A bicycle odometer (which measures distance traveled) is k9 i fxkuf9,-f kvy6,aattached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wikxiyf6f, ,vk-kua9 9 fth 24-inch wheels?

Suppose a disk rotates at constant angular velocity. gyki ty97od32Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increask7 g o92yyid3tes uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vi*4.pm23 yz60pk*t r+ nkd (mt zfudk9uste3balue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greate6po1g3d, 1mwyihjle 9fkf-;u r 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 behi. j)y;7ouobq e4f4zc cnd your head than when your b7f .uc 4 zc4eoqyo;j)arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets a/kou7l 4p1fykt 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 to pedalpk1 uykfl7/4 o, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower lesekajp*zqlva5 1 +*5ygs with flesh and muscle concentrated high,k l5jy ps*ez1v5aa+ *q close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?) zg*),9wo bybwa2bse

If the net force on a system is zero, is the net torque also zero? If the no+j dy fn300*bbpog (vet torque on a system is zero, bo fg(3*o yn00j+bvpd is the net force zero?

Two inclines have the same hei9ba ,)+ b/qh1y,bi day-skdyuvd +fm/ght but make different angles with the horizontal. The same steel ball is rolled down each incline. On which inclinyv/,kd ,dfmyh a)u+9q1s da /ib-b+bye will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from restgd)js a69e/njhfm tngk4tpzv-j- -7 8f7gk-t ) down an incline. One sphsg- f/ -t7vjhktzg9te-d7 fg-kj)mpn68 4najere 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 bottom?

A sphere and a cylind54 /njs :gakyshc06a io5:wurer have the same radius and the same mass. 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 hayshu455:ij sg6/ara:ocn wk0s the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mo)dx3x.bm s f:rtx r,p-ving or rotating objects eventually slow down and stop. Explain t:spxd)rmx xb-.3f r,.

If there were a great migration of people toward the Earth’s ef9 b(d+xqliv *8.eqwf 5,i9fsnh k7xlquator, how would this affect the leng(q59s ie7vxq w9bf ldn.+,*ixkhl ff8th of the day?

Can the diver of Fig. 8–29u.- onoga9l*/isdkj k xa-,f* do a somersault without having any initial rotation when she leakl ,d*9k*ixfaa .go/nuo-sj-ves the board?

The moment of inertia of a rotating solid d ; isyqrz.)5m+,gs+ ceo iqoi*isk about an axis through its center of mass r,oqeg s+yi;icz+s *io.mq5 ) 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 each outsks5;rsf;k+ qd 1npv 9ftretched hand. If you suddenly drop the masses, will your angular velocity in s+5kfd;pf ;kqrns91 vcrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same m;j4cfg.xqxnhd 2 ,lf+ass. However, one is hollow and the other is solid. Describe an experiment to determinedfx.2qg 4ch+ fnxl,j ; which is which.

The angular velocity ckuhx91z,l. ,binp q)of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a pointcu9q ,p b n)h1,kzlx.i 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 turntabl8f*bo, n(jn ple. What l,bop fnj8n(*happens if you walk toward the center?

A shortstop may leapi;pokmr/r(h o.k- s8d: ps4g n into the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look qui.ihpr : - ;4mg oskndp/(8rskockly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of s wl ed0on7s/;angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Di0w /;s7lnds oescuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radi5g;0(odob wi vans: (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 coin3n /rauykpc50 wpxo,v(95v gf 1 6bcnrcidence. Calculate, using the informaponufpv 3 55wg/x0yrrvn19kb 6c ac(, tion 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,n.lra2oz k ytmg g5s.(3 tj4 s1a2tz 6kfdb,:n000 km from Earth. The beam diverges at an ayman2fd5 t4at(3sjo6k.2z,n1 gbl z:ks t rg.ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rf:4s d8r4nltbtw0zdc qr(3 *h(k *rahpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is t bc (adsn:rr8 *q(4ft3 0lzkh4w htrd*he angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a levq xd qae8:(ze fjiyp:zos00 9a(z ,lc2el floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is itsszap(d o02a9cqje:iz, el:8xzyqf 0( diameter?    m

  A bicycle with tires 68 cm in diameter nqkj- ty2bidza4 l01,travels 8.0 km. How many revolutions do the wheelskzjd yit1b,2l q0a-4n make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anut+mtf 9 ox,jexr*9vku/e rp)8gt64 sgula*trpx 9e9o)sgu6 ,ujf /k +t er48xmtvr 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 re(i1 ,.7s wrxm.3mkmknk,vcbsvolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the c.kxmic(,s,r 7k3k mmwn 1bv.senter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a)cl znpm5,+ibbyaobajc/ +8 3 0 in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speen-o6trt2 jkjbg d3 v e7dapuzw,* o04d-0,q fbd of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle b8pkpwc abh/p+m h)55 7.0 cm from the axis of rotation is p8ch+kp)/ 5bamhbw 5p to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acce6jwh0g r*, pduu7nrc 7lerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. De,c gn rph70w6 urd7j*utermine
(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;t *+/f t. b0y*qyz avumtbfu0 $R_1$ 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 them7 n 5oyf xvp60r*ewb;jselves into a slow rotation to distributv6;pbf*n75ox er jwy0e 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 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 restphn )*mwr16uwp1)rk qqnc7+ 08mi4zjgk9b iw to 15,000 rpm in 220 s. Through how many revo+)qwr104 i*1 h6 q n9kjznmwigwrkpcm)bp7u8 lutions did it turn in this time?    $rev$

  An automobile engine slows downrrhd(ryx:g,6 d il1qz kk,b92 from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of fljp l76 ba(222ruokhbb ying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speeru7a6b2 p hjb2(obkl2d.
(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 unif3i *6 ttix*0q -jiqoinormly from 240 rpm to 360 rpm in 6.5 s. How far will a poiion3i6j i*0 tqx*-tiqnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min I bw.jnp7;oc9s t turns 1500 revolutions before it comes to a stopj.bw n;7cps9o.
(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 ungca ztl ),l)0tig-i 1uiformly from 95km/h to 45km/h The tires ha 1cl ,lt )iat)-zigug0ve 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 reduc y/xr9x6byh)z es its speed uniformly frox zyy) b9/rh6xm 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

  A 55-kg person riding a bike puts all her weight oioq t6zjlq, gjq14 )0vn each pedal when climbing a hill. The pedals rotate in a circle of radius 17,41oqj0zvj)i q 6qtgl 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 door3 zg.s)e/kp tj 74 cm wide. What is the magnitude of the torque if the foeps .)3 zktjg/rce 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 thc 5gcai c.y::ke wheel shown in Fig. 8–39. Assume cc:ca 5.yg ik:that 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 attached to the enu f93p8zmd1mkp bf* +mds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizonm1 kppf+ubf8 9mzm *3dtal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinp 0svex 8b e(sjq,dm88der head of an engine require tightening to a torque of 38 0jps(sbex q8 e8d,mv 8$m \cdot N$ 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 p v1x rgkq+n:1sphere of radius 0.648 m when the ax1 rkqg v1p:n+xis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameters.kg 7 xvq9de64xt s) dxv;w7w. The rim and tire have a combi.74tq 6v v )79wxsdedxw;gx ksned mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatv:d)vxgm.)* 2djs ap ked in a horizontal circle of radius 1.2 m. Calculp.k) mjd):dg *avx2svate
(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 gt3 +mrnhe4t1dr/jxu ( k .xu)bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay /tx+1u mgjnx4 uhkrtd3er (). 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 inertizbb0gzrq l y;pg i056qtqt9 .4a of the array of point objects shown in Fig. 8–43 abouzbpq. yg 9t4q0zibl6gqt 50r;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 oxygen atoms whose to cuxc(tok+)2 2i ilty9tal 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, n)ve) :dpj pa6.hpiq; blv+5ofi:hc4 engineers fire four tangential rockets as shown in Fig. 8–44. If the satellit:pb;cjia hdq5vv)f +p4 o.6eln:hip) e 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 cm n-bcy d 9t7bq.n -t:b0ag;eo yand a mass of 0.580 kg.b:.antcydq -g 7 y9tb; e-0onb 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 bahp mq8qi;d4;opsa /l zu 0r*7d8 ae-xzt, 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 tano19uda .ekvx8w 3c (gfgentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two ca.8dg vu9 (of3kwc1ex hildren (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 ssc) vu 9jwz3y:hut off and is eventually brought uniformly to rest by a frictional torque of w3cv :)s ju9zy1.2 $m \cdot N$ 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 babuw)wld 8 a4 9d;v 4uur6j3)bdcgi8svll 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 theu)xok el4vfl* 7 8*gwi action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball ii7wf*olx 4v u) 8*legks 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 ,azf nh67 8qf*7xbogvcan be considered a long thin rod, as shown in Fig. 8–46.8b 6 x,hfqnf*go7z v7a
(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 co(4fqwl6kqna.h+y iw:p 2 4hmd nsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fouhv mdkhagb:43g(y j9,o z ;*vdr full turns (revolutions) and releases iag:zd ;joh39gbdmyv4v* (,k ht 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 23:3zzx/ ooea pyntf1d;(kt ,dbpu c(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 de *kmx4q+x+xf /9qd 3z;pwq v9rgyvx3.h;r spnvelops a 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 rolls witho6u ; q9zymi0ozut slipping down a lane 6zqumozi09; y at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun 0txt q)2b3ckl9p( iqjb y)pmj 9u9 9bvas the sum of twojvjq by(c0tulq99xb3 ) bpipmt99k) 2 terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass hhjioraxi:./ l,h s 24of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from ,a/4.rhsi2ih :hlj ox 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 starts from rest and rolls w lhm9trk.;n. sithout slipping down a 30mt r ls9hk.n;..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 stisae3h rk q4a0m46rs4 arts to fall and its lower end does not slip. What will be the speed of the uppere h63 44i aa4ksrqsm0r end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a ;xlv7-al3t 6 bg f 5n2ievumd)efq6x(0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10g;)ut3 f-lq n xl5va v(e2emb6df76x i m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of dv bni/ ,ju;1w*)gvoayap(n 2 a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpd wv/ab)2g ;y vna, no1ji*up(m?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hi 0;q)ui:vc bbss 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 db:qcvsb; 0u) iecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inerymw 7/( 6 rzt: 2fdpzj(tmf5istia by a factor of abou7w zf25pzt/tfmm:(ys j (rid6t 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 position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rot-a+c5t ge* qbqt8ww1q 1 ssr*cation rate from an initial rate of 1.0 rev every 2.0 s to a final rate cqq-+w*t bgct8a1 qw *ssr15e 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 atmhqmnh azh)6lh/ c7 oq9-k-6x 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 the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it? q7-/)ma6 kh z -chqhxn9m loh6   $rev/s$

  (a) What is the angular momentum of a figure skat2j j-lq s j5,skwyd9y1er spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ear7w;.0 7exmf l4ppwy(97cm xit iom7 fjth
(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 mqbhl: bady/r:c .,q .+l *i(/siorylof inertia I is dropped onto an identical disk rotrib/+sq *iqolay :l ydc./bl.( : ,rmhating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 63ca c1aw:b*uc-ds * +waq-sq8za eyy $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mg- fezvbjaik4gj8+ f4ok5 zay1+ n-*serry-go-round platform of radius 3.0 m and moment 4i j*ggbfv+n5-o jfa ekzkys a z8-41+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 vz6dl0ukk 9 m1melocity of 0.89z6lk0m 1kd um $rad/s$ 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 coll/o0scivz1e6pfowpr* )uv edep- ;;: aapses into a white dwarf, 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 momentum, / 1op-edzca f* :0eeor i6;p vuvps);wwhat 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 involvegta (0: +:)kuymbmewq winds 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 kam48.fv 6dmc)dt 2qg$ 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 freel xn k:*d:b9mxb f-:;zdo+yseny without friction. The moment of inertia of the person plus the pla x+ydkm9f- :n :*zx;bn:o besdtform 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 stand dx5qtw(l ;u.ts at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of x5tdqt;(wul . 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 thew /z f+*:fe2ixir 5rhw7evq2g ground with the end of the rope lying 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 without slipping. What length of rope unwinds from the spool? How far does thze+we:*hqf rviigx272 w /rf5e spool’s center of mass move?

  The Moon orbits the Earth s -2d1 kb: upcty1tq7reuch that 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 mom:k t2r tbpce7udq-11yentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from r*(p(tu3 d cimyest 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 bp2dd*m/j:dr t+6zzrshape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s inter ptr2z *m/dd+bzj d6:rval 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 o* uc4n+ f(dwbmf 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 energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-lo4(ncfb*d wum+ng 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 rxf z1o-o m.x-swi9x- )v1l zprear 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 w)m;zgj* 8nnyn8e1y/- swro(xu k: czkith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass z; o1j)kz mug*-knn 8/s8e yy:xwn(rc 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 energy storyoe mk 3zia/4h)5ixmb 4rs42eed in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of dia5mmk2yeoxb4 )/se4h i rai43zmeter 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 a2/eezma.+ tt4 dm9pra n $H_2O$ 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 r bhr 65uevsm2*olling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignrkd/ ,,2 iaxkqgpl1q1 ore friction) about a hinge attached to a qp/1idk1 kx,grl q2,awall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the 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 thehm. 6 r1oxy9 ulc0zb.q floor, and we want to exert a horizontal flbc6r.m uh.0oq1 zxy 9orce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with v ;vi5nh96( ajbu g5yoa radius Thyn gvji5uv bo(;h 9a56e 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)k dve841f,z) 4bfmn,duh flwtl- v)j 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 fea4ld, fzn)1tvk8 4uj,m f)lvwhf db) e-t, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyli m8;;,wvkvx xfnder 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 with arms hvxx; w ;vfm8,keld tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrig 1 8amhr zevvek 8xc*(gj)l;: 3zeih*cal plates, of m i eeh zjm*8g)v;:xlc(g1ez*ka3v hr8ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsa ms0o ;f95y0zm(osv y along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the 9os(vzyy s;a5m00om f loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume+ hu mpbfua2/j6sh sx7;u*pbbv238ct $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions, pzwx op7s*ol1ef8ck)0 (m qg as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diew)78g1fz cmk(,0lp* ops oxqameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s