A bicycle odometer (which measures distance traveled) is attached near the wheelfydqlz0 .-k)aovi .j 8 hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with fo.v.za-qil) 80 dyjk 24-inch wheels?

Suppose a disk rotates at constant anguv(6dw+ u lk( dlsf fynf/tl.),lar velocity. 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 tanw(fu,k+6ln df.l) tysd(lf/vgential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the ag2, 5v ezxq( p* /24deebu)kc jye,zhingular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?oa*djljtvwo6 3d n:ju) / );ib 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 behind your head(za gkn5) y/a u7lypk- than when y7 pa(-g5kaklyu)z/ n your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has j zq-th)w e6yh/88aovseven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sp/w et-v88)6joahqyh z rocket? 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 lowevozpxjz3r2/l- + l-1l5a,sw7zcttu w r legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On t zusczvat3-,oxt/ +1lpzlw7j2-wr5 l he basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.gug 5qr; jzb*a+0j*w d 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net to 5b(u3ssc s/j2c wd +a1iq1mmdx/b c,urque on a syscsmbmcj1 w ,bdu+3(5iaq 1 sx2cud/ /stem is zero, is the net force zero?

Two inclines have the same height but make different ang b. (p)kl9u*iqa/zxtytwu e* x.7m dw6les with the horizontal. The same steel ball is rolled down each incline. On which incline will tp xt)6 *buzxa /9*l wq 7te.(udwk.imyhe speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start r-q;7e j ea9f,pjvlpxf8cn77zr ho +3r olling (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 kinetrp9 7l r, 8fj+3f7ovj -xc;nh7paqzeeic energy at the bottom?

A sphere and a cylinder have the same r;)dg(e lpdpe*n9 vdv1adius 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? Whdp;n91*pe vd)dvl( geich has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are cox03 eayqdh(5:mz,8 exuq w6brnserved. Yet most moving or rotating objects eventually slow down and stop. Expl5x uhqqb, ( wz80dexeym:r 36aain.

If there were a great migration of people toward the Earth’s equator, c1 ucv3 /ftpyj3qs ug4 54bva;how would this afqf54ygc3bvtju p 1sv3 ua;c 4/fect the length of the day?

Can the diver of Fig. 8–29 do a somerhp q8.xl ( pp.,mcteqypx4+ 9c)r q:qusault without having any initial rotation when she leave . 4e:+qchp8qpq) xlqt cp,m.(uxyr 9ps the board?

The moment of inertia of a rotating s45m4r q gqnsc ow7vxj(+)g0qwolid disk about an axis through its center of mas7r +4v) gcwgm qo j0wqqx5(4nss 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:zle s aocrm886n+z 3o outstretched hand. If you suddenly drop the masses, will z+z8 3rsmenl8ooac:6your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However g73icy * .gn(ghbuq7j, one is hollow and the obgyhi c7gg(j*u .q7 n3ther is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotavvl 6q.14 hpyxting on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, l1p x.hvvy46qdescribe 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 q qig8,p7e (gaz, zz(7xvoda4large freely rotating turntable. What happens if you wala74exdvz( qzzq,ig7 p ga(o8, k toward the center?

A shortstop may leap into the air to catch a baxh1ygb-)w 3cxll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice th1w)-cb 3hx ygxat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of cons7x6 3 pgkz+wnsuxo1 avj :a.g+ervation of angular momentum, discuss why a helicopter must hav+x7n+ uza kgxj36v :swag .po1e more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following 43o xcqle+ki2angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate4 ( )0adlrn-e wewbo q9exr)j hbbt;.4, using the information ib)4x0)9 hnb.werqbewlre-;oj( t d4 a nside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from 9tep p7xq -fb m,at3x.f;)flc Earth. The beam diverges at an angpxxme;pf7t,3 tq.a flf - 9bc)le $\theta$ (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 o*u7 vrhqcwudm92q c,4u )2ra 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades sld 2oq7ch 9auw*mv uur4,c2r) qow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to atz;pdqrh, .+nnother child. If the ball makes 15.0 revolutions, wha+ q h.nr;dtz,pt is its diameter?    m

  A bicycle with tires 68h d 51e;nrhd.zfcv6l;9x yd0 n cm in diameter travels 8.0 km. How many revolutions do the we;hh vz5x.n1l nc6d ; 9ddf0yrheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 psu+ jfv)iu d32u1.cwrpm. Calculate its angula1s3wipdc.ufu 2ju v) +r velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig.og q ;vr bjh1ij-6bj:d2(/joj 8–38). (a) What is the l j;ojq:o r2db i/1gvjb6(jhj- inear 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) in its orbit around the;1e okw0e)2m ho.v nazmn71nbqt.n :h Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed utb2,grg5f -rc22l ytk4iq 82akqo :eof 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 particlevf 7p9c/ctio + 7.0 cm from the axis of rotation is to experience an accelerationf/ 9iv7 c+tcop of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates b 7p8 xeeph.5e7 ucwt)uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine ephpe7cuw5 tb7.8ex)
(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 gshh:w (vu0kmcsth 1y7(n- ,n$R_1$ 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 abo,2s()t l 3ksitx0dwf sard 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 interval. The spacecraft can be t,lfs)d( 30iwksx s2t 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 acceleratesbf5jwa op,/hm zde8* oc 26t5t uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? o5 ptb,*faje/c 2mwzh86to5 d   $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 sodzms tvepf/(*2 1(za. 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 hia817vhx ln;z(udkj 8l;ou;(x f* zk cwghspeed jets in a whirling “human centrifugev;z alkjfhux*u8;7 o lx( kczn1(w ;8d,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm iuao h-t p2wnuyb 0o/+o:-i0 hqpf 1e1pn 6.5 s. How far will a point on the edge offbo hueyq0p-tp 2opw-u0/ 1h1 nao+ i: the wheel have traveled in this time?    m

  A cooling fan is turned m/9kbro;h mv ku3ip 7*xgii9,off when it is running at 850rev/min It turns 1500 revolutions before it comp/o3*,g i b7hkm; r xim9k9iuves to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces icu5x ) 7efp o96zg*v2owbwm+ bts speed uniformly from 95km/h to 45km/h The tiresxw*g bw+5uz7 o mec)v6fp 2ob9 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 revolutiv1xv69dpsezpu4ijh )bdc t3w.(:/ op ons as the car reduces its speed uniformly from 95km/h to 45km/h The9wo3/:z udx v4phpvj b1.te dcis6 (p) 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 pedaezpl x8(ug ;/3ylq(eg l when climbing a hill.g8q;egpxuez(yl l3/( The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the endu svw( 4ghw81ql :8bqjd0)eb9nyt,h m of a door 74 cm wide. What is the magn,9h:vgqywmlq h (dun401j8sw8t b)be itude of the 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 torqu1 cv dwfmh935qe about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.c5q m 1w9d3vhf4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the enii k5)( h6trqn*th. 5i ttgom 6alum0-ds of a massless rod which pivots as shown in Fig. 8– inu qio56mh(r5-k6th.tl 0)*agtimt40. 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 engine ch0 5b++z91dx udb-imqz 6ukdrequire tightening to a torque of 38 dm 6qh1dz5+- b 9xub0+k iducz$m \cdot N$ 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 w u4yru0 :d* eq8tx:ofu*shjj.hen the axis of rotation is through its center. . j x4t0*:e*:hsorfudjy quu8   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ipmwcnr+1mb , *n diameter. The rim and tire have a combined mass of 1.25 k+n mw crbpm1,*g. 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 rotated ipzpdiqht 6ghx: s9ry* h 93u5:n a horizontal circle of x6q3h ur5:hhp igy :t 9s9zp*dradius 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 6n :9vrkfxkh 84ckr,y’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwee v,fyrr: x49 kknk8ch6n her hands 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 the arr (hz:1am 3f kwcexr.k/dt4/whay of point objects shown in Fig. 8–43 abou.(ecmha z:w/f tkr1 4/3hwkxdt
(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 whglzlw h2rr9cd ,71*)es jstd3ose 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 corrln.dfqe2aq52 ect rate, engineers fire four tangential q2 da5 2.nefqlrockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniformhm0e1c t wa89v cylinder with a radius of 8.50 cm and a mass of 0.580ta1chmwv8 09e 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, accelera)q 0*ommx7 h ckttqg9a6xd5y0ting 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-driven merry-go-round and is abl+lq- swv 9k9sagn.1ece to accelerate it from rest to a frequency +lwe.kssn a -1cv9q9gof 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 is evelh6h. qr (f(8ypsnqh 0z)nz1kz4xou+2ki/t lntually brought uniformly to rest by a + /(py h4f1qkzt(lo0 .u)lhx6nzihsrz q82nkfrictional torque 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 acceleratesn(qfbrxt8 * -x a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg b/w0j8, qdl d-t .z hza;9chhjiall is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the tricepsa.90z- q8h/zlddwt; ji hhc,j muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown inmc9 gvwtno:+kss2k+h g9dk (. Fig. 8–46+gow.k ks shd299g (+tkcn m:v.
(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 consist eh52rt t9a0a9itbz:hs 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 withi5 crtu+;zsrp tr m:o9na27+rqn four full turns (revolutions) and releo2s:uptnm;r5r c+zq a7r+t r9ases 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 mc3ma;p bs7s72z dl x*yoment 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,thlm 4l2td i. 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 rollsc-m tw-dq y-v4 without slipping down a lane -ymd -v-qwct4 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of theazw8u+ iw.2xuka.i);ocx2n3 c0iv zk Earth with respect to the Sun as the sum of two terms;ia2x.zwnk0ia.)ouk8z2u cxi+ w 3 vc,
(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 kv :q86d(6 hfv3 vbzanu1och,work is requir,cbq(vkhh86zu anfo dv63: 1ved 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 mas9,ef*s 2 fwphjs 1.80 kg starts from rest and rolls without slipping down ah we*,9psj f2f 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 toahs at4r7 sly,z8)t/d fall and its lower end does not slip. What will be the speedsh87a,s dl a/ty4 t)rz of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0bsy r7apa8*,vqi +9hh .210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular sar97 *+p ,hhv8bsiyq apeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg ydw f.)jov 1uv2)ly+duniform cylindrical grinding wheel of radius 18 cm when rf2 .1+jvu wy) vd)yldootating 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, op:yk:3 +bhzhmn 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 decreases to 0.8 hb:z3:mp y+kh$rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in F1lak*o 0py ,apig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when ch,aolpp0*a1y k anging 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 init4su gln u.z k2rnu/-0lial rate of 1.0 rev every 2.0 s to a final rate k- nz /0.rnu2lgsu 4luof 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a bo2uk02qq(xo frequency of 1.5rev/s The wheel can be considered aqox 2ubk02q (o 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 momentum of a figure skater spinn7h es1ak9x. loing 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 ofig h6rrole1 .f. a8zu. 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 fpyfnww44 ghkv hc/4 844xva,disk of moment of inertia I is dropped onto an identical disk f hk4 ww44 4 8pgy4nhc,v/vaxfrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns af7djhqec. ztz1(skupue )rq 76 t+9i*,/s b pct 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 rotatin hub. w)bocp8 ):jt;iyg merry-go-round platform of radius 3.0 m and moment of inertia 92bh. wc: o);i8)yu pbjt0 $kg \cdot m^2$ 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 isnw:-z9wk +an 9hfk(x yv((h yh zvpy+/ rotating freely with an angular velocity of 0.8 v(n:- hnyzhapy+/k+ wvxky w9 (9(zhf$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its m .i1fjtpaq*l*e,4m u zass in the process, and winding up with . mzu, i*lejqa*f4tp1a 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 in excess of 16pz2*h+2.l zpv .o,kmsevw awio,d:v20 $km/h$ 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 rftg /cp +, clgl8f50spv7z an;uy5;f$ 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, .sdn ;5gwoor1 yw3a87p8aa j athat can rotate freely without friction. The moment of inertia of the person plus the plat 8owajarpy8;n1g3sa5.w7ao d form 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 1pbu lfio0 ,6zthe edge of a 6.5-m diameter merry-go-round turntable thi1 plzuo,b f06at is mounted on 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 o w48u(0:+mjgan/oxmrm mq t- sa6 n*ithe 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 tsam40 r8on:w -m( o6tjq*u+a/m xgimnhe 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 al4: k:5cl miwshkjhn.4ways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abouthwkcj: n.k4l 4mhsi5: 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 rest at a rate fxmr)15/uba9z) s(kz e azp v2of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.2km m b.s+fm po8lgqyv3 .i;*.y0 m acquires a rotatimsmgk.;lobyvq . m8f*+ iyp3. onal rate of 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 ofim2)cyye5 yzf1e.zn*ss v +k6 two solid cylindrical 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 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 fr2e)s.z v1+y* neicz6 s mk5yyfom rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angxhcc0fou0lxtg5j ; l /) m..dhular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times otf*b *):lm*rniot*1nie 6 bbas 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 in the process, what would its rotation speed be? Assume that the s )* nt*n ibibr1tomf*oe6b:l* tar is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for ad ci1(5tf*no (ea 1ohk81mepa low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such ac81(f(o i ot*en1mea1 dap5hk car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratesvh u)(r k0 vzj jp125g7c-jcob 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)51 4hm2nszzba7p(a6oop sr o - 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., we ignore fsqb wkk3a+ 31rriction) about a hinge attached to a wall, as is r1q+w 3kakb3n 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 ve)vxu a j9z 23in-jh,fzrtically on the floor, and we want to exert a horizontal force F at its axle so that it )u-jzj x29h,3 zfavi nwill 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 o-, i t79vy+ gnpwmg*82mdw6k e/ ctecon a flat road is making a turn with a radius The forces acting on them v-t9gdiec*pmy7 8n kt2+/, ewg6cwo 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 begins pm: s q,akp ao of-.yth-c*7p/whirling the rock in a nearly horizontal circle above his head, accelerating it fro,h:ospyfa * c--tp.a/qmo7 p km 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 come from?    $m \cdot N$

  Model a figure skater’s body asgnfzsro*w z*+bdx e e 1-/z 2ss1c:uo6 a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body *+xz sreoezcz nw*udg/sf1 o :b2s-16.   

  You are designing a clutch assembly which li:6zl /:/vqxs*tca rconsists of two cylindrical plates, of mas vsx/qal/z6:lc:t *ir s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra:j0b j9 (vzbjddius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that:b jb0jz(jdv 9 the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumeojgw nkkv)6/ 5 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speedo 5oh+/tgxzb*/d m4qe uniformly from 90km/h to 60km/h The tires have a dimbx /5 /oo*hd et+4zgqameter 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