A bicycle odometer (which ;,q o 2korj5ydnp 85jtqm6p: cb;cbu /measures distance traveled) is attached near the wheel hub ando/,morcb c ;unjb8 pqj2yqpdk 6t;5:5 is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. d6l2/ct mm sm3Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration tmmm6dcs23 /l change?

Could a nonrigid body be described by a single valug -z2-ry,(n juec1subi 73 koee of the angular velocity $\omega$ Explain.

Can a small force ever exert a gngg:v.- vlv 8-xt,ln(xy m -ckreater 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 behind your head than whn3 vwjr9tmrk4redb*gwhk e;y 8 q6 ;99en your arms are s ekn98yrjk;w3v6 4;dt h be *mrr99qgwtretched 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 t b **oz3w;iotbu3 61xb2rda:fn)wljm hree 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 pedal, a small front sbwu 16ndjif3w2ma o b*x3:obt; * lrz)procket or a large front sprocket? Why?

Mammals that depend on being able r3sn *f wf ts 4o,g/v3.fp6pjgto run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dyn/p,v.w*6g t4 gf s3jsnorffp3amics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Figt(xgrk ny1t+u4zbm-*kbx/ 0 a. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the nev/hpo e z hw/.mf-o)6kt torque also zero? If the net torque on a system isvzp/k m6.eho o) wfh-/ zero, is the net force zero?

Two inclines have th6 w7 y:apzca89dp(pq fe same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greayqa8: pdf9a7(cp z wp6ter? Explain.

Two solid spheres simultaneously start/ ry4s0yy4krm pf7.i* zui2y b rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic7b ukp0 mf4y2yrry/z* i.yi4s energy at the bottom?

A sphere and a cylinder have the 9 :-0epzq olrgsame 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 g0e o pz-l9qgr:reater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or z6heofu7o 1;ju*ym;j rotating objects eventuay ; u67;oz mjujh1oe*flly slow down and stop. Explain.

If there were a great migration of peoplempyygh2xc2 *) toward the Earth’s equator, how would th2yxmc 2p)g*yh is affect the length of the day?

Can the diver of Fig. 8–29 do a somersault with ;tph9d.wpqy-*x 1odg b/(zjekmkqx;dmt8++ out having any initial rotation when she l1q;opxptm. 8 jek+bgmz+/9w hy(dqt dd;-*xkeaves the board?

The moment of inertia of a rotating solid disk about an axis through its cent8 v k+7vt)e+bl2hj zh7tqx8koer of mass is z 7l8x v+e87jh)obthvqkt +k2$\frac{1}{2}WR^2$ (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 ri07s5vmom:9.o 8r 6uka, qe2aoxf ss kotating stool holding a 2-kg mass in each outstretched hand. If you s9oemxr m6 osk,i us: o20sf5a8q.7avkuddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical arbf*xu c0ua.f0g wnx7 -q ;)nrnd have the same mass. However, one is hollow and the other is so*b 0 g. nxaxw ;)urrfun7cf0q-lid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poi b(xrix4 y;mc-9hhjy/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 tangen- m xyrj;/4ihy(bh c9xtial 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 turnt,3qa 0rjbfvd) able. What happens if you wvfd)ba 30r, jqalk toward the center?

A shortstop may leap intz+(qxs33u ev.arel3,t44 zgfqt pc+lo 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 quickly you will notice that his hips and legs r4+z. tprl+ efxelvq3g3saq4,u zt3(c otate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of co dhffl seys,6 uz6t:xu+ .72qenservation of angular momentum, discuss why a helicopter 6y.2s sd+7fxf h lq:teuu6z ,emust have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesc fx3 :p-)zsly 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. Calcula3+u alnf)0y uvte, using the information inside the Front Cover, the angular diameters (in rad+)fnv u 0yl3uaians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directepv p)xd02f;hyb(8 vjoncn 5 6wd at the Moon, 380,000 km from Earth. The beam diverges at an ;0pnj bxc2 vhwovpyf)n65 8(dangle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Whenb9 zqv4(:i 9vck1 n: unh7nozo the motor is turned off during operation, the blades slow to rest in 3.0 s. What q b7h:v ozn n (kci149uvon9z: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 /l+ :fz;apu 9nr6 jwu9)mj(b nhv;chy8 bb3hf makes 15.0 revolutions, what is /3zrnb9vc(mf; hhpjl; any8ubh fuwb9 :j6+) its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do 5 vzoep5xh)(th:y htsqii n)oc,t:b61v9j7 q the wh5 pt9qvtq7 vho() :hjin,5 ioztcbx) 1:sy6e heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calswzyvr9 rx)03 culate its angular velo vz y0)3rwsrx9city in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (5-+ a4psv77abf vwkv r7rm6qsa) What is the linear speed of vss+7makr5f r vw6p a4q7b7-v 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 velocitky og2x2nnfw3p8 qx;.3bv,w ky of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point dd8/.79gew; mv x dutm
(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 7.0 pa;i5khkwwff qin(h(qgy473hvw + 5, cm from the axis ofwhwg 3ip5;5, (hvwqy q ikf(f7k4h+an rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratef8jkc;frngv9/r+q wo q ;c4( y wf88sts uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determinwfk s;roqt+cc49v8nfgf y8wj q 8 ;(r/e
(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;j we :dzwm 38zxx7*vq $R_1$ 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, astron)7 ah-yu xb /ebh+-dmdauts 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-mi /a h mby-d+-)dhuxbe7n 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 rest to 15,000 rpm in 220 s. Through howzdni/g/z*ur:r6 jx(3zek0 sn many revolutions did it turn ine 6 n z(3z/rkudns0xrig/ j:*z this time?    $rev$

  An automobile engine sg2jz td+mt (;klows down 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 flying highspeed jets in a wh zlx tx..gh1e .h.bxz uq5s2.pirling “huma .xhphqz 1g5.tb..e.s 2 zxlxun centrifuge,” 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 rtx : vlxwzs*8k *:v3nppm to 360 rpm in 6.5 s. Howtxp8k:s*3 *xn: wzv vl far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned ofcoqru / h (z12:cul6jwf when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop. /6wz :2hlqu1oc j(ruc
(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 un0euyqcr6e -+e 2s lu+niformly from 95kse+0een2yqulru +-c6 m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unsur+*.qnq l-*x7x)e8gmdrq +hz +y cliformly from 95km/h to 45km/h The tires have a diameter of 0. ) hd+gqyrclu-rx*xlqs 8q+zmn+e *7.80 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 whes8fc:5yi : kid+mkx4j6v,up ln climbing a hill. The pedals rotate in a:kki:,5s +8 vdyixcf 6lpu 4jm 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 end of a door 74 c.h.w): va mar)hbg1mr m wide. What is the magnitude of the torque ivrhm..ahbwga ) r) 1:mf the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel p)pg6u+h4 -ej+u lkgv:c pgy8shown in Fig. 8–39. Assume ug+gpc kg 68l-h :+uyepj p)4vthat 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 ends ofxdfefqv*f -zgi)q +a7e z a-h7b:vx+4 a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horiz :xbqv +evafez x fqa7*hz-d)+f- 4g7iontal 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 require tig,+,yi ;l-reni4gq ctzhtening to a torque of 38 ti +nez-,l 4;y,q igcr$m \cdot N$ 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 zv-kl:xn/x 4x18248ka st xbmbz xv7a.648 m when the axis of rotation is through ita44 n :v/x xtzmlvas-bx82kbxk 71 z8xs center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whe7-mk8 ivulf 6e4uaqf5el 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can bele4 76umq -uffkv5i a8 ignored (why?).    $kg \cdot m^2$

  A small 650-gram balllka9 *v/gxi:l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 :lv/ a*kgxl9im. 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 constant angular sp gow,(z e7 /s(.6q,j2cmnlysuzf, ccr eed (Fig. 8–42). The friction force between her hands and the cj re snz/owz yquc. m(g, ,27s(l,6fcclay 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 o(/b qvkkg( 7xioda-w) w f9wca -((bwrf inertia of the array of point objects shown in Fig. 8–43 wfoc( k(7-(a/dgbw (r i)bv9w wx -qakabout
(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 consi29-dqm5z 6pqmt key ;vsts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satio 8jk4 qw9p2ot.-ocn ellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 ip8jon2cw.ko t-q9o4 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 8.506h epe+l.eyl ) cm and a mass of 0.+peylel e.) 6h580 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, acceleratin 9shxn-1d3og (r,hek y rqsj51g 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 tange9r8sjo f*v u ;ri ehcboy:-+a3+6uqdjntially 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 children (each with a mass of 25 kg) sit opposite o8fbi9*vra3c:6 jd+;qsoe+ -j hy uru 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 r(:z.5vug pwj1l wqs0 lotating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 wluz 5lj:gs1 p0.qv( w$m \cdot N$ 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.8 * zt,sps9,ylj joqa 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 ty e.1x5keu;w ghrown solely by the action of the forearm, which rotates about the uwky eg ;51.exelbow 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 shown in vc j*p kkl ;a z)bd 14kfd(bl-y1iun2*Fig. 8–46.*fp14(y1cudjnb; *k i2-)kbl vdakzl
(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 consistp1qn pyepf z0c 95c6/ws 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 xl7 .ohtr w.ie.2r 3hc from rest within four full turns (revolutions) and releases it at a speed of 28o 7x. ech rth.wr23.li $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of zsb3e 5v4p i*f$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqy. 37urw-zje zue 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 radiusx s2u ,.klwem2u-jjs5thx; u2 9.0 cm rolls without slipping down a lane xt. x;hm2u -s,w2ejuslj5 ku2at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofp 3iel. gzc cwy j,u2g,8pb-o 7nc,4te the Earth with respect to the Sun as the sum of two c4 g nptc zjy,- cge,bpi,l27.3e8owu 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 of 1640 kg and a radius of 7.50 m. xtoqwc(v n u4,9 dzj8;How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume t(8n qc;v4ud9owx ,zj it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.x1 yx6n x* 5mesu8e9ndz/t)xf80 kg starts from rest and rolls without s6f 1tx m)exy9edn8n/*s xxz u5lipping 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 itscp o:u 0nv8 zl2;l0b 6vmegep0 lower end does not slip. What will be the speed of the upper end of the pole just before it hitsc v0e ;68lemvo 0pgn:2b0uzlp the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentumt :3rjngl*ecvn3 4zss yn+k: 4 of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angu: ks33z4v:ngtjr*ly4e+ ncsnlar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifoo+ 2zziveb8*h- -( ;fx bvjewygy(nw +rm cylindrical grinding wheel of radius 18 cm when rotatin-xv 2 ((-ow*fiejz yhwnbev+8zbyg+; g 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 hisz4/ /ufbxz w/er4w 3 ws8.p jm7yestl3 side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a hou/4wm83b7 ss rjpe. x/efzy wz3tw/4lrizontal 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 shown in Fig. 8–29) can reduce her o7xkl)a h71eqmoment of inertia by a f1a 7xh7qokl)eactor 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 position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can intldsc 1eom 39v5li-o2crease her spin rotation rate from an initial rate of 1.0 rev 9ilseo1l3dv-mtc 2o 5 every 2.0 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 rog)tys+y9uhv8iz q ;my+ m b5g(tating around a vertical axis through its center at a frequency of 1.5rev/s Thestmyib vy+gzy()9 5h+8qmu; g 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?    $rev/s$

  (a) What is the angular momentum of a figure skapr-ze4 3 1brjzj2fv3gter 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 mome -vrxbh9* 8)v4 i. du-8m hprtexbvz-cntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of momelnajs 4 ;5ekawb2e67 ent of inertia I is dropped onto an identical disk rotating at angular speewj aaee7ke5;n6s2b 4l d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 2pf4,b .xmzefp8, lbm $rev/s$ 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 cente/6 v.bjp4m99q jrvugfwrn+4zr of a rotating merry-go-round platform of radius 3.0 m and 9rjv qubjvg6m /.+nw9 z4fp 4rmoment 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 1hg j5m2oqsn+ xvg.hrn -o,b 5rotating freely with an angular velocity of 0.rg,j 55bnvno12 smhx-+qo gh. 8 $rad/s$ 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 vc.sq e1hw e lc8 ul;f(5hj.8ba 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 rosu;elhcw. h e(q1f5cb . 88jvltation 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 excesbbc 9g,pr /zoy2zk5n 8s ab7:ys 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 2y 2ltc0kv.pr$ 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 cang);fnxom08v/f uwa h (8 l;jtk rotate freely without friction. The moment of inertia of the person plus the pla /;vgojxmlaf0nt;8f 8 h)wk(utform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the egba t 91 auf0e/9x,oj j:yh5 wdz4c7cedge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 7u:f whg0z ec5 99t1xaajby o,/e4cdj$kg \cdot m^2$ 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 edgm2o;qpepr:j( s. ( qnd of the rope lying on the top edge of the spool. A (2rpjoqm :;s(.eg qdpperson 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 tzb he/8by0bswv 84f. such 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 anf0e8h.s/bt 4yvz8 wbb gular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ragw nelw/o(zh5( k,rj8 te 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 ,bzb r w9dr:s22m1pr9luy va6 uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the 29s ,wprbuby2 l91rza 6vmd:rmotor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0)4cxqf hns*+y(fyw 2hs yr j;0a:jes/.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 conservati()hfsw j*j 4ay2qrsyh0e +cx/y:f sn;on 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 an twe7. qa7g*tgno1zl 8rjm,x2 b*e+xsgular 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 wf3 g -gnim+;zvith 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 rotation speed be?-z+g; vfg3 mni 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 encb 0p1bo v5:kuergy stored in a heavy rotating flywheel. bp1bc05k: vuo 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 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 a cvv;iajq:eo.*2e9y tud1 +: krx;dgdn $H_2O$ 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 rolling on a horizontal surface rozi0f8uo+ ,u 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 ata /wzpo ;q+3qug x,xw 7s apa(5avd183dc kl)nd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release1,dz wgt3ako;/al+87q x(pwp ) uaasd 3c5vqx, 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 vertiqqq w* v v+z7d c/:;pv*ayj* o(cc8cpzcally on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step a 7y q v*d/ccz: cv8wp;+v(jco* pqqza*gainst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/.0p1ks. fg wkps on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the no 01kfwpg.skp. rmal 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 0.50i)zufq0z+ ykq 8-p +czrp 9g;mj0bk3j -kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle abpqz93 zjf+rymu ckj;q +8)z0p 0gki-bove his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as k: oa1sb/ u,jpkq8te9a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spikps 9/tbeu18ko :jaq, nning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consivtc.d+/yrr*2 iqm a :msts of two cylindrical plates, of :c2+.mi rarq /d*mvty mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough j2 iaq6 ,twx/upa7mwnx)48ig track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the trx67a)tjw/x i4gmu n8i,q pa2w ack? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assc2i,/eflw (gk )bfm9s ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduc/*mvc7n r2m t6*9wa gt)rwvwrk* u;azn)o .mfes its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the an2kmu7r9 cf tmrn .*ao*)m*nt /w6vwva z)w r;ggular 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