A bicycle odometer (which measures distanc ig7f,,drw2 (ezjqulr *lk 3s6e traveled) is attached near the wheel hub and is designed foz qldws, u3rr27ik*(fjg, l6er 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotat7zrl7le tc 31rkk9bqk 5 ;9rfqes at constant angular 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 tangential acceleration? For which cases would the magnitude of either component of linea ;kr5cklr 7tbz7 9f9lq31keqrr acceleration change?

Could a nonrigid body be described by a single value of thh( 1u6k0gkjh4l:n; nz-*6k stuk autqe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger g6u.k(3bjp sr2(jb:bt 4rjhw 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 wit gz142yh8kh vkfxgv( 5h your hands behind your head than when your arms g1g zk vhhxy8f(v5k42 are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets esg8ldjm 5w kb6;vg9fzc)4h/et k w5; 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 sprocket? Why?m5vtbk k; ef4g/ )w8d;cezsjl65g9hw 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 lower legs with fld5jj hyvyoct (-+hr)y9k w2.fesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribut ofjk9-hytjrwyy d( .)25h+vcion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry (g/wjwa)2f,f1ijs ,h9/mjt q wf2ukb a long, narrow beam?

If the net force on a system is zero, is the ne:.u:ndsb tpdtrf, ,ralfa7 tx8bl/2 3t torque also zero? If the net torque on a system is zero, is ta,n,bu axs: tt.r8 pf2d7br/l:ld3 f the net force zero?

Two inclines have the same height but make different angles wr*m0 ,vml h hk*dwetn0+wcb(*ith the horizontal. The same steel ball is rolled down each incline. On which iclh,*+m(e*0hdm *r 0n vwwtbkncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously wt ksx8hbt4d81ku,*fc53h u0n j0 zqvstart 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 g 80u 35k1 hkbnhd*xvf8ust4jzt,q 0wcreater 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 svzx1r-)nqvj4sbav 7;w/ u tz+kn 03vutart 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 7 a1 n+) ;rvzuvvbj/u-zn0 xw4vt3s qkthe bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mos-u(m jw ;*9g3vanp kfdt moving or rotating objects eventually slow down and stop. an3vpw m f;9j(gud*-k Explain.

If there were a great migration of people toward the8zq9x6xcd//fdem :l ,x v4k)oi n(un t Earth’s equator, how would 6nl4 : zv9tox,ded/fxui/c(mnk xq8 )this affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having a jp2 cdo7/l uf),ot-x5kkh8g hny initial rotation when she leavetx/gkp2o,k fj5o)l 8h du-hc7s the board?

The moment of inertia of a rotating solid disk about an axis th1*afm h7* qsyqxzz**vrough its center o zx7qv* q*y**z a1hmsff mass 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 sittinb- nc-;nw qg;xg on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will you-qxn-;;gc w bnr angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass.sv5gs3 xy8mtiiy*xann9v*8*)fa v go6b(fp 8 However, one is hollow and the other is solid. Describe 9fg fb3 6yyinto*via8ss*pvav8m 5x (8)*gn x an experiment to determine which is which.

The angular velocity of a wgfc5qeh6cd 98 heel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the96hced qgf8c5 wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely r r/as72 zkd(lfotating turntable. What happens(rzaf2/ s7k dl if you walk toward the center?

A shortstop may leap into the air to catch a ball and thro qkdws )xzh/5 sw4v1q3ln7 p8pw it quickly. As he throws the ball, the upper 74pwlsnq q /pd 18x3 5khz)vwspart 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 conservation of angular momentum, discuss why a helic:d.nfk7 d(2g2yfpwc jopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller ca.jfw 7dckyn(f 2p:gd2n operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30d13-+wkrw a b-fhwpsst *f+6j $^{\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 cn*rvk:q-i,7qa t f (xsoincidence. Calculate, using the information inside the Front Cover, the angular diameters-fq*v,n:akr st qi7(x (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00008:isgg yevmc)pj o 06 km from Earth. The beam diverges at an 8ggv pie0:jos)m 6c0 yangle $\theta$ (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. When the motor is t*qlypj1d oo 0,urned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladesqj,*1do o0yl p slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil9f/z ws/1,dhpf hhr t.d. If the ball makes 15.0 revolutions, what ds/9t,.zf rwhhhf/p1 is its diameter?    m

  A bicycle with tires 68 cm in *1r9sl ms-mysjdvmd-2qxx)p-(p )msy *t tp* diameter travels 8.0 km. How many revolutions do the whe-- my1 *yr)vx*std 9pmmsm (s qpx)d-2 *tjsplels make?    $rev$

  (a) A grinding wheel 0.3 tww djd3wu2f2fuf58(5 m in diameter rotates at 2500 rpm. Calculate its angular velocu8(fd jw t 3f2dw52ufwity 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. 8–38pw.j7ixq+xtgy *.2wq (dctd7 v.y5ai l.pv4 k). (a) What is the linear speed of a child seated 1.2i + jqpcxl.2qx5*yka v.wp dt4d7t7v(g. i.wy 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) inj-ezmy z4):s9 jph;v y its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee8/f-j:w9 +kgr: ohruf kgw) sid of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if atmdh/g,z:x5,zg,k1vw bz p/ -dd pjp* particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,,,tzv kgz p:j /zhb1-*5xd w/ppd,mgd000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformp k-+iz dqjivsx9j.kkw a b7y2)a9g8*e 2oe4oly about its center from 130 rpm to 280 rpv a*s). op i+xkz9bjik9dowg7k-je2eyq4a 82m in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusic 5,5qwi7 c wu*nu(v)x nzx7z $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo ; ju mk .3nzni49tv;-bkwm1vrspacecraft 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 tho-r;tki3 v1 ;uzb9w nm. vj4nmkught 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 15az8s g3t6,stvk1aoyafm kj ) ) xj7(:t,000 rpm in 220 s. Through how many revoluksg:j)stakyxf t3o( m7ta68 ,vzaj )1tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm i3k trk 8w :palgj;3 /u-myfk;fn 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 vw2 bzh)dbj/ h1wc,+7lpr0aijets in a whirling “human centrifuge,” which takes 1.0 min to turhdwl,pv7zw h1+cr0i )b b2a/jn 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 acceleraydsw j/:+lt2ct-qd( rtes uniformly from 240 rpm to 360 rpm in 6.5 s. How far wilrlct dswyd- /q2 :jt+(l a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at , lsua8m +vky. pm s7u3-ol;xm850rev/min It turns 1500 revolutions before it comes to a st o,umv+ xkl38mmas7 ;sy. pul-op.
(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 uniform 0-lnz djxxb.(ly from 95km/h to 45km/h Tnl -x.(db xz0jhe 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 re:0-gbki(zcmif3 ktw. w bo u32volutions as the car reduces its speed uniformly froi:kck zb-otu(3i 20w.bm3fgwm 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding (-lsy cufz:3ra bike puts all her weight on each pedal when climbing a hill. The u(sf-r l 3yz:cpedals 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 end of a door 74 cm wide. What is thnepn5u9scc;u nqum2m2 9 5q k3e magnitude of the torque if t3 c 2pecu9qq5mm2;uun95s knn he force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume1jhbtf8y t21 +bv2shz ty+ tz fb2hjv2t1 bh8s1hat 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 en b,brf v/t;25 tqzrkrtxee,6(ds of a massless rod which pivot rt,tk6f vrz25/bqbet, r;x(es as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requirep gg(2.qhy()qk0a,mj nomdg(g15ay y tightening to a torque of 38 ,m0a(1y.)qa(od yygh q gngj pmkg2(5$m \cdot N$ 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 radig )qe9t8toj6(usnx r6us 0.648 m when the axis ofj9u6n8(xgesq6ttr )o rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inerti 4g)ysd wj7n+ma of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of7mg)+dwysn4j 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizont.puh2oioef , -al circle of radius 1.2 m. Calc2u .p,fehoo-i ulate
(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 rotatingj3+7p: hsw5 u 3g+dorf ugvup5g(1uup at constant angular speed (Fig. 8–42). The friction force betwvu( rugu 3pujgdo+wup7sfgp 5 :35 h+1een 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 array of point objects show/ txpw;//gyqszxne;q s q;pyi*k; 8 9cn in Fig. 8–43 ab zgwn ; x iktcxpy8spqqqes9;/*;y//;out
(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 whob le p):8gej4v-i 20spa yz3fnse 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 correct rate.( +)1c3tsfadvd9snt p;qzmdq8k o a,, 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 m, what is the required steady force of each rocket if the satellite is to r kav o9dafqpq d1t(mz3;sc ,+.nt d8s)each 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder withr.p/wew) +mub a radius of 8.50 cm and a mass of 0.580 kg. Calcmwpw/b +.)ureulate
(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 *jt,(8 jqnsdy bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-roun8cp, /p5xz/4oyrqn 36huedz k d and is able to accelerate ituz,/doq pxep 6yn534hk zc 8/r from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broughzbtjfs56c( 2 .brrdl;,h.tl o ,cthz/t uniformly tzlj,6(z2bhbl/;c ots5,tt. hdrrcf . o 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-kg ba;zguti 8pl rnbt9tmc0td5 p3 a/ n,wf9a(l2y;ll at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whic(xrwka;+ b1*.mht boch rotates about the rtamh;kb+1b xo*c( w.elbow 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 Fig.rwl-hyp(zmhef7 1x 5: 8–4w(z-p hm exh1:7fry5l6.
(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 consistssf *8rda*rco 9+1qw fkqjr* ;s 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 accele f)kuv ovw(pkeo67qc//rn: r,rates the hammer from rest within four full turns (revolutions) and releases it at a spe/:rwq(kfuo )krn7 6vve o,cp/ed 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 haw381k p;ute gss a 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 develops a torque of 2 6mz1)kik*fpao-qyoy 8dg 1 c+80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping downmgdo po-/7 my6 a lane a/op7gm- o yd6mt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth pi-vo7b -)k htdp1z kecz8;0dwith respect to the Sun as the sum of two tp zce-kob1)dzk7pv0 i;d -h 8terms,
(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 bn+gk55 wil4q mass of 1640 kg and a radius of 7.50 m. How much net work w+ b lk5qng5i4is required 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 starts +vy+n3c z/t* er30ljhr 6psj bi 7nl;kfrom rest and rolls without slipping downhtvj;s +e p i c bzyrk6+l7l3r/*n03jn 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 verticalw(xo j e;zj4r9ly on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conse49 jro(je wx;zrvation of energy.]    $m/s$

  What is the angular moatf- (l h.w0f o9sxcb jq+4dc4mentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angfx4wqls 0 o+j h .4ca-9dcfb(tular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical6yp0qcdr7v m a)wg(k 4l vie/; grinding wheel of radius 18 cildqk m7) /a46c(vy epvwrg0; m 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, han njzhj -c *1wnir) kx/slq/1yyde;/o)ds at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases nywsio-*n/)j rc y;dk /ze)l1/ q1jxhto 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) ;v8iiijzf 3:k-wkz)r can reduce her moment of inertia by a factor of azj-r3iz;kik)vi :8w f bout 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 in/u4zb i. gzwvf qdgf*z -;8w;iss*/ rev every 2.0 s to a final /z -ff**uq;i;wzv ig s. gd4is/n8wb zrate 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 rotatint9fbe0, yb2-0u i tgtsg around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. Whattbu0e- iyg9ttb,fs 20 is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular t7s)0nuu0l d xmomentum of a figure 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 mom3sv 2 hv/+n2gawgon-eentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto * 9ma4dyt8jyfey9w 7xan identical disk rotating at angular e9tay8xj 7* md9y 4fywspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ij/rzhj0ke2x 7l 8)hzat 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 rotating merry-go-round platforkqy1thqx a-:; aev1g ;m of radius 31 ah:qgyqxtvke;1 a- ;.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is roti3hskl* uk a*.2f/ecfn)0 ljcating freely with an angular velocity of 0.8 0lufi.3f/a*kk ecnjh) lsc* 2$rad/s$ 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 collaps5 nn+:tqio no(es into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the l5in nq:+tno o(ost 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 exces))5pcdqg myf t2r* fd*s 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 5g y dpemvd ydk-.s4fls38.;g$ 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 canfjui hp38bcab q5j)7 0rsiv 1b+l*x 6c rotate freely without friction. The moment of inertia of the person plus tbx 0fi6cj+bb uc7 l*p sjri)qh8513 vahe 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 stands at the edge of a 6.5-m diameter merry- k0s5adosh1y s8u )upmp e0:(ygo-round turntable that is mounted on frictionless b( uysup85a :dsp syo1h0 kme)0earings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rovy3ktf et 8). .bnsp(upe lying on the top edge of the spool. A pers .n)3tkb.ut (fe8svpyon 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 side icacm;bz)p p7; 5s,rf/y:dp ;jr+sy z always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its +s j:c ;psar/;ifp)ym; crz7dby z5p,orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at1m7e g46 1lik i- amgsibhq(+z a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of ra 8*nsec dwy6wek h39; lj*jox+dius 0.20 m acquires a rotational rate of froe y96j wcke *+j;lnx3*od 8wshm 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 of4ku2 1g9 8qad7evtm2kl 1dvd23zhdzc 2gfk2i 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 8 zdd tkgdul4q2via92dmfc1 k23zk27h g2ve 1to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reaa: a2wm12z qd l ll:9q;jr,dsm ejpw4.r 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 as great, werbxi/ybs7t j50km , fd,e rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of rat /,sfdimjy7 , 0b5kbxdius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use en v3kv0 l .)ypkk43ilywergy 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 should be able to tra v3k .kli03wy4 pyv)lkvel 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 illustratesdes2os9 q c/ 2m -y .at78xjpcstc:gf- 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 rolling on a horizontal surface at speed v=3.ulm..d qs:fz3 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 leni 9azmg oy3(4cgth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21ga3m (co4yig9z .]

A wheel of mass M has radius R. It is stand cfh vu;u09c5,bez+d ding vertically on the floor, and we want to exert a horizon;dczv 0 d,uf5c9ue+ hbtal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling l tl myts1uvpl0:k 8jp2.8r:qvdf 3b4with speed v=4.2m/s on a flat road is making a turn with a radiu:2lf v4dvt:p 1 ll bs80yrmqkp3 ut.8js The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock intv79+ qfyule opr)6-/.9e*uemra, q wjrd2aln o a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torqu u+uqr 9yd7w/)l9, erpjq6ne r2 *oe.af mlv-ae required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’n8t (x7mrskcqnwd 5 66e4pnsr4 jpv19s body as a solid cylinder and her arms as thin rods, making reasonable estimatesmj7r6skp w9 1c (vnx qtne4ps6rd58n4 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 cylindf nzi1f gp,eu h)osqb7,9 s-h*rical plates, ofzs1 , ,o*geni )hf psu-9f7qbh 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 track of Figahuppg kcb-,.2c 9ix- . 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the i-kpca9,. c p-bhugx2 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 assu/7q+g+vjvn gcme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th q)n ,fvk7p7t fad gg0s+d).vle car reduces its speed uniformly from 90km/h to 60km/h The tisvgq. f,7 0kap+ nf )ddvg)7ltres have a diameter 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