A bicycle odometer (which measml x1,b,xih s94wteu(vga -oet78 oz. ures distance traveled) is attached near the wheel hubg4ohv8iu,eezso mw1,. 9 (tbxt7 axl- and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vfag7 ag -.yvwy:oq ,;bf bd30wbn.i)l elocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incroal;dn-ggf7.qyawvwb b,fb) .0 y 3i:eases 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 describup a4.);o vddbfo+ 4nted by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert/xuaq bf6f0 vw /j-6uw a greater torque than a larger 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 behihqq2* / ig 5yf-zj gqbe(wslyjp9r6*. nd your head than when your arms are stretched out in fr-s*y 6*qw qb./ygh(e5z2ipjr9 lj fqgont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three atjdrr*gc9 d3 *n 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 to pedal, a smallrdc* j9d*grn3 front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowbink p5fhb, a(1xe0 (cer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageou ebk0np5 xfb (h,c1(ias.

Why do tightrope walkers (Fig. 8–35) carry a long, nar p um5dkrmk2qs6 hg*;4row beam?

If the net force on a system tpk efbxpw) b.c*w4qr6z7hp,3tg e64is zero, is the net torque also zero? If the net torque on a system is zero, is the net fo*pzt3p epefc4 k7 bgw4 6x)r.hw,bt6 qrce zero?

Two inclines have the same height but make difv:vc +qgmg v 87d+erg7ferent angles with the horizontal. The same steel ball is rolled down each incline. On which inclir em+v7:q+ gcdg8g vv7ne will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simu2p de)b*dk;.( um3dq ta juch,ltaneously 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 greater speed there? Which has the greater total kinetic energy at the botbc3dh;jtmdd .u2* )kqp eua (,tom?

A sphere and a cylinder have the same radius and the same mass. vem-q c730o .ypmbl x*They start from rest at the top of an incline. Which reaches the bottom first? Which has the greatermv3 b7lop0.-c ey*mqx 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 cons v7.ut*ky:dw/ f(p8 ,x:khl8fq k hzggerved. Yet most moving or rotating,(kg7h* w8: k.qdth fk zxyg:/8puf vl objects eventually slow down and stop. Explain.

If there were a great migration.i ztlzt)a7lzuz 0/fay a*lpw6cu h0c *m281n of people toward the Earth’s equator, how would this a8/ iyh.l6l0)*2*atw1f zpccl anzz7z t 0 uaumffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any dl/g0 qrbk7 a9initial rotation when she leaves thegkb/ d7la0qr 9 board?

The moment of inertia of a rotating solid disk about an axis through its cente nu) gzh4(c7oqr of massucnqzo4hg ) (7 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 2n:-1q + nbqyfn-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular vbf- :nqy+n1 nqelocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hav(rlj 5d808mbodz r iz19g9jk pva*td6e the same mass. However, one is hollow and the other is solid. Describe a9zjl0a5g bvmz8kr( 8rtd 9j6pd* ido1 n experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poingp3f kjzxd3ek 23 h;me c;q83gts west. In what direction is tfhkpx323g 8 e3cm3;zj q kdge;he linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating t9dtmso -lg g 7vrezn +(l4/rq)t9d9uw urntable. What happens if you walk toward the censd+ugl97q9gdrrwn t(m z/9 -4v)et ol ter?

A shortstop may leap into the air to fa2m iybq07/2zo t3n iriz+5 zk 8d*z7w5tmch 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 the opposite direction (Fig. 8–3dar wfk87mi5h7imzbt * z/25 z+3qnc0zyio 2t6). Explain.

On the basis of the law of hajhj*82 )vyt1l)ljw+d m tq+conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelllq mvtj28w+t)1dlj)ay h+hj*er can operate to keep the helicopter stable.

  Express the following angles in rt; * 0g)4njub;y awdks+4-q ppxdczw;adians: (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 slfd-s94/nblfk ) /w qof an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diametl/-d)sb 94 lfsk/nqfw ers (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 dilyj0kvr4 eh7x-tvr0 ; verges at an0l 4;yxk7rev rthj v-0 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 ratewilpo4 88:jlj7fqz,y of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelerati8lyi7 z 4w j8pql,ojf:on as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 r-(-cym:u4ra g0yyo dm to another child. If the ball makes 15.0 revolutions, what is its diagy4-0 y-acyur (o:rdm meter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How man-)ec0 b ixkxvq,ms32ey revolutions do the wheels ma0qkb 2)mei,3exvcsx- ke?    $rev$

  (a) A grinding wheel 0.35 m in diamet-y -phfbhp5 :ier rotates at 2500 rpm. Calculate its angular velocitp:i-pf by-hh5 y 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. 8–9/ agzvua a xc 1qq)0yoh30kx038). (a) Whay0a90qak/govzux3hc0q x 1) at is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in0ykcv1 *30k tzf/o/ir rx6yzn its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of klv/0qny * .cyirn6h/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 pa- hfcchx*ximgt lsp8e8:u :yj* 8um: s.ho5(zrticle 7.0 cm from the axis of rotation is -huspj5mm tyh88 cx 8z: :(*ugs.flco*:hxei to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceletu 5m5 mk6d;errates uniformly about its center from 130 rpm to 280 rptk5md 6m;5r uem in 4.0 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 radius gs:n og z:s5g-$R_1$ 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 Moong:1+p ms// q4 otg*q9 jfhgmqk:lw:ay, astronauts 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 rotation to 1.0 revolution every minute during a 12-min time intera q/4/+g9fsp:okyh:t*qwlq :m jmg g1val. 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 rpb g/f1iubn,*ir+sa x 6m in 220 s. Through how many revolutx+asiin6/1b b f ,u*rgions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm i n/ 6hh by2w.ubq2oc/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 ind zlj3ft)zv5+q/f w +m a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reac)z tj/ fvlf++z5dwmq 3hing 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 360nx x/rz)aak0 6 rpm in 6.5 s. How far will a point on the edge of the wheel have travele 0xakrn6 a/)xzd in this time?    m

  A cooling fan is turned off when it is runnupbtu-hhmzy t8 g., k ba6-(v2ing at 850rev/min It turns 1500 revolutions before it comes to a sh - 6ytbmkh (tv,8ua.p-uz 2bgtop.
(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 uniformly f;u 0k0bn3/xi*u i ifd49kajqkromji4ki0bun /k f 0qa9d;ix3*ku 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make .3n x9s*akrhs65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8*x hsr.9 n3aks0 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 weieg n. 7.fnem3aght on each pedal when climbing a hill. The pedals rotate in a circle ofm.n7 .e3nea gf 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. What isf4jum 7fkxe 00 the magnitude of the tu0740fx kefmjorque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39.s .6i:vl 9 *( ijer60cvhhfo 16enzhqf Assume thal9:16f 6vs(6hhiqe *r n jc.eh vifzo0t a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atw tetpmg37n3t;0utw8 tached to the ends of a massless rod which pivots as shown in Fig. ttwu73t 8p 3g wn0;mte8–40. Initially the rod 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 of an e-f 7dg/lhlc rkj8w .i)ngine require tightening to a torque of 38 8-cg 7 r.lflkhi/w)dj$m \cdot N$ 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/mqs+ f7e- bs ,hmecawgkb9a.i50l ijds((1 p the axis of rotatio5e +a- aiijd9m w/ qegsh p(s(b7m0flb.s1,ckn is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm cem:8ujenv0*(j m/x0q cg;czin diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be 8m cjnuzx(:cq/v e00g *j m;ecignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light yw7fu .x2x)xyrod is rotated in a horizontal circle of radius 1.2 m. Cyx2). f7uxx ywalculate
(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(+emfv :ti /ga8 z5uu8dhb:gx22hg yo constant angular speed (Fig. 8–42). The friction force between her hamgvdzuf :ta22g 88yhxo(h g5+ib: e/unds and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of theql :mp5d- nkv9 array of point objects shown in Fig. 8–43 knm- : q9ldv5pabout
(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 consistsb25.mg(p 5g 8xgjflww 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 spinninif4bi2r scget9. y 7 ::ift3xig at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44sff:x3i 79 t terg :ciib4iy.2. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 pe 4f(mn9:qp8o 1 8js*cpfrmncm and a mass of 0.54psnjpm8 q mr8ne1 *cp:of (f980 kg. 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-or : qwwo* 5ot.1y;ez*ayobx it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-dril inqbr. + nacp(d9u*o9ii/ 0hxv:el1l+7tsnven merry-go-round and is able to acce l *qlo9ir/lcv0bhd nux ai(ti9p71n:+.en s+lerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and ij.5d huck3s1o - ;t iuyangc64au l-g2s eventually brought uniformly to rest by a frictional torque of 1g ucs3d.- k g6 52i;oluuy n-catjh14a.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 baly+gluj pd43t*t x np;krv-5- cl 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 acti(v1dl y8.2s+zg u 9rdpbkzg5z on of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is.y rd1v zlb8dz+p(uz29g g5 ks 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 k;fm(:w 2q.lo:ydu:fc 4diby -po:zt/t*ecd rod, as shown in Fig. 8–46yfdoz(u k:l p m y2/tcd;.i-:bedtc:o4qfw:* .
(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 t.zggjn-yq: q:fy+7 c ewo 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 fulf :g)kvwl* vrb7f7hk4dvw7c1 l turns (revolution g hwv7dvf1rfbkl:) 77v4k c*ws) 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 a moment of inertia c ,/uk5nbomo ( 38mbflof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a toc ,ip (u x7 r.7:gyn0hngepuc9rque 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 radyue1yw ) t40atius 9.0 cm rolls without slipping down a lane at 0ty1)awtu4 ey 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resqa ow g)(y5o .qty.jr2pect to the Sun as the sum of two tt2qoqaj )yw (oyrg..5erms,
(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 m ghb+oa4jhsa5 y+0q8z uch net work is required to accelera5qhz4gjo+0+b 8 asy hate 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 1l 2d8vzl6cyvq) ey7 6q.80 kg starts from rest and rolls without slippi)l qz6v2e8yly6vc 7dq ng 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 3;deno;y9 iy, qd0jsxbalanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation ofsd;j;n yid , yo09e3xq energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on t0outdz sa ,kbezqn6 o m9y/+ g1ss-9u6he end of a thin string in a circle of radius 1.10 m at an angul -qoast,0ou m d+e6/6 bzsyuzs n919gkar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheelk kuop6o6 f9a,)s/kev of rs6a u epk vfo,6ok/9k)adius 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, hands4w icz3qvenn)5y/)p;fx 5 p o4n- gagu at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decrease 5 op)xun5 ng4/zv4ni3;pfwc)aq-geys to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one sjk(xuj .t3kit q6n 0s wp.sah fd66p; 1*h2vvyhown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck sju*6vkid 6p .w6( hvh.qy01x;n3 kt2tsajpfposition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can in)/h-qs gyvkz z6 ejq(:crease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a f-jh 6ze) sqg(y: /qvzkinal 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 cent2ffqu z (6,.nb0ig2sygwxm 2ger at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto w f usni2.y(2 2q0bmfgxzg,g6 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 atbgo ydk:9yw1a1fdq4 s2 0k). moaq2;,m qarwg 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 angular1.qb/e c 8*mtdh ;aj6 ;w8 hgjzfvjy7f momentum 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 olwi2j+ z7(-zt x6bbllf inertia I is dropped onto an identical disk rotating at angb z2l6zi-lltjw(7x +bular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns au:ev9ncf he -so1:1gxt 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a turbjjl; f 8;v+-ezzg tv08d/ rotating merry-go-round platform of radius 3.0 uvve;tjf+;08 d8zb t rg- /zjlm and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular veloc/ah l-4.25diqru3z f gt 8mz ol9og5teq1vg2 iity of 0.8 q-5qi8g19t u mof2htar3d .2ligze/glvz o5 4$rad/s$ 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 diijs0 m9 v;l.8dwhfw8warf, losing about 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 momenw 0m.d wj9v siil;hf88tum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excesr d 6,iahgv.yp-non42s 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 q i19 tz:l4hzc$ 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, initiabf0,a1: anm8x)xwo tmug4o,ally at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform itxu mw) ,m, xaob ngf:a41o80as $I_P$ 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 m+aoc 94 l1nrc8*qnh roerry-go-round turntable that is mounted on r n4lc+oho 9*qarn8c1 frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope odeuz b*54q o8lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person witho8*dqboz 4uoe5 ut 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 ,kk q7,871or0zcx zff96x:8tqy p ysuy :jvrwthe same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angularuzyq kv qj7frf6tys ,kp:8 wxz :1r,xo908y7c momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 -)a3sa ei*dvuttgh g67g ezfa0:s* 5fm/$s^2$ 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 ofp(ztymu 940y a a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate t(tp9zy uyma40he torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindriz3er:k )mr7tp cal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energ7tr: e3rp) mkzy 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 ex: bamb3t :9lbq,1s nwheel ($\omega_R$) 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 grea+bpsa,i 1*z 2paapnad ib6 3z7t, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neu7+baa1z,63appi*i zps n a2bdtron 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-polluic;1u54qvops;p( bbb tion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total masbobi1q p; 4pvb( ;suc5s 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 illustrateszyu78q ,njmx+ 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 rolyuk-:7zwp b7d88cqc h ling 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 j4 ndbp3*rd7p and length L can 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 acceleration of the tip of the rod. Assjp*pd3rb4 nd7ume 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, at jz rd3)9rebcr( l5i0nd we want to exert a horizontal force F at its axle so that it will climb a step against which it rerbzc0l tre9 ()ij 5rd3sts (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/gq1n;an8mti8e3 im4k s on a flat road is making a turn with a radius The forces acting on nn q8gt41ia3mmk8;e ithe 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 c2eg7 3enac.j, dk:lxinto a sling of 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 feat, and where does the torque cojk7 gexle.2c,dac :n3me from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her olq+o 9hzv +qzq;:6esep 6w6l arms as thin rods, making reasonable estimates for the dimensions. Then calcul q z6w6lqq6p+el;9 :heosoz+v ate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of tw1qkprf2ou( o ,o cylindrical plates, of mp fu, ro2(kqo1ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls alp3:isfyjaob++gg6j4wh ,i s: ong 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 wi ,i+s :i wsfohg6+gj3aj:bpy4thout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume ofij7y2giy2 u8tpv *($r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly m *+kpx*o8qljfrom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a)**o xm+q l8pjk 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