A bicycle odometer (which measures diz8ts6 nlw /*)c9 i b(qnsfyjm)stance traveled) is attached near the wheel6 izmq)/stfjw9 b)c8sly (n* n hub and 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 veb wbr(m u;dl56locity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increadw56 r(ul;bmbses 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 desxal z3 i*gu; g-e z:)agwp;2ghcribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg ko(h(rbul.o 7g ex19qer 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 difficulp)hkbv c *a3m zi-y9 ;h.ukw+et to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagramv-uz )p+ehbk k.9*wyh3;icam may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at thmk 5:/p u5 ua5y3,jqjlm*foca e 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 pe53: /y5qm5fjuolapkjum a,c*dal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flee6p q3nb -6;gf cf6sfh *fqtlt2htz9nk+vr/)sh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advp9f/-hf+bqt l t3czht*kegsn; n vq6f) 6 26rfantageous.

Why do tightrope walkers (Fig. 8–35) carry a kpe, .-, a oibs(a.puylong, narrow beam?

If the net force on sn7ozjf: ft f/17/gnzq;ano5j3x0 epa system is zero, is the net torque also zero? If the net torque on a system is zero, is the net fonf5zf 37s7f/:o; n tpnj /jo ze1g0xqarce zero?

Two inclines have the same height bua : o wa7q8v9:l mjh3lz(jk1dut 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 bq7:vk aaulhj 1 9lozmwd(38j:e greater? Explain.

Two solid spheres simultaneously start rn u(wj4ml lcw7:n n73tolling (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 gm(jcnw l n77l4n uw:t3reater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the s yr ;ykp)qbf0jls,31 hame mass. They start from rest at the top of an incline. Which reaches the borysf1)qhy ,3b; jklp 0ttom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservefm mcxcgowd..tm6(*lv :6:i xd. Yet most moving or rotati(.:xi c6mlvcmgdot*6m x. f :wng objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Ejikxtti s5nwhsk 41 z.1j495earth’s equator, how would this affect txki4.i w15sh9z jtn tk e5js14he length of the day?

Can the diver of Fig. 8–29 do a somersa - jc7se0r2j qn(y-qv t+o(tdgult without having any initial rotation when she leaves the bo0njqy7q 2t((-+tes dv j r-gcoard?

The moment of inertia of a rotating solid disk about an axis through its ce mylkb,n1+f/k nter of mass is, +1yb/nlkmfk $\frac{1}{2}WR^2$ (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 ix+a4li f as,0y)li9 lbn each outstretched hand. If you siai9y,0fbl +a 4ll)xsuddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and ff )0q-*wi ,w suc/ivujg-1c fhave the same mass. However, one is hollow and the other iq c* ,ivusf)jwc ff-/ iug10-ws solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points westlasn)n -3; cri. 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 linrn)s;c- 3 inalear 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-zljvl,6 euq*;te,.m uaw, cv a large freely rotating turntable. What happens if you walk toward thuva,,,lm zuve6tc*q -ewlj;.e center?

A shortstop may leap intn-lohy gayfkk85 :)/jo the air to catch a ball and throw it quickly. As he throws the ball, the uj- hol/y :f58ak n)ykgpper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserv:4twrazxy le ue -. iy,b(ko;r ;r:ah9ation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more o9(;y;zrwra 4:ibxae.ut : h l e,ryk-ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: - tbl-- tj(opz(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. Calculate, usineo ig 4njou1a 37wq-8og the information inside the Front Cover, the angular4egoo-w nq37 u8a1i oj diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directapzzc)0v9*/d:yo0 y7 emxm vjed at the Moon, 380,000 km from Earth. The beam diverges at an anglexzm)dz p/0v9y a7yvmc*jo 0: e $\theta$ (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 650a(.ny l)8py9vig (u x:lzr tw10 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow dow(r .vnyi9t: x )lw8gzply(au1n?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Iflojq4of w7,x u(s,/ ed*g mv2j the ball makes 15.0 revolutions, what is its diam,o,j d(m lfj2ue/oq4vgx7sw*eter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many re1h,nzz 1n 0pep-fosr.-1s72qnf g yhnvolutions do the wheels 71fo yps hf nnhzqpg2, nern-1 .-sz10make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atb wrmn d77vg u9c7t hgu6)p).c 2500 rpm. Calculate its angular velur 9b du.76g))7gncvpwth mc7ocity 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 complo wt*x-xo7sq 2c7rwg2ete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the c7qox rc 7wwot*g2s-x 2enter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (g7k r l)f)yh-ka) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speeyfn z 9f p1:4mqmkqr0.d 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 rota7y2plodpn 2b: cdtbtq:5u * v,te if a particle 7.0 cm from the axis of rotation is to expervbpttn2d :: y,*uol qp7 5b2dcience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about ite kj 0gt,64ub/nllwg4s center from 130 rpm to 280 rpm in 4.0 s.jt4kg/6 l4lug b, w0en 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 radius td6((iz *+izh 7e qt7famq 3oh1kcxy/ $R_1$ 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 abgxeauav t7iqp986 : q.oard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time intxg6a8u qvt p7iq e9a:.erval. 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+ v 7ve8wepy.y,000 rpm in 220 s. Through how man +pew7e.vyy8v y revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal2a1nok6bkf;* qf8qlpugf6 ;v culate
(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 jepp.5ynq qe (g.zyx)q:ts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolnez)x5q.gpqpq:( yy .utions 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 acceleratvn *zu 5.bf.sfes uniformly from 240 rpm to 360 rpm in 6.5 s. How far w.ufnvb5*f .sz ill 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 850rev/min It turnsl -dbb.t+ *yn0mgm v)r 1500 revolutions before it com*m0v-) + m.bgynd blrtes 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 revolut8jrkf ;d .l0diions as the car reduces its speed uniformly from 95km/h to.8;rl ik0jfdd 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 rdh-dh6i.8ew geduces its speed uniformly from 95km/h to 45km/h The tiresdw ighe 8hd.-6 have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal w90ausrj m9;1 v q9ykcq(o7 nfihen climbing a hill. The pedals rotate in a circle of radius 17j;ayqrik1cqo( 0s9nv 99m f7u 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 enqg 0x5vlnguxi2 (.+hzd of a door 74 cm wide. What is the magnithn giz5gu.2lxv q+0 x(ude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the whiq 3u1y 0+1xtvcx lrb-eel shown in Fig. 8–39. Assume tha-qtlvc x+i3uyxrb 110 t 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 of a massless rod whichc t22i5 iu +kpo7uws) e/x:haz pivots as shown in Fig. 8–40. Initially the rod is held iki )ezs /atwu2i5h:p 2x+ u7con 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 tightenia 0uhuai 8s4exq 80bbxok+.; zng to a torque of 38i xkha x+0 abbu.qe 4szo80u;8 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 mzen( gj72qgw0 when the axis of rotation is thr wjgn0(gz 7e2qough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle f ;z1 e34voi2yw 3:nw*hzka wkwheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg:3y oz fakwi*w ;3eh2k1z4nvw. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the;he h1 a6itwbc34jqg/bkm: n4 end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. twkh4agec3 nq;4i6:j/hb mb1 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 rota7s1rk d-2e4 uc pbfpp.nop( v9ting at constant angular speed (Fig. 8–42). The friction force between her hands and the cppr. 21cfd4b(-ppo79n u vkselay 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 f(kct xa4jk+ p3f l/k 69;vzwlshown in Fig. 8–43 ab64z/l9k ;fjf3 k(cltxw kv+a pout
(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 consist4 +v bsg/u9mp: qtzq+u*cecs1s 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 satellite spin-,2 (ftkwrrwyti,h )gning at the correct rate, engineers fire four tangential ro i -yt(g,2rt, wkfr)hwckets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 c)ko (vo6a*jxj9pb9 dfm and a mass of 0.580 kg. Calox dbfjjkv*o)p (9 69aculate
(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 b9a c4 e4dae9asps,uu7 at, 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 tanges-2;vs*5z j6gpedmxi 0g rx0 bntially on a small hand-driven merry-go-round and is able to accelerate it from rest tog rs- m5s6d2zejgv;x 0 b*p0ix 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 bro(vb b2vj14ama1*utvt 5w aur0 ught uniformly to rest by a frictional tora5 u0ur(2b*1t b1wvvamvjat4 que 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 ball vk;:+i42dfi ow t, fvbgkp;e7at 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 act,mtj)3s6ssk f /tz yzhorq.2 )ion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleraz rzk.s,)f t2ym)3/ jtqs s6ohted 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 s+f w/b.dmlwg;hown in Fig. 8–46+; d./ mbgwlfw.
(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 g5ca95 9l wlybppk7 wvk2qk/ex 5d; ,lmasses, $m_1$ 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 hamm;lr+ bdy/ v+jqer from rest within four full turns (revolutions) and releases it at a speed o+vlq ;j+/brdy f 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of d9yzft5 f,2)t q-x ,pq8cygnz $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop jzyfad+9( klidb;2;ys a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls bof2 gx+ez- y3xo)f9q without slipping down a lan oby+3-e 2)qz9og ffxxe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth w iz/gi9lt93 koith respect to the Sun as the sum of two terms t939igiz/kol,
(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 7.50t-nnjb s863f t m. How much net work is required to accelerate it from rest to a r ttnn38 6-bsjfotation 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 from rest and ro n oxv( 8;.q)mphc+8;edgmpjalls without slippidmoeq (;gap h88n c)xpj.;mv+ng 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 vertis3f6 g xx9t3nzcally on its tip. It starts to fall and its lower end doe3nsftzg6x 3x9 s not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating;ws *a0 7jzppl :x9xfy on the end of a thin string in a circle ofz:yf09s;xxl p w7p* aj radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mdys 1scyogwc(v (/32tm. 4am aomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rvmgdo(ma1y wcs 243acy(ts./pm?    $kg \cdot m^2$
(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 pla d vtz mkgc0hl m,)up.*+7r-zatform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig.mrkmdlgzt0,.p+a 7hzc- )* vu 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) caoxl1jyc t.r44 7rsof /n reduce her moment of inertia by a factor of about 3.5 when changing from7/s l4f 1xy4r jctroo. 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 ra(s933 jxh yeaox7z e .3+fbauk dtyk(7te from an initial rate of 1.0 rev every 2.0 s to a final rate o xjkykexo(a(9y3u+a sdb t7.73h 3z eff 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 verd3b1.b );lu43wjniqr( lz wvmo+4f eo tical 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 flatbz j lo44nbvf+3o.1m3dreqil(w)u w; 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 momentu2 qhj*or,f9 kam 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 m9 o3b :mbp,wquomentum 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 /2zc9dl 7kq:my*q9hylmz8h h np)td;is dropped onto an identical dizd t 2/yc8hhq* 9ly;7lpmqm hkd9:)znsk rotating 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. e sc*(41p oz2o5rigjb4 $rev/s$ 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 of4,2fq,ita4tay nsk5:;7 kd k zu0ivvh a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 n,di5 vt ait ,akkuv7;fsq 4y 4:h2zk0$kg \cdot m^2$ 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-r kw*l94tv/ kd+j)jg7qj a1t fround is rotating freely with an angular velocity of 0/v7wjtfk*gjl1 kj+ dt r9qa)4.8 $rad/s$ 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 a whit/)j ,a*lddf iue dwarf, losing about half its mass in the process, / ijd*la,fdu) and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120x9as :e0leso2 $km/h$ 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 tg+g: no- jv3d$ 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 platformycj7- xqds:0 b6f* 8*z vfxsah, initially at rest, that can rotate freely without friction. The moment of inertia of fdfyxsb:7 s608*c- xv *hja qzthe person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge lr uyi0i-lei )r 594ffof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a m9f)l-0lr4ye 5 r ifiuioment 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 rope lying o63+ c imrbxn(n/ ogs6cn the top edge of the spocn66+3x i nosr (cgb/mol. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth su dg,vhe g,mev1.ts0m3 ch 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 angular momentum. (In the latter case, treat the Moon as a particl3g vm1sv m,ged,.t0he e orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1/ua 8vutwx/3:;jx r 9 b;kaagz 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 rvz43+4kx, cm p l 2of+z0isfvwadius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constaomkfc4x f3s+,0lp2vz ziw +4vnt angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mas1o/qb.hwy ; +5 xg:qexf)snjus 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 reachuxs5g qqxn;y jwfo.e/) b+h1:es 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 rear wheel (o8 6j:ka l .h,pibg4hp$\omega_R$) 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, q-k:ff s2;esw 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 th k w:q-es;sff2e 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 enf tb3tx9w9 8f 4h5j65pve jn sq1/hbxgergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flybtn1g3 58pjj fw /h4sh t99e 5bvfx6qxwheel 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 an 2+z(hu wxa,qw 6ig0wdt5m -mj$H_2O$ 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 /k4e .tm):5ph1tx/jyw(n u wuf rw+gvspeed 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 (vzk p:y gv99qM and 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 heldp v:zkqv9(y9g horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radsd(lqv5xw/ 3 jodj 6+et5tun0xt -pp1 ius R. It is standing vertically on the floor, and we want to ep q1jnp 6t /+v (jx3s 5tl5-detxodw0uxert 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 h

  A bicyclist traveling with speed z.;7tbgnszf1pepm 1 b 5zkw: 9v=4.2m/s on a flat road is making a turn with a radius The for:nb z9fw 1pzp kb;1. 5ts7emzgces 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 sling of length 1.5 m anday0c k/ iqon;68qiqm/qm:-j 3y a ns/y begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest t/q/ m0yqnms-aa6ic /ki3 :yj q 8nyo;qo 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 i g mkyxg*j:uoct3x9y 0r:55ocylinder and her arms as thin rods, making reasonable estimates for the dimen o::yxry9g3mjxg*k it55c0ousions. 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 ash.7a8hc 0.e 7ey+uel v +kittoni+7tq sembly which consists of two cylindrical plates, of mae+ a.t.ou ceh7 i7nety 8+q+ 07hvkiltss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fige1i cj .e,-hfd0)gbep. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the 0gj-ef,i)cpb ehd.1 ehighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu7/yf./qvaukbi bz*j +jy x ;j.75mcrrt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make-ldg6xccu( x 8 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) W6( -d 8cugxlcxhat 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