c2c v lcp r7wdp;e6.eh 24-inch wheels?

Suppose a disk rotates at constant angular ve (axdkr/sv-bf :x(u3 k; l1snxlocity. Does a point on the rim have radial andkxxl (/ v uax -(ks3bnsd:;r1f/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a gwy6 rtwt /g8,single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Ey2ky3zl fv ,8ad-;r/pxs5(i6 h pqebh xplain.

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 tgl1gxh: ; j 8mzdmd(/rk65lqio do a sit-up with your hands behind your head than when your arms are strzhdi/jmgg6lxd;8m1l :r q 5k( etched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprojfb5p5 h;bnxx4 yrn02ckets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder tonhy045nb2pr ;j xfb 5x pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to runkgqhzp ,id 0tf5 eot,*/ ic65i9cj(vx fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribu5ihp0z,9dj vc ktcq go5(i i*xt6f,/e tion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n;8c4ety0 uvcuarrow beam?

If the net force on a system is zero, is the net torque also zero? wte4 grqnlemj),cd99.m:p (oIf the net torque on a system is zero, is the net w9)t :,r (poqndc94lj.g me meforce zero?

Two inclines have the same height but m3;ysfjy 8, j;qz/ *+uq3m7h :aq qpqe8tpw pgnake different angles with the horizontal. The same steel ball is rolled down each inca qwmfep nyj;s+;q tzq :8p p3u7yg,*q 3/qhj8line. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simult-9mjs9iny ph6 aneously 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 incli jy 9sm-6pih9nne first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. The lrwe1x83*zchy start from rest at the top of an incline. Whic x whelrc38z*1h 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 momentum are coled(l 90 y,nsigky2 1wnserved. Yet most moving or rotating objects eventually slow dowy g9kwyll,s0 d 1n(2ein and stop. Explain.

If there were a great6 d/.xw:dadqzvm s 94j migration of people toward the Earth’s equator, how would this/69sd dz aqx:4vwjmd . affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without haviq*yw8 *5kw1jf+sf d obng any initial rotation when she leavks1wj* f5b*oq+8f wd yes the board?

The moment of inertia of a ro-fz8jl.s w mguehq4, s(z;w. atating solid disk about an axis through its center of mas- gjza8wz hwuq.;fm.s el4,(ss 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 rotati yv9 ox(9h 3yws cb/9s;zt-yaong stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angula bs(v y9zywhxo3 /-osc;a99tyr velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and . 6/.wa;yq/ ymm d6k .ky o+xcufkqbt3the other is solid. Describe an y c+;w d/x3qbuf kkma k/6oym6qt.y ..experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. 2* r*j weg)ilm7h3i4cspplqd2)(ti dIn what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangentiaqe* l7p3i iw2*td ps)gc hj(4l)dim2rl linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. W+lfpd9wam u/p6+r xf,olpo 5-hat happens if you walk topollmpfpw a6 +rx/- 5 ,d9u+ofward the center?

A shortstop may leap into the ai4sq+- v*l 8igxj0swgbr to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly yoqj- i8g0+l sb* vgxs4wu will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of chdt x++ds0s5n onservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Dishn++dt5ds 0s xcuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles inhkyiu( kkzn44(. +,) k lp21yk izxpxz radians: (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.4uo-u1 6ad gnb Calculate, using the informu4g6d na1-o buation inside the Front 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, 380,000 km from Earth. The beam divergs: sk/o9i,r rta zf:t8es at an::krs,a9o fzs/rt it 8 angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender ra iiis2ft g)/vig1/atci 2:5ukfi ;+qotate at a rate of 6500 rpm. When the motor is turned ofi / vki2gf 1ai/)q+ua2ifist5: tcgi;f during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball maktiwk*mm)oy:uy 7vw ntqh./quuswy)+ 1k+-n 2 es 15.0 revolutions, what is its diameter hyn7tky s)uu/tui+vqnk:12q)m.w + w-moy w*?    m

  A bicycle with tires 68 )/e;flhm46 cqwly3bqvs w -z2cm in diameter travels 8.0 km. How many revolutions do twvyq/ 2)-mq ;b4flhec36 wzl she wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter r0 uj(+h/tqhypw l40wnotates at 2500 rpm. Calculate its angulauqny(40jh +p w0h t/lwr 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). h)tfrgc )49l r(a) What is the linear speed of a child seated 1.2 m from tfh)9cr) gl4rthe 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 t3z8wyd:c2ye bhe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed w0,tvs;g0mh6 5fuxzd 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 par/zn.s hpyaimoas/y,0* 8xu31w cn)xlticle 7.0 cm from the axis of rxn a, hoslyx08*c/)s /wp a3u.yzni1motation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 28wf3 j3w w1sra(2za-aq0 rpm in 4.02rwa33f1(wsj - qaaz w s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius8rn: jcxn z;k5-qhrazfq569 b $R_1$ 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, asn645k*ao 79a cz xfesdtronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotat*s 5xok dez76a 9cf4anion 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 unijq ne, *1 7:ikbuhnu7ml( 8vnpformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?1(kuhnbem 8 , lupn:q*j7ivn7    $rev$

  An automobile engine slows down from 45s)i+x-)31 *vblmgz e q cz9icj00 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 s8uican(c zd w.f;)fp4 tresses of flying highspeed jets in a whirling “human centriffdwuc().iaz8;c f p4 nuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly ny m,vhrnd8): from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this timr:n hv,dn8 y)me?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revone/6vbrok k *;lut* n/;6vebokrk ions 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 revolutions as the car reduces its speed u2v gj3b v371e.tdhf(3y jvnvzniformly from 95km/h to 45km/h Th32bg3hzey .dfn v37vj (vvj1t e 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 q0s ury )hhcg9ypm mj99,aq/5revolutions as the car reduces its speed uniformly from 95km/h to 45k)s0h5urp/9qcgy 9,hm a yj m9qm/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 her weight on each p gsh+dh tic58()4os cmedal when climbing a hill. The pedals rotate in a +ms8)s(g5 chih4tcod 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 a door 74 cm wide. Wrwm 7x.hos8j m; c2d4nhat is the magnitude of the torque if the force is 2 xon74wh8mm sjrdc;. 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 th u/go7px )157 mtkltwwzor l+8e wheel shown in Fig. 8–39. Assume that a friction torque of 0.4/)o7ww tkgm8l7o5lr+ 1tuzx p $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends ofg9t+*dd yl f)p 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 direction )*tg9 ly+ fpddof the net torque on this system.

  The bolts on the cylinjp3(p/cviubr.4) c dj1c- w ecder head of an engine require tightening to a torque of 384i13pcpb-)d(j wce/. cj ucrv $m \cdot N$ 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 m when th l05q;xww 5irr+g5u bve axis of rotation is througuwv 5 ilbrw0 q+5;5grxh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel bcg/esgu x (2d8dvyv) )b;.0wdzwt h7m;zpm)66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?)dbp))mdb 0mde(g .gww / 2;8uvhc; xz7zy)tv s.    $kg \cdot m^2$

  A small 650-gram ball on r zxfx vb4t+(0the end of a thin, light rod is rotated in a horizontal circle o(f+0brv x 4ztxf radius 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 rota5x .e+mw34yp/rio) 63;uxog uerm njrting at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N toype6+mgew ox5u/3.i;rn or ju4x )3mr tal.
(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 inertiaxg3yk ;ie f:)q of the array of point objects shown in Fig. 8–43 abou3xkeig f:;qy)t
(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 oxyge nf((jd 3w,g-6cscl j u,5dgmwn 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 satellite:yr yvjw6)r)v0b vyh1 spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg an0b) y hrvyj61vvyrw: )d 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 uniforviu6g w4 ;8ul-mosj )pm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatei;u)g -6o8m wup4 sljv
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest tg:ywbg l73uqg.h 7e e -rq3or47c3rmho 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 merryup x/fcz1d y*8-go-round and is able yuxz8 *f1d/ cpto accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, 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 ojyq4g+;owjod/o*sf0w k7 qh ,ff and is eventually brought uniformly to rest by a frictional torque *7qwj 4jo/0 qgk hfw dsoy;,+oof 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 ball at 7 :o0 ddi .+)qc/3c)bzv qhlec k$m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the ks te*y pz-zgrmhbl0 53*n483zxcjy0forearm, which rotates about zzj-xzpyyeb40l s 0m3nrc* 83*kht5g the elbow joint 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 sh nc ,gv: 7uqyr9s,ee(xown in Fig. 8–46.u( ,rygx e:nqve,s7c9
(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 br9. c4xvidxc*0etym: x1j7v 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 resbol9 6/k;wk ydt within four full turns (revolutionbd k69lw;yk/os) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has* wccxe9u0i5o. cef(osmn -) c a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops ao- f,qpdj;xv - torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm roll :b fr5bra0l83emqnc) s without slipping down a lane at 3.3c5rma:)f3be lr08 bqn $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy (h.v7v 4u.m2 st gkxshof the Earth with respect to the Sun as the sum of twoh gt2mv4vuhx.(k 7ss. 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 maj)zt4cjs,8 zz x/ye 1ess of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 r /ee8jjzxyc,t)z4z1 sevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k*i.vqb) 7jfxj6 krpl+ g starts from rest and rolls without slipping. xijf+bj pvlr)6 k7*q 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 balarbwh0ran-;-dc z ;r+xnced vertically on its tip. It starts to fall and its lower end does not slip. What will be thezxh; rdw-cra 0+b r-;n speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en*g8q1f lr wn8 8jtbogp1 ev(b.d of a thin string in a circle of radius 1.10pgfwne 18b*t( qvljr81bog. 8 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical8gxp j7/bit9l grinding wheel of radius 19/ib8xj7 l ptg8 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on axomzhd a6or7gm2) x fs252c5y 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 rotation decremfhycs2a 76)dr5xm2gx z o2 o5ases 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 reducrs uz0j0*w t/ovyl3d 7aa. y7pe her moment of inertia by a factor of about 3.5 when changing from the sa30 7wypzl o/0uv*. ay dstj7rtraight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation r6z fu 12y-nqbb56m)tw: lntfn ate from an initial rate of 1.0 rev :6fn6n b1 bztw2n-utfyl m) q5every 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 vertical axis through its center at a 3oyi yjd( 6l,nu1zqb3frequency 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 fq3 1y,63zuib oyjld( nlat 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 skater spinning at 3.5 0h:lt pv gu99xuvm* 3u$rev/s$ 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 t p;2yzt.h:a iEarth
(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 dropped onto anm.b9amr ez+q hij iuyn 41 6ju7kw-61m identical disk rotati1m6-zhaq7m b +k w j4.6nju9yi1 irmueng 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./z*w: ct.oyj jtu6l e(pzb0 o)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 rotatinghwpkq pm mo9+)- 2k z9i6frrx( merry-go-round platform of radiuswko6k ix9+hq-)z rpfm( 9mp2r 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:coaxf ay e0d j92r7o- angular velocity of 0af07 eojdyxa29 :cr- o.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 half itsw6 - oc tbcobho.,s8-x 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 would the Sun’s new rotation rate be?(round to to-bs, x.hocot8bcw 6- he 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 win5mp )wqfi5;w;b / x st8wuxons sn8a2(ds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass zhlwznr 27rl. etljkm:596 .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 p:l*br) buz wp0 z)v,pir3z 9crlatform, initially at rest, that can rotate freely without friction. The moment of inertia of the person pipwz, c)u*prz:zbvbr03) 9lr lus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m dim*,w:n vey .bg12gi:kmye im+u4ykl 8ameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertiagib2y:+.ymum,v 81mkl e iye4 :*wngk 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 lying ogi43hhb5) 9d/v mweb en the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–5b4bg d9vweeh3 )i /5hm0. 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 we ldg yd.8*knn-65 xythat the same side 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, treat the lny.egdd8 x6-yk*n5 wMoon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a kkco j,r)k:gbjtosw :e xr/( 490h9ra 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 unifgj(/rqa ozpg+6a/8 dt-:ui/d.rjum b orm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0g ibj zuqt:dru8(+//gmdja/ 6 -rpo.a-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, each of mass 0aux xz*gb 77j4.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diamete*bzx74a7x uj gr 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of ts/kbvgnn, ubi.sp55h:1f 4 ae5a+ np jhe 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 greatorbh2-u a2xr bblkoiw y;9f67 48*typ9bo ,y e, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo grav2 t o ,y-ikh8ul76fbra94ob;yyr o bp9*2bxewitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use enerk -9g,ktpo f,k6d.qjzgy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 .o kf zk-pqg,tk9,d6jkg, 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 illustrates an5yv7e q mqn64t $H_2O$ 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 (h32n2uv, 2que1dk ycpauded7 8oop) is rolling 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.,n7y1m q0;inmw4 ibl8j+0p y xk-u+lbf we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At t;qx-0lu y0 mbfn8k +inw 4pyl+i1bjm7he 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 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, ande 8r8gkmnf a;:fmx6l0 we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). :a 6fmr8egxnlf 0 ;k8mThe step has height h, where h

  A bicyclist traveling with speed v=4.2m/s ou8 ,o l/4p;-dkwtmwl2jh ,0rh+zrvnqn a flat road is making a turn with a radius The forces act ,nhmt;lzhw8r+4o,dl 0/ w -jkqpr2vuing on the cyclist 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 begi4or4lgq ,i6roxn9 9 yxi7e:nhns 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.q6xon 4gox,el:ih7iy 9 49nrr What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a s (jjyie6m 16nsolid 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 outstretched arms, and w16(6iyjs nj meith arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pg dktpe;(;e .dab. l,*7-wgju fj xf-elates, of masslepf-e,xf .wu ;jgjeb(dg.t a*d;7-k $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiuirna;(75zuu z/qr7e+d std(j s r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertire(i/zj+nr z7u5q ad(7;t sudcal 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 assexqp m5f3fg;m3gko m u/2m 0:(mfh opm+x**i pume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifo qe2gam;de7ft6cvbt05 5xui )rmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. ef 0qti) gmu7;xvat2be5d5 c6(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