A bicycle odometer (whz 6*s6va +vsw2cga kt3ich measures distance traveled) is attached near the wheel hub and is designed for 27-inaz *6v3wta2k 6 +svcsgch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant an-h1+h5z*.kzs jro uy ygular velocity. Does a point on the rim have radial and/or tangential acceleration? If t51j+z-rsz.o y*yhhku he disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angsae oab5j, i*hebws. , xs0*dhd4m:e j3fz 8n)ular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fox0fej9g 4- t2xo3d- t5b:unt; zvihkou nh9/trce? 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 diffic*phctqw5i1 os eiq0:,f hz6pis0- g ;oult to do a sit-up with your hands behind your head than whenhi-0 0wtqhs:sgzq6* fio,c1 5e;pi op your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal meb y4xe*0 nvlr z*oc:zqo 891cranks. In we4xn*z8q1rv 0bloeoc*z9 m:y hich 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 small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to ru. iuuz,6ewm z,4fe *nwn fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rot.,,fz n6z iu4we*wu emational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b +iyxb:0*;ksiczbzk .eam?

If the net force on a system is zero, is theiddzsqco9q4fgu2 r595k*20 qe1mp j gg a w91x net torque also zero? If the net torquea qg2qcw10*9d jx9q5 fi142zs kuoedp rg5mg9 on a system is zero, is the net force zero?

Two inclines have the same height but make d)snxqr(3 pfbis 9(o f4cvq( f9ifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline c(b (ffxp)f9 nv qq4ri (3s9oswill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling ( p,fquk* +5mdget5kp1p qyj1 /from rest) down an incline. One sphere has twice the radius and5fpdqq/t51 1pkuepmk + g*,j y 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 cylinder have the same radius and the same mass. They startoz4):i x2etd-kq;4ntjisiz+x;t5 h h 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? Wqoz:s 2ti ;diext)hhx 5i k; 4njz+-4thich has the greater rotational KE?

We claim that momentum and angular momentum a/ hx5)o n8jmiad ieb50re conserved. Yet most moving or rotating ob8/ i)oj0i5 m x5enabdhjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s cr5 -5-.hs2o9kkgw2f eya zomequator, how would this affect the length of the day?g5a sk-w2-eoyfr 5m29kh. c zo

Can the diver of Fig. 8–29 do a somersault without having any initial rot+33pq/ffhwta4 bh n4yation wf/+4f 4 3nwq3ybthpahhen she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center fhc tn1gyg41) of myc)gfh g11tn 4ass 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 outstreo z gztqx6q8+(tched hand. If you suddenly drop the masses, will your angula8 gq+qx(ozz6 tr velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same m5pb ug.h 5miv 5tkl2s-ass. However, one is hollow and the other is solid. Describe an experiment to de-lg .v5k5bu25ths pim termine which is which.

The angular velocity of a wheel rotating on a horizontal axle poipsfm7mdb znzii:/4x8d+pq yq nv.)1,nts west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangentidbfp,m+ispx7m:8 )iq1n zv4 yz./ dqnal 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 turntabw()l2* x f0akgnk+f xtle. What happens if you wa0 x fagt)wf+kl*n 2(xklk toward the center?

A shortstop may leap into the air 3 oa29vmgy9rmto catch a ball and throw it quickly. As he throws the b2rv9g39mmoa yall, 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. 8–36). Explain.

On the basis of the law of conservatioslp2iyb (n (xqulh +*agc75 8 t go(d7v6uoh1kn of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Distcdq6y72 aobup* is5u8lxhk(n7g1(h + (ov glcuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians:4b7 stw( 3i :nmjw /cuo-1;hx1 rxbpyp (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 coinciden1jao 59d53tx smry/ydv8s 2cace. Calculate, using the inyosmvd2 xy/ 3t5 5 jra19acsd8formation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fromvi ;8s.7t m:(sfnjmkhlhne:6e r+ 7 hs Earth. The beam diverges at ar fs67mss: 7+jnek (vtlm :ihh.en;8hn 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 ratec m pfq* tdchs,ce1:5z ,0vgt+ of 6500 rpm. When the motor is turned oe tdc+5g:fcc1q v,0mh*,zps tff during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes x/q psxn :8x+eo9 lvm xp*0dl0cpb;;u15.0 revolutions, whxx8x;0lnpo /q xu+ e9:vs *;bdp pl0cmat is its diameter?    m

  A bicycle with tires 68 cm in di,ye+or9 ebwb0lg 0 ys)ameter travels 8.0 km. How many revolutions do t , o0 wr0ys9lebebyg+)he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates bq6ohunb-cxrpk c;:f(b 999o at 2500 rpm. Calculate its angul( fu cbb-9hcpr;qoon: 9b9 k6xar 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 onwe+5ce 1 fop9qe complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear q+eo pe59f1wcspeed 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 s7 uon6itzv p7i8ko emd(f,w* fu r8*q+v6fh1Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed off rg36f 8(xnrp 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 pamvq 2ys -y2 /iegdjk48rticle 7.0 cm from the axis of rotation is to experience an ac /sdqy2 ke-viym8g42jceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acrx2hz (; *rp( m;wmrxtcelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 sr m2pzw xm;h*rr;tx((. 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 radiu2gcqru4(8o ly+e x4 wrs $R_1$ 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, astronauzhi + fp(7lw3wts 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 interval. The spacecraft can be thouw (7liz3pf+ whght 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 f(uwjdh;)m wofi6g- ,g rom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn ,(miu;- wjwhf)ogg6 d in this time?    $rev$

  An automobile engine slows down from 45002r9mi:f gq+br 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 flying highspeed dzv v2asen9efb.a,w7u* :t;sj* wvo jx), u2mjets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionxu .9b* jo:wvf)m, awd;2vu*7z tese2,jav sns 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 acjp l8edx.xh- 9iasi 0 5cbvt4,celerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled854i9ldp habex 0-xjcs ,ti.v in this time?    m

  A cooling fan is turned/t jjth;;.+blwy0jsm off when it is running at 850rev/min It turns 1500 revolutions before it comes; .j;bmwtt/jys+ h0l j 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 makdzqwu e:,yil3e1*-4ort0hn m 43 cphre 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires havwe-4 4tn,1yrr 0m:izdul 3ohpq3 hce*e 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 reduces its speed wg/;8;5bu:;/dpfap mgtlz dluniformly from 95km/h to 45km/h The tires have a diameter om p zwdlgt 5b:pu8fa/;/dl;;gf 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing+)9 doy ljgizy bh-3:q a hill. The pedals rotate in a circle of r3 l :9)jhy-qd gioyzb+adius 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 fr./2x4 t2yj8q nz xgtdoor 74 cm wide. What is the magnitude of the torque if the force is exerted82.qf4 2 /tytxjgxnzr
(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 Fitb b(6p02m f mpz)ybs(g. 8–39. Assume that a friction torque of 0.4 m6((pbmt 20 y)psfbbz $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a y)dmg odw8 m958gku4 dmassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal positiowoyg mgd8md4 89k5ud)n and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require rxn ;a1 ixw+v:0d)inatightening to a torque of 38xdiv 1wiax)nra n0 :;+ $m \cdot N$ 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 the a:2 ()*cts;qz1ov sdagte zg i+xis o zt1sgogczt( ;2q a)*siev d:+f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6667g,)4ub. 1)tytuebgor pc rt.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Tttg,y1cgr.)o)uebb6 t rp u4 7he mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, l,az nnkt* b- y18o-emkj 6hyv dp30fe)ight rod is rotated in a horizontal ci01y-)e8d yn,kp*n3a ekb-t hmf ojzv6rcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at cpglau(w8p(: a(yp, :h m) oxwuonstant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totamw(hxul:p 8yppo()( : aagw u,l.
(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 array of point objects showbz p,;nqy.x0k-xkk /2vi(inin in Fig. 8–43kx/n- y ,qk i (vbp2z0;kinxi. about
(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 total mass iz;vhw aoej(+(-*irc/gc: anmo f 82wis $5.3 \times10^{ -26}$ 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 corre4:)srg-jepkdz8l g/ ect rate, engineers fire four tangential rocketg)4zp8r/dj ge:lske- s 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 uniform cylinder with a radius of 8.50 cm and a mass of 09w ;vf nag884ggrq: ksjz7z7n .580 kg. C;n qgj8:r7 87n9gz4asw fz vkgalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest ton 3nfzkv 7*s)n*bd2qa zit7+o 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-d +r 9 zt:dp47njij/ljariven 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 children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to j/ t7ij:z+r9pnaj d4 lproduce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1u-*ix jy/x3u z0,300 rpm is shut off and is eventually brought uniformly txiu*/uzy 3 j-xo rest by a frictional 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 accelerates a 3.6 ea.4y tpvo3e6-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 ball is thrown solet.9 ; (ynfoec q dz;.3l 0ehqaz1ca.uoly by the action of the forearm, which rotates about the e3z.a9ocy 1z(; fl0 nc.e; at qoe.udhqlbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shownpn*z /(r9a7n;tjk ms h in Fig. 8–46na(m*; p 9shrt7k jz/n.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse ,k qjlfj.cn62s, $m_1$ 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 accelo swlt l6yvt6a)d z,ck(u( c1,erates the hammer from rest within four full turns (revolutions) and releases it at a speed of ,act s,o( tuyw z(l66)v1klcd28 $m/s$ 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 4hrcwbt5xe 2n1un sm*s5-/ , ccw)uas moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine yrle02*p:i cv d-- seo,nlhq0 develops 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 c2/bs46 xws pmbm rolls without slipping down a lane at 3.3 4sbp2m b s/xw6$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sg+l+nr -aq(;gd cyug8un as the sum of two t rcd-ul ; gn(aq+yg8+germs,
(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 n +r,. fep(xth tzm8/peet work is requir p,heztf t.8m+(x/per ed to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg staz:za),2ld ns4 u8 lb*fzyftg2rts from rest and rolls without slippingyzlt: *lb),a fnz 2zu8d gs4f2 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 balanced vertically on its tip. It starts to fall and itx*t ot r923r0jgse 5sbs lower end does not slip. What will be the tr5x3b2 jo0rg 9*sset speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mom,u4hl)okn py fsi9j67 u0klr4entum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 k4nrl 9ijol,u 40fp)s6ky hu 7m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cy07 e.yc 3lmv iq(tfb,ylindrical grinding wheel of radi3emqyv7l c( tfb.0 ,yius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at b glr(h k,)s:ehis side, on a platform that is rotating at a rate of 1.3rev/s If he ra:)s, rebkhgl( ises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one skkb 6 1gylpfcc84z 3n3hown 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 po y348lk1c fngk cp3z6bsition, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from a 4ic cs(.tpc:xn initial rate of 1.0 rev every 2tcs c p4ix:.c(.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cen2 * z..0hprc,psmtaw vter at a frequency of 1.5rev/s The wheel can be considered a uniform disk osr .0p.*ph,camzwvt2f 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 , j6rqw6l1vh(r8kknet.h c9n skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of tehxv1k( /zz0,w rbdg :he 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 moment7c.6rihhl3gw e p n34e of inertia I is dropped onto an identical disk rotating at angular e.7 4 pw3h6ehrnl 3igcspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at yjr:6+qo(/fif sj33 rn pjw6h 2 0rptt2.4 $rev/s$ 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 rotating merry-go-ro5rmwqq; :3 xieund platform of radius 3.0 m and moment of iner:q53i; wxemrq tia 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 v)1 rg3oj 15 bl,mltugdelocity of 0.gt l1 bor,m1lu) gjd358 $rad/s$ 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, gdmjjug+h+ ( 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, what would the Sun’s new rotation rate be?(round to the nearest integer+,h+jj(g d ugm)$\approx$    $rad/s$ (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 u81hqvaq8 :xl-mhl2 wod5r 5e 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 b/-,jamxzf z53uh3 ga$ 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 plat ew7t xb) 2;n:d waxmt53b9eqqform, initially at rest, that can rotate freely without f ebtxxnaqd:2mw;t75bqe3w9 ) riction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person statgw. ,d h l -h8wsfwi)if nrwsm+84g-7nds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on fri hw.tg s+8w )-riwd7wifsf8 4lngm,h- ctionless 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 or z eiztm dq) j0d(,huuo-8f(:f the rope lying on the top edge of the sf-etm:z zho0(i rdjd u(8u,)q pool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sbc/ad .1c y mz8jywma xj01 )zo.fyi2 c6lu83aide always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moa 088m 1 jyf1 c2zyo.m6bwjydaiac3ul.)z x/con as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raoejn jgcs/sd1xynn9 k*-l em1 ) 56f/yte of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of rxyi+wsh96z ni3f.cv z:7h p8cadius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motohwxcyi z7n 8+vspch3. 9 :zif6r.    $m \cdot N$

  (a) A yo-yo is made of two solid cylizhqdlqno*gvp u/* 0uamw5 q9q+.h o14 ndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined ho9q4 *qv+l5w*o mq g 1.uaqud0hp/nzby 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 from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular 8uewuiz8d 68 tspeed 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 ouriup 9c jahr9;c3u6o2 5i ew)ut Sun, but with 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 in the process, what would its ao3cir2h5t 9 c69iuwu)up je;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 6xysoa b(2j62 imxt3k automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and ko 6(m62j i2ytbsx ax3should 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 illustrate* 3bie;drsz h,s 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 rolsd hhpv/) q(rc3 (jlfx9q/mw.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 and length L can pivot o+h//qb01t wlq bcwt9 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. Assume that the force of gravity acts at the center of mass of the rod, as so9lbq qtw hb+c/0t/w1hown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we wstb1 o1p8+7aoypq0 r0uto q n6ant to exert a horizontal force F at its a6sq 11t oo0u7yt+p bnoq8pra0xle 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 tuda4:) l))0th+ d hseb5lej7qt b4j yvorn with a radius The forces acting on jd+th)oablvyq4 :lbedt)s4j 75 0h e) 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 lpp 5xv )1uab1 ;sify*w9jg,btength 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 requirewp9bvbf5txs1g*) u ,ip; ayj 1d to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and h*ag4jxrw/o6n er arms as thin rods, making o g4 a/*xrw6njreasonable 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.   

  You are designing a clutch assembly which consists of two cylin z+b*ot;lb:* tnul x-zdrical plates, of masb*+xtnu z-* :;obtzll 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 radius r+ ix p7g2h8otk rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble xgpi kh+87 t2omust 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 ashl, hg*d0+ uk3qot rc+sume $r\ll R$ h =    (R-r)

  The tires of a car make i*71eidp aj4n 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a d*71 ijn 4dapieiameter 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