A bicycle odometer (whid: l+lgi gp1-gch measures distance traveled) is attached near the wheel hub and is designed f+- 1gglpd il:gor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pof mwnm r)aha3r4hv;cuzj8 kwh/9 b; 22int on the rim have radial and/or tangential acceleration? If the disk’s anguhzavj;2wf92 )m3 rn hkm w4ucb8h r;a/lar 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 angulafvs ec5;f3i zd3 k.vfi9a*d7pxfnp6; r velocity $\omega$ Explain.

Can a small force ever exercs- w/j3cddz 3fgnn-7 t a greater 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 h qf0xnllx 3gdr27n/p y1.j 5xvands behind your head than when your arms are stretched out in front of you? A diagram may help you to ansx 3 570x2 l/qrv gnydf1njp.xlwer this.

A 21-speed bicycle has seven sprockets.ya+/(kijpwi ,fq mv* 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? In which gear is it harder to wy+/(, a.ikmqfi *jpvpedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have s une7jkap:2t)l+j h3,gb-hgmlender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of ro mjg -:)ajkl+hbe,t2gu pn37htational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cad7chs 1tgbkyy 8ap:/2rry a long, narrow beam?

If the net force on a system 032dgl k yxt6t1z s4rjis zero, is the net torque also zero? If the net torque on a system is ze01ydgkz x62 lj34strt ro, is the net force zero?

Two inclines have the same height but make differentriw0 ,i wa8o7 i:ckb d ;hdbt+tt12:by angles with the horizontal. The same steel ball is rolled down each incline. O2c i:,a8+itb biwtkybh7;rd:d01wt on which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start r8a*kf qqscd uee1/x+9 olling (from rest) down an incline. One sphere has twice the radi19*c+e/ xa8dfku qes qus and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have t6tnmk :. x+ 6h5 fxi+e7uqsigphe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has thifm 5th 7+sx+ .xkpn q6iug:6ee greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moozb q n j*r./mq:xp,v)ving or rotating objects eventually slow down and stop. Ep)x, qo jnq r:b/v.zm*xplain.

If there were a great migration of people toward the Earth’s equator, how wouldwap8fo ,wg vk p5.-t-d this affect the length ok8g wvp- -a .,5fwdptof the day?

Can the diver of Fig. 8–29 do a somersault wi49exls1 wye/a* ke u3zthout having any initial rotation wheny/e9w 3*exz u4 kle1as she leaves the board?

The moment of inertia of a rotating solid disk about an axis throy92dx2c gmp7m ugh its center of mas27g9d xy2mcm ps 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 p)a u)lo a ;- -u:6y1ik mfqtkbjhg+w3in each outstretched hand. If you suddenly drop the masses, will your ang6a)) t;kg:qp3m-f-ohi1bulj +yu wka ular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and ha2927mby a5matu 9aqq)p5p cl uve the same mass. However, one is hollow and the other is solid. Describe an experiment to deqmp)y 22a7uaau bm9qp59t c5ltermine which is which.

The angular velocity of a wheel rn7( x:ynbnq3) tum6ccotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wm )qxcu 6: y7n3bc(tnnheel. Is the angular speed increasing or decreasing?

Suppose you are standing on thex/tdg eo. kx -/4(vag w(1ibcs edge of a large freely rotating turntable. What happens if you w x.4-sa1tgcd /b(i v(gke/oxwalk toward the center?

A shortstop may leap into the air to catch a ball and throw itcb g7jx5e/ocn1 qc h 83te.b0fo r3w)n quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his h t/5h c jbnfnxgo w7ec3 1e)3q.oc8r0bips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatisnz7 0 m.az .p;pdj0ap e7w.5fhm, mmjon of angular momentum, discuss why a helicopter must have more tha wf;7majzh7p.s0mj pe0am. .d, 5zmpnn one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a uyy(xq a6fo*9c 9ynd3) 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. Cal9nx fiv6 .,nyzculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the ,inz vf69ny.xMoon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed atwaw z6y (uvg;r1 ua0-o the Moon, 380,000 km from Earth. The beam diverges at an anglyw- ;ovu6auw1r z(a0ge $\theta$ (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 tra/ -u(8u i+dezrkp; ahe motor is turned off during operation, the blades slow to rest in 3.0 s. What is the a p aukui+a-e;rdrz /(8ngular 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 balbj /)qb2b;zvtiri- t/l makes 15.0 revolutiorzbqi)-t ;bi/v jt/b2ns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter trav 5*mjd7mxuqi+; o c3ieels 8.0 km. How many revolutions do the wheelsom i5*jumi3;7ec +dx q make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates3m3btbmj. (fl+ vb(cc at 2500 rpm. Calculate its angularb(.bc+tlcv 3m (bjm 3f velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.cy)p/+ 3fb sobbodj g*r; 2c7i0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centebgpybb/r cc;d 7*fj)+ o2si3or?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in itnky u5 tglo+73b nzo.)s orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee3lccyc h- 3hdu7 g (pk)6g+i6k.sddind of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0q5n 6:l:iiavy(v1 slb3uj/ a i cm from the axis of rotation is to expvn(1 iv3: 6 5luyqbasa/ji:li erience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to -2gm8 c ng jp9y1or9 :/qsealm280 rpm in 4.0 s. Determingmey j-:pqs/1r9ogc2a8 9nlm 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 radiu6 y. iu .jqb9sksni i-*0qlhb9s $R_1$ 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, astronauto2da/:hl el*: t)szchs 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 mhcz/:ol*s)ad 2h:el tinute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from )u- ba upiwri9aq *q74yz 3zj+oa5*kl rest to 15,000 rpm in 220 s. Through how many re*l urzi5bk q+ya3aa)iz7pq j-9ow*4 uvolutions did it turn in this time?    $rev$

  An automobile engine slows d9 ;qt2,spqk; q uiq:c3tfzq+nown 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 testedp.ig mrvu m/9ech,. kpp)sj9: for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takrhvumi g:p)e cs..pj /, pk9m9es 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpfmxy-g87r7x k: a cl.um in 6.5 s. How far will a point on the edge of the wheel have traveled in tgu7- :a8fx.r 7xcmklyhis time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnfosseo92z4.z ww0 h)c s 1500 revolutions beforwe) .4029ozws c sfhzoe it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces it)c1wztd-t. p8khwlgyspx 39, s speed uniformly from 95km/h to 45km/h The tires have a diame8p)k3z1sw 9cgyw tt l,phd .x-ter 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 uniforml3-8qih p4pvu4k o9 bbqy from 95km/h to 45km/h The tires hqo43kpi8ub p-4v9q hb ave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her dp .+lg:ll1 5 ugotsh.weight on each pedal when climbing a hilll .ou+sl1.htg 5g: pdl. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74n,n kt1kq j7.r 8l4;5pv8smk0tvbh d c cm wide. What is the magnitude of the torque if the force is exerted bkskmj d17c 4lvqh5np8n;t8t0 v, k.r
(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 i -sl r5mrh 0c,rlsvft(/mai(:n Fig. 8–39. Assume that a f(5cmm fasli,htvl rs 0/:r-r(riction 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 h.jm9 f3auegb67yj pkui( -j. the ends of a massless rod which pivots as shown in Fig. 8–40. I9 6(bf. u-7ukmijgheya 3pjj.nitially 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 require tightening to a torqu bi1rl:q ubyf5l nzk95 .sq/s5e of 38/9qusf5z iq.lb5sl:k1 5n bry $m \cdot N$ 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 0.64rt 5;kn25w*xp 4iyhko8 m when the axis k54kxtno r h;2iwyp *5of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66hxu wsk 2s;wly .yoeu5(:lxj:1n3bo91t+kt k.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. T9yo1ej3t s 5sl1uhb.+okku(: ;:wt kxlx2 ynwhe mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a tj.uoz-q5paf v /8w0 ,w5kia sdhin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcula0jpka5q izvw/8sfod .-,5 uwate
(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 wh r79x5aucnxt 1eel rotating at constant angular speed (Fig. 8–42). The frictiont97nuxa xc5r1 force between 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 objecviab4b8- r-n9 /sshy gts shown in Fig. 8–43 aboutsysbv8 4/9iha br--gn
(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 masseaf/v 2g*a :gs 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, engiya/np5me+) :lt q-dxgneers 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 thmq:x-tdn) py/la g+e 5e 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;6 wjgu ldz2(,i o; c sh6-13adigxce qnx/et5 and a mass of 0.580 kg. Calcu;o-gu2sl6h 5d c 3xii(jx1 nqe /z;taec,6dgwlate
(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, a3q,j13efa*enk2cim j ikea4j (sdd7 3ccelerating 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 *whkqdy: 8nqz5-x uw;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 produce the acceler:h;*wkqxqy8z nu-5wd ation, 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 eventuala6 i)hnb.5tn lly brought uniformly to rest by a frictional to.i )naht 56lbnrque 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 rs:ba zr 3)t8 s0jxsc6ball 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 fo 2isrj)iwyoy1 rz s:0 uqfk(5.rearm, which rotates about the elbow joint under the action of the triceps muscle, Fig1.i(fu )0y5krs yz:qsior2j w . 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 blad/4wvm5n*d;oaik mn) he can be considered a long thin rod, as shown in Fig. 8–46.)a/mm45k * ndvi; onwh
(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 mhv,, , garfs lu1:np eatg.(5dasses, $m_1$ 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 frovopw)dm jp79;24fzub *r0xm c m rest within four full turns (revolutions) and releases it at a speed of 2 9r; uwzm2jbmx* pc0vf op7)d48 $m/s$ 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 aj; lz,z pqqa)r2)k6k0su; sog 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 oqt.2v iwnh ;a+f 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping h+ -c l4hnfdf0q-qh j0down a lane a 4q- dchl h-j0+hf0nfqt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit; sncjcwf 4(h+ tcnb0;h respect to the Sun as the sum of two terms,(f t nccc4bs wh;0j;+n
(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 radiu1k.)tcwneq4swl5q e,s of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? eqc 1w.4wesnl5,kq)t Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starwre rm8 4p- ee gsb514en6hq m0xqy4*l/ph:yqts from rest and rolls without slippingw/ x rhm5 6eyg-esqrp1m84l he q4 n:0*pb4yeq 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 startoym1 kzp.fz.a sz/b-wfj3 gy ij*57l 2s to fall and its lower end does not slip. What will be the speed of the uppe jybo*3f -7 ywzsm p.af.z gi2lk51/zjr end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum ;(mzan c kxmtzg326y .y+q*hp jw6m.hof a 0.210-kg ball rotating on the end of a thin string in a circle of 66. m.k;2m(jyxhh tqaz+w 3cny z*pmgradius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniforyub,* 5wgfo+ ;p-bafzm cylindrical grinding wheel of radius 18 cm when rotating b5*f ;byuza+-,pfwgo at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thai7ns s 0jgj2m)zr7, fapx64yk t is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, ,jg27sn)xzi64syp0 k7 fmr a jFig. 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 l b3bh1(,wcjqilh- f*one shown 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 position, what is her angular spee1fwblj *h 3 q,bhc-l(id ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation raxp)y, e htk79kte from an initial rate of 1.0 rev every 2.0 s to a fk7kx e)ytp9, hinal 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 ae6d7 ug.w -zypvz3r 8axis 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 they78wau.-d z 3pz rev6gn throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater suc6 * vii(lix2-l9dub pinning 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 t ,6v6wssw y(xdhe 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 an i5yupxsu. 2 z3jdentical disk rjp 3uy 5.2usxzotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4ro-y2;u *ux3fdy gm 4wt ;d((xa u3upj $rev/s$ 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- ,gi55zbv,jbhgo-round platform of radius 3.0 m and m, jh,v 5g5izbboment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular :- qxjxyh5x*j9ld.x9x u:xwnf .gqk* velocity of 0.8 -y9 fgxjx.*qd :h.9jw x*x:qu5nl xx k$rad/s$ 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 eventuxo e, 2mzs8gm s2j)kj8msr6jfju 2u2.ally collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries awauu s22j fs6kog.8 r2se2mmm,jjj) z 8xy no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess o(3xp7*(krxbov wm l o4f 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 e (uj 1ys tck*15,;ghtbzgpq3$ 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, initiap9-zi f *k x09ju4fw6gqo .roully at rest, that can rotate freely without friction. The moment of inertia of the perjq6irkf9 -opx.0*f9z ow u4gu son 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 stands at the edge of a 6.5-m diameter m*h nyjz5/ gkrb(d;j v7a5f0xr erry-go-round turntable that is mounted on frictionle j ba; (5hf5dyxv*/7rgk0 rjznss 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 ofui52x szmrpk-ka.: s( the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope p5-u:(xsr2 kak.si zm unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always3l6f. xxwejg8q5 b(n;g,tytd faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In egj,t.w6 3 db(lg5x ftnyx8 ;qthe latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate o -j:m 8i1mlyigxbmy.fiz .+/ .c; knwmf 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 shapenl* twbt4ljg1j.pw; / of a uniform cylinder of radius 0.20 m acquires a rotattj/pl j4g.wn1w l b*;tional rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricpcb b( ; :vb*.nzour 1l56wrmeal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is r*u;elo.rnr b:bzwm 5 p vc16b(eleased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angulacez) l :l3qya+5j 7whh2li y2yr speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timesliw7uvhqyuquxk2 ; .*+x;)gw9 fgp-t 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? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentumkiv xt;;.9hq7p- g+ )u u wq2lx*wyfug.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution aul1oow u6dpkw) v15nxx4cu6p(tomobile is for it to use energy stored in a hewd(xw1kv x u p6co)5p6nu1lo4avy 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 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 illustraen:-9xk-q fxpb v)l fgc :7hrgezo/2, tes 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 horizo(zd,a pp;ox0(rf.nc2y lawqw 7yt.u ;ntal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i3rciko jr)6bu 6w blwmhel9 l ,a2u k)y-dd2*(.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 a bel9)kl rum2 wl3-) *jibo(u2 6 rk,hyddcw6mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floorv ;q+hsqyh d0;r4i g4d, and we want to exert a horizontal 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 gq ; iq+hhd4d0;rsv4y h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making+5d+ /ra*ilwaazre, 4 xt8ll n a turn with a intawraa+4r+z* xdl5,8 lel/radius 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 into a sgf.svlagrr 0*pc-(b 5tfm30lling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it frocfr0l mp r-a lb*v 53(0tgsfg.m 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 a solid cylinderqvk*qkj9+ 3)7r6 j.mpgfvfnjb0 r0w u and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate thuj*nkpfgv q+)f6b r0jk q.v 70wr9 m3je 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 cylindricaexqm1tc( et t93b 3e,sl plates, of mass e9txsm,et(t3bc31 q e $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loo.(6 0c, dtmwegs ryd2yped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of thet26gd.m(,e sdy0rwy c loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumexii5(d v./lu d $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolu7+mpdv0gl6 p/lr jns(cc1w 9z tions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of dp01m7/pzlw9n(c 6lr cvgs j+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