A bicycle odometer (which measures distance traveled) is attached near the wgop t is1+;kn0heel hub and is designed for 27-inch wheels. What happei 0s;+gn1opkt ns if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloci ev-)wsoup 5)bty. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accelerationue bwp-)v 5)so change?

Could a nonrigid body be described by a single value of tsd4 y4t8 fol/bx(w/ jthe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largeu2muf ghe2g*) ,dcy0fr force? Explain.

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

Why is it more difficult to do a sit-up with your hands beh7cd db)2 xr lu)qzoy6;ind your head than when your arms are stxd27)bo6 ql)dcr uyz; retched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocka m1 (vizmof6;8z,6tyxmck 2cets at the rear wheel and three at the pedal cranks. In which gear is it hxz1a(,m; m8o kyt mf 6vc26izcarder 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 run fast have slender y nzmrdw9wh*rmy9,e,tal4; ,lower legs with flesh and y ,wm,9hlr 4ney,daz r;9wm *tmuscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longd 6ps1qp:p.jc , narrow beam?

If the net force on a system is zero, is the net torj cni72v-j,dtiou -4 yque also zero? If the net torqu y jcv4,d7-oun2it j-ie on a system is zero, is the net force zero?

Two inclines have the same height but makk c kpn ty3jm82*pujds.f* ;)je different angles with the horizontal. The same steel ball is rolled down each incline. On which )m*jk k.fp2ncyj*;j3t du sp8incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulta.6s tykj;.lngneously 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 thntj;gs lk6y ..e 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 dt .0 a(uggn)pmass. 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? Which has the greater rotational 0gu).apdn(gt KE?

We claim that momentum and angular momentum are conjt+0 n s 5lij. 7al6a/js5qndoserved. Yet most moving or rotating objects eventually sljalqj5a+/6s d.0toni l7j5 nsow down and stop. Explain.

If there were a great migrationtk*vsk,;q, j j whof31 of people toward the Earth’s equator, how would thf,1vstj* ,ho ;qjk3wkis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having anyny7vups/-94rv v; lf kno *-yr initial rotation whe 9lofryyn-v/k;74r *v pnsu-v n she leaves the board?

The moment of inertia of a rotating so n8ao*w(pm vt-lid disk about an axis through its center of mas p-mon* t(v8was is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in ea;,v , w,bqnndn2p1 d*dq 5oyfech outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrd,v2wdny, e1p*nq qn,odf b ;5ease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howe4pd v(ip v8r v)9b1kkjver, one is hollow and the other is solid. Describe an experiment to determine which is whicpk84r v b1 )jvvdki9(ph.

The angular velocity of a wheel rotating on a horizontal axle pxpvd kr* 3c26woints west. In what direction is the lin 6pk2*dwvrc3 xear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wh1mbevf/5n: d bat happens :e5mb f1 bnvd/if you walk toward the center?

A shortstop may leap into the air to h+xs4e0o(9e3rg df/gcyv9 qacatch 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 the opposite direcgcr(9ev3o yqh s+xa409ed/gftion (Fig. 8–36). Explain.

On the basis of the law of conser(sbz .7ngthy3vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one ozs.nb g3h7t (yr more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3 og ;r24w.vy1bga gnwy5vr6g)j1us 9f0 $^{\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 coincidenceg6z*om bb.c6u . Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moo bbz*go6.c6mun, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergp c5/-cz7f5t odcphq1j pp8w7 es at -1pd8 fj/ qz7o cpccph7 tpw55an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When t2 sos6du j,kk16iovk g -pjr96he motor is turned off during operation, the blades slow to rest in 3.0 s.-u9266gjojok6psi k1s r,vd k 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 bal;th638qw6 qmtv8n cgi l makes 15.0 revolutions, what is its diameter;3w66hc8ivmq t tqn8g ?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dog avkl.;d4sm9 the wheels md9alg4 vkm;. sake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angulyntsx8x4c(ojy(d* b.ar vby (ysnx(xjd*c o4t8 .elocity 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 (Fyn 5j+t)urj 0ubh5sam c*b+-hig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the ceutn-ju)sby *+5h0m5b+hrc aj nter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earthh+x4 6pp3dgs-oqa: veukn7v85o k9w t (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of )y rv1w xwm5r2a 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 ced9op/.n dj) vfntrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleratpj.ofdd)v/ n9 ion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centergh4a; a1 -eos1ny hk8f,k,gtmt nr4q3 from 130 rpm to 280 rpm in 4.0 s. Determkqn gt ysh8 , e4mfot31rkga,h; 1-n4aine
(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,9bnu 9m;lh s$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselvexxb+puse 22:6qunq- z 7u+psas into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation tozap xuqx7qps-2n s e:uu6b2++ 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 reo 2viuylpu(0 )st to 15,000 rpm in 220 s. Through how many revolutions) 2opyil(0v uu did it turn in this time?    $rev$

  An automobile engine slows down fr g;ywtr:8runnw jr8c8/ dv (;zom 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 sti(o0 o9hz,eq ighi267ka5zk oresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reac o i072oq5 iihz9hg(kok6a, zehing 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 24 b5q82xsc wkgsu0(a7u0 rpm to 360 rpm in 6.5 s. How far will a point on the edxuq (kuss75ba 0cg 28wge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runni;jqo9 fy.vb5 ing at 850rev/min It turns 1500 revolutions before it comes yfjb; iv9q .o5to 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 revo6; guw7pns mfh *(.ssjlutions as the car reduces its speed uniformly from 95km/h to 45j s ps(n*g; 7shwum.6fkm/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduceqs,+v*( 7) bptv wjt 0huqh1(peg.fwas its speed uniformly from 95km/h toae hv hq,tb(svjw gf *+pqp1u(wt07) . 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 her weight on each pedal when climbing:wl/uwnuvl ( 13 jdi3m a hill. The pedals rotate in a 3u iwd/ul3 w(n :lm1jvcircle 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 551eef2myii-smf w.ah:u4qi +( N on the end of a door 74 cm wide. What is the magnitude of t iheews( i4f.1yua+i fm- :mq2he 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 s,yrxq(dr21gb,miotg j (ik .(hown in Fig. 8–39. Assume that a fri((rmdi1 t , bgxg.j,i2yk(qorction 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 yj rwlb187df,massless rod which pivots as shown in Fig. 8–40. Initially the ror1,ybfw7d8 l jd is 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 oah3s0omcezqy32k 1nv 90 vy 9kf an engine require tightening to a torque of 38z3qn9evo2c 1 k030 h9y ksvamy $m \cdot N$ 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 ofy2xx,,o8lp qqsa r(s g7coj9 3 inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is txr jayss79xopc q,82( ,g3loq hrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim *8*)ctaaqxnp and tire have a combined mass of 1.25 kg. The mass of the 8a**n)p atxq chub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balr0tq d,l(n2s 0d,wrscl on the end of a thin, light rod is rotated in a horizontal circle of w,2s ,dr(l rcnt0sq 0dradius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constan 2mbud uegc1o +o2 b48mqf+uo(t angular speed (Fig. 8–42). The friction force between her hands anfub m+ 841go+cub2d m 2eu(oqod the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia o rt n,rseao7/z)c;2lnf the array of point objects shown in Fig. 8–43 ar n2t nlc)zs7roa;,e /bout
(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 consis+ zg.+s3/ owphy gzwp9ts of two 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 satellite spinning at the correct rate, engil m1- 7vfkakquf- )en,neers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of7aenf1)lv ku-k,mfq - 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and ab)2hdjq j31-0 wxq ib7b ua:ek mass of 0.580 kg. Calculata0du1qbei wqb: xk -2bh37j )je
(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 swi k t)p)xg24(k 0tpnejings a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small han86ho;ewp ( jrubz4vdc 6/h-dnd-driven merry-go-round and is able to accelerate it from resv ro/6w(ebhz 4-6ndd p8c juh;t 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 and i yw6 w) +zbylpkgf h-4(bz:ic1s eventually brought uniformly to rest by a frictional torque+(lf4h w: 1byzbzkgipc-y)6w of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kgk841 n8naaxp i ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm,x elt1i0mi-q- aaz/n; 9 pfa2t which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The baxaf02p ;-li1z a-tent/maq9i ll is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fiby2(w. h8keb yd z-vn*i m-9szg. 8–4b( m wyn si89yd.zevzk2-h -*b6.
(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 two mas r.ckob 6vnn7)ses, $m_1$ 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 qd l*x*xjldl:,6la/ i four full turns (revolutions) and releadx6,dai: l *jllx*l/qses it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mom r6jmsvnu*/hs6 0 5rlgent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 o -v10tkyoeoqyn9-zzx0 21 pp$m \cdot N$ 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 racuaxp-u gx - *d:m(3a lane a* p xr:mxc gdu(3-au-at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to to7d5tgzu: he2 vqtb/ 0he Sun as the sum of two termubvzo :/20 httegd 7q5s,
(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 of 7.50 m. How much net t- ai e4wde(ay nyy40)work is r(4e4aa 0-ydniwyy t)e equired to 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 cm and mass 1.80 kg startl w6enfm5h hm w071+nfs from rest and rolls without slipping down fn7 0wm1f+6m5ehnlhwa 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts x5bdf:j *gh u 4-yyn2lmtp* g+to fall and its lower end does not slip. What will be the speed of the upper -5 *l+gdymun g:y4f h2xpj*b tend 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 ball5eg2 1imp ;nyx rotating on the end of a thin string in a circle of radius 1.10 m at an angulare21i5 g pmxn;y speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical .8vuf2f wd 2kr5djcceio 35 e(grinding wheel of radius3ckfrwdvud.e52io (jf 58c e 2 18 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 a platform that is rotatinl:vo5o82v b. xzpdpy 5krid, *g at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatio v2l5bo .ki 5y,r8:ppz*xvdodn 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. 8–2r x4:jgwe uf**9) can reduce her moment of inertia by a factor of about 3.5 whe ju4g*f *x:ewrn changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what 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 of 1 79nw/;i*e/fyqno s9 fkfg q*c.0 rev every 2.7feqfs9;* *wo g9cyq/in kfn/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 ahus mr/aaiavp2r6* /2t a frequency of 1.5rev/s The wheelha*a2 uv ma rp/sir2/6 can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentu n q-y2s* gx6ryp.r2xum of a figure 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 momentdtrw6oid/.w du;v.; oum 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 moment of inertia I is f3ya a)o/- skydsc om(5lg56zdropped onto an identicalo-af5y) ksoa(/lmzg3 65sy cd 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 2.1xo*r r;0qzadab ccpmz75 e.qb 5g7 i+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 rotj /uhszwcx+ i ca6),9tating merry-go-round platform of radius 3.0 m and momen 9 +6uihz/xcjwst,c)a t 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 merk-u6y(3v hap rry-go-round is rotating freely with an angular velocity of 0.8 y(aup r -3kv6h$rad/s$ 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, lo/l1ji)i+ tft wsing about half its massi /+)tw1it lfj 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 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 -z2y/ fty s:g)k0 ;ee bnuaqu4in 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 *jt +7znxd ddu)6-xszs;dit;$ 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 can rotate f; zn8aeip k)mfz-p)z:reely withoutnikz8)z - ;zmf a)pep: friction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-gpnte) v2 9 g5v1nd0ggoo-round turntable that is mounted on fr9gvn1ovt gg edn25p )0ictionless 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 roenw*m ax+8)b spe lying on the top edge of th8 sb wmxn+)a*ee spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. De-j/ j 9pzph55cfysgs 1termine the ratio of the Moon’s spin an1s/p5z- 9y5s fjhcgjpgular momentum (about 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 from resg5 .7/wo26rbfuzepe at 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 oa.92c :+d xlkoqc/ est 5pj5ru(e3 asrf a uniform cylinder of radius 0.20 m acquires a rotational rate of 5qaj.c( 5tl:exrask+rouc e3d/9 p2s from rest over a 6.0-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, eac ;p*hm (1mh ,p91hl(kd srfomqe3:pqjh of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 k jpph:m,1fhsp9(kqd1r3 m qel *;m(hog and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheezqxr px)95 diuo+-tw 7l ($\omega_R$) 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 gr6j mnea jt9( c p.je6xk1)qagdc-,xp 2eat, 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?q 26mex tp pj.c6xjnag)9k- daec(,1j 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 gjeikt c;7;gwy-- wo(to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter -yicet k7;go;w-wgj( 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 illustratesui n7r1ok fovn0f3k+t ) -s*ne 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 (hog nj5 nfcnyzvc)7r d0ks9, h92op) 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., cu:fgi kr*6kl 4bzhm.d9 n;.bwe ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At theg.b6n hc4bk ifl;um*dz:.r9 k 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 isk stqnyxm* -eo.v66*-dliu 1/vk+ a oq standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it re os6d* lv *ya6. nmxq1ukqi- +v-/kotests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with e48ogyek*4ch::*wz7rz m5e 2tad o hj speed v=4.2m/s on a flat road is making a turn with a radius h:kg*h4o8zro t4ewz2 7 5e*jdm aye :c The forces acting 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 bejduneys8l 59c da9)5g gins 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 tord)d5e 5scnljayu 998gque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bodyg zhpi tm5e8,2 as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the g8 mi2he5 pz,tratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembl(4oc+s/ ox +mcijz, vty which consists of two cylindrical plates, of m(z+js,o4xc /t o+ivcmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius rb0g r34 bmns6 ca8ge1jvo3 /wj 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 t ranj sb1/b 6ewg33m0jo4vgc8he highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu7ox/vw 2nwm. 6 zaq+.k,mnwy jr 6m0iut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85w) a-ck0pkmf f9vnx79 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acce-p)kaxfwm9 970kfc nvleration 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