A bicycle odometer (which measures distance traveled) we3tvu6 b yzo8 .6c-hf 8lwq4zis attached near the wheel hub and zzb q-v4lhc.68fwy3 otew68uis designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hav,bqryz-ut y9*fu:d p3u qm*s; e radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/p3us-:9td ;u*myb*fy,z qq ru 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 v j0wiia/nh4 ht;zshrc9 z/-w8p 4pl4the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thas d6 /cod6*volkl24bcn 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 / )tz buac/j4kwfks3 *do a sit-up with your hands behind your head than when your arms are stretched out in f // )buk* 43cjksawtfzront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thre.ctfu95:vy plf.5rnok m o8rc z*3cl /e at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why: zctvo.8kp 3co /uly5lcf9. nr*f m5r? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend ong i;r0 qce.wbub4;e;2wmx6 xi being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dyngu;e0cxr .q6bwx;ib4w ;me 2iamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lo5*u g rv l.hja)es:fu(ng, narrow beam?

If the net force on a system is zero, is the net torque also z1+yrhotu0f+(wi7 i ve ero? If the net torque on a system is zwf0( vuiie++1 7rhy toero, is the net force zero?

Two inclines have the same on( *7k ujsg k )xy3rq-d0r 5yjazz1u,height but make different angles with the horizontal. The same steel ball is rolled down each innuk5jq*,y z1ar(ds7 yrxj)oug0 3 -z kcline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One v+*l,yvf9j6q(g rt9j kpohs ; sphere has twice the radius and twice vo yrjjvtfsk9 p 6 ,9;(+g*qlhthe mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start,b/zs 4491iija a)urb,wxb ye 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 the grea ue4 s1ibbyxb9a/z,)awj,i r4ter rotational KE?

We claim that momentum and angular mo0 g: ;* zftl8ggjscpu(mentum are conserved. Yet most moving or rotating objects eventuall(gs;gjc:8fulz*g 0 pty slow down and stop. Explain.

If there were a great migration rq0:- mlk3-oc:8nwlbydg( ie ww. 3tq of people toward the Earth’s equator, how would thisyre biq33 c:- twl(k-q .0:dnwmwol8 g affect the length of the day?

Can the diver of Fig. 8–29 dqkoh rk d7svo4)7vjy2d0cm - evb4g6 1o a somersault without having any initial rotation when she lo2j0hs r 4 6b7)y7e1ovvdvdq-k mkcg4eaves the board?

The moment of inertia of a rotating solid disk abov19 s6s7bfdgvut an axis through its center of v s1fgs967d bvmass 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 if1 /v4b-g+n 2btotxmd n each outstretched hand. If you suddenly drop+dfm1- xg4nvo/ tbtb2 the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. How)jdr,kyncee/11 d7lhvgf(/blpu0 9u ever, one is hollow and the other is solid. Describe 0 7/1e(vd,p k uduhe9gbrfy /jcnl1 )lan experiment to determine which is which.

The angular velocity of a wheel fl)2of:-+ 4p9 1ybz,(lliqyyf n tevvrotating 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 eastvpf-+ni vzy4yfyt 9b, )e1ofq2ll(l:, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stan8 ra8oi8le6h xy/w 6jtding on the edge of a large freely rotating turntable. What happens if yo8 wji8y8 e6hr/l6axotu walk toward the center?

A shortstop may leap into the air to catch a bal 4j tj6jpu dgr+ wg v)aliw91+,0nyb7tl and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotatybnrp0 td++u67ij alvt) g,wj94jgw1 e in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a heli j bnimzf.8g (3)ok1ovcopter must have more than one rotor (or p .z)bgvn om3i o(81fkjropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radifhz (:yiy,ke* ans: (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. Calc0i3hodxsrt jg) k1i.t-(u z 5dulate, using the information inside the Front Cover, the anurdjdti)5og-th1xs0 z3i. k (gular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at t0gf 6qg*rinzg4k0asr *3j;uehe Moon, 380,000 km from Earth. The beam diverges at an gg03nkezg *irf juq*0;a 4rs6 angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned ok*c8+fyy k8b36e ho-yhvbh(jff during operation, the bladesfvk8yo+3jk(c*ybhb h8y he6- slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball 8bgei(ik6i j; on a level floor 3.5 m to another child. If the ball makes 15.0 revolu6g;j8e ii(ki btions, what is its diameter?    m

  A bicycle with tires 68 j;* v)cfrs /szcm in diameter travels 8.0 km. How many revolutions do the wheels mak;fv* )zj crs/se?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at skzjx ,v6bs0kev ,u7s9t 0p 5s2500 rpm. Calculate its angular v7us ,6 vse kv90s5sbk0 p,xtzjelocity 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.f v/qlnq g/(mnfazc*zh. ;h3: 8–38). (a) What is the liq.fh ;nlchfmq (n*3zav/zg:/near speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) itup mpk/cn6o, m5gr8)n its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a j ifv((0 tyd/gpoint
(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 centrg( r3u84/; 0 mmpakxx:wro ofiifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelerationmw /u 4fk(0grom;8poxar:x i3 of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130w-b)me9* j1 lnz nxrf- rpm to 280 rpm in 4.0 s. Determinee-)*xr1nmbnzw j9l f -
(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 radiush k)i+rqk /kc *t *)cm1o*ouzu $R_1$ 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 7 -c+jn0l5y akp02huy+yhsr l the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceleratenlhc+50 alu0 jhy-+rk72 yspyd from no rotation to 1.0 revolution every minute 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 rest to 154b9mwnu .(.nq ftc(xr,000 rpm in 220 s. Through how many revolutionc b 9qm.4 .rwxnn(u(fts did it turn in this time?    $rev$

  An automobile engine slows down w9cfda98v ; 1d yfabrggrdu;f3r9 vy7:: (b eafrom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “hlqx-9mh42hcp/fq a g s ph*:q9uman centrifugem qg *f -l/cqsxh: 9a2hhpp49q,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diamei16fr6- ounpp ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hanp 6pi o-ur61fve traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns0q0q 15jq mwvqk2nxg6 1500 revolutions befv6kq qjxn5mq1 g0 02wqore 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 its speed unikokpl.jgpp *f9h/3-o) b;zzeformly fropgbpo-;l k3jz*ke ofh p/9 .)zm 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

  The tires of a car make 65 reo(.h,+dubnewl/ok m1 ( bn7a hvolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tire hw d ,.(mol1b k+a(n/ebuh7nos 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 when clicv2/ z2371zzn8bjhc8ag p iek mbing a hill. The pedals rotate in a circle of rad1/ b7zi8nh 28zjg3kcezp 2cv aius 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 o2elf e8k/oy4tsvz ldk (.dx59f 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forclke48xfz.oys5ldtev/d2 (k9 e 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 Fikurg*ln: * kd0g. 8–39. Assume that a friction torgrl:kuk*0 d* nque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachkrm;jtr,8jyerkh, *) ed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held rk;jjh,) yr tm8,ker*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 tighteninl g2a1meo2x,ng to a torque of 38e 2ga1o2lxn,m $m \cdot N$ 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 10.70anhl qtb)db f clnd)hx+ 80;8-kg sphere of radius 0.648 m when the axis of rotation is thx0)fna 0db;d) tclqnl8+7bhhrough its center.    $kg \cdot m^2$

  Calculate the moment of ind *i2yf : 0h)pyy77wf/3ejblhw4jtzv ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire h7f/de*27j0 p f3jy yythhv)wz :biw4 lave a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ballfln o) n9- tg zfw3;3ieujnx x4/(i6gd on the end of a thin, light rod is rotated in a horizontal circle of ra )gogfx jwd il3x(;eu6/nnt 3f4iz n-9dius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping f 0l5wk gv*xfvv4-rufpt2x) 1a bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwee vfk 1xp25v4fv*0gxl -u) frwtn 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 objely- cq39qj4 mfcts shown in Fig. 8–43 about-jm3f9 4 lycqq
(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 tox ;ezs yd3de):tal 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 s*9m ;,na) s boqp:lsv)u0fhw gatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. Ifps0mov ,9ns * ;)g:flaqh)uwb the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a m0c- hb* byb.y lyh3y lk(+wz wadtq60-ass of 0.580 kg. Calcu+b(* balwczk0b.-wylh3d- ht0yyqy6 late
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3id8cq( ftyq(;w)3 snw(hv5;t hsuf8t $rev/s$ 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 h +xq2eowy nz1xo4nq:/and-driven merry-go-round and is able to accelerate it from rest to a wy+x:4 eqzn2xo/no q1frequency 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 evvwy th-* 3op4ientually brought uniformly to r*4vy- otp3 whiest 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 ball at 7 rsz410ij h38thda)z3aoo ;g0 vr -ihv5qqs(h $m/s^2$ 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 solelyvgkq 2g)w m2 av6j2q9l 3jwlm.xh,u4 m by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from restm24 qwk)g.2vu9l3lg qmhmv2, w ax6j j 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 considerevng a8r)lv 6.nzarc0;;o4 ceh d a long thin rod, as shown in Fig. 8–46.c;nl06c.h8v)a g4earorv n; z
(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 masslgzgo3ohc h :c4jux-7)y *muz b7v75wes, $m_1$ 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 q :v c)hqn(rrn tj0l88from rest within four full turns (revolutions) ann h (8:njl)8qvrqrtc0 d releases it at a speed 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 has a moment of iner wg)b*y /tzf,atia 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 torqvz*;dyb9xneg+k r;3 l ue 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 without slippingz4xi6d(mjs y5ronx 86frjoz2p 7a.ud d5s+q1 down a lane a zyjs8x4. frm2r67 pjnd+ zi (5561aqoduodsxt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th;x fw+-mj e3lvca+y7 ge Earth with respect to the Sun as the sum of two ;7fwlmegv 3x- y+ +cajterms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How.l/ lj 27wcaw-z- it(7 2j 0nihvpooao much net work is required to accelerate it from rest to a rotation rate of 1 7a0jcjht a2n.-zo(o-wl lip2/ wvio7.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg y4c()o.aie bh ql98.xtgik 7 nstarts from rest and rolls without slipping dow.kaxo)bh(c7lgnie itq489 y. n a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall qvbrv v1n w9b/g-r 50iand its lower end0vnv-91 bwqg/iv5rr b does 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 on : q 6aqhigvp8z/ lf.m-the end of a thin string in a circle of radius 1.i6v q-:ha .zlpm8qg /f10 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 uniform cylindrical grinding wheel/ (94vlun, b0,z vtegrhquo.a of radi, o0,vl t9a(zeuq.4bvn/gh ruus 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platfob;98 y9tmi qio) +pepw2wi;y hrm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, F+ w;2po8epy) i9 i hymb;tqi9wig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment r+zx6 k l:x03nkegl 9qof 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 spee xlkz l3+rk6qx:n9e0gd ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ranlk(hq+udi; 3te from an initial rate of 1.0 rev every 2.0 s to a final rate of q+k(h3dni;l u 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 axoaz,:u h u,p2(sdgio +:gvln +is through its center at a frequency of 1.5r2ush,il:+u (zg,nvao:d +go pev/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. 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 spinning at 3qadq f+ /deq*agr:7i7.5 $rev/s$ 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 momentxnh 4-l h0bs;h rpi+3lum 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 ofelot;1n fm( qxbwjg +u3sn)v;m70t9n inertia I is dropped onto an identical disk rotating ajqf; g (n1vxt0bn 7m9 m)n le;u3tos+wt angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atyxa2 d1gw0 5ql 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 .x4v t + zaosnv:iq76nf1qi0rstands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of:i+fx arsz.oq4vti 6n7v1 qn0 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 angulaso;rc,u ve 6w9 1.hx5xut n,jbr velocity of 0.cresb69jh ,5;ox .vxu1 nut ,w8 $rad/s$ 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 wh 8(z g;nstk8nz3px6 ni2 ,oeqzite dwarf, losing about half its mass in the process, and winding up with a radius oziz3z8x8q 6tk; npne,n sg2( 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 win*v:s9 bwzoyw-(9ap0 kazb9to ds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass d nb w7178jrht 2tjwr6 fv;0lf$ 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 can rotate f lk5p*i8q9nuqb 1 bwi+reely without friction. The moment of inertia of+* i1q b9np85ublqiwk the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diametl yqgtm3:eb :7er merry-go-round turntable7y tgm3q :le:b that is mounted on frictionless 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 of the rope lying on t ctw vhum)21-t1cl0eqhe 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 of2hc0)q tu- v1wcmt le1 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 always faces the Earthkavt(si p7 01 k0yto4e. 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 particle orbiting thea(p tsi10etvyk 0o7k4 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ouez z3x(v 1m3ef 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 tboqc2l pszd k; *f9o*7he shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest9bzk 7o2c *sfqd;lpo* 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 cylindrical disks, 9n ps:peydl77:zsk nj7fj2b8each 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 reachesbsp8 kf7np:js2nze79y7 :jld 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 ant5 2axcq. 3yrwgular 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 masxh6)j- 5,llwpwb. shms 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angswhh )-l jwpx,6m. 5lbular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile7l3zo.bm 2wf, pw/ tau is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has/,wuwz 2t mf l73baop. 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 illustram1g1eeh ;h a fl-qu0c3, 5rlzutes 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 xrcvsy,ci*, 6surface 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 caxnv 5j b(g0lgw+s:85f wr s9bxn pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The r 8:slj0(w5fgxvb9bsw +ng5x 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. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and.) h l :qjno1;rhwfv-y 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, wherwqoh:yln-v .; h)jr1fe h

  A bicyclist traveling with speed v=4.2m/s on a fvzlxfdk,e9yp 1d *z;5lat road is making a turn with a radius The forces acting on the cyclist and cy*;d,dkze5y f9l1pzxv cle 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)n2 goo;+ ztedu/o eh9 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to ac;)noo heg2/dt ouez 9+hieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods8,qs *bl wxyu4, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outsty4uxbsql,8*w retched arms, and with arms held tightly against her body.   

  You are designing a clutch a:j wc;+rdohl6 or* o(ew*, 5ens+ntnassembly which consists of two cylindrical plates, of masse ntdjhe;(o o,s65nc*rol na *+w: +wr $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 lons, 1gbumus1 n*l3dwio7s9- qoped 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 q bo g1-s ns9i3w,1su*ml7dun the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumjx0k*( f7ub-n/p p mvje $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a.htyx5 r/ vco6s the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) .ycv/ orth5 x6What 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