A bicycle odometer (which measures distance traveled) is attached6,vrj el8t8bo9d+w vt near the wheel hub and is designed for 27-inch wheels. What happens if you use it oelj9,o r dv8wttbv6+8 n a bicycle with 24-inch wheels?

Suppose a disk rotates at . a hw8wfig u84:o2gn5f1m,jt*w-p pr zeh k5wconstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceler2pw,ke: h.4jwg5aogwwn ht-i 5fp8f* 8zr1 umation change?

Could a nonrigid body be described ;jyu y t1 8eeyt6y((oqby a single value of the angular velocity $\omega$ Explain.

Can a small force ever exertpa/gl :ycjx lf . 1.8oma;jqf4 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 hands behind your head th;vah7,rxu 0n-gg8 ;, kt wvsntan when your arms are stretched out int vva 78s;nx kgrt;-,nh gwu,0 front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the)-c qmgsca,u+ pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a smalc ,m+a-gqs) cul front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hm;oqg z r783xwave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the barg7 8om 3wq;zxsis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry 9)5n qrnxn5d4wu6wg oa long, narrow beam?

If the net force on a system is zero, is thbnt c*(rtav5 w*.jr x.. *xfeae net torque also zero? If the net torque on a system is zero, is the bftr.eca t w.*xarvj(*n.x 5*net force zero?

Two inclines have the same v so)tozm i5nr9; zo5/height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the spe;9z5/ o 5imz stonvro)ed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. wdhvtkjt(w xnq ; -1y*9e* hc.One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the .evn-w1 yqth(c k*td9wjh* x;greater total kinetic energy at the bottom?

A sphere and a cylinder have 4u, yw8z 5m kqll6smzb2cod) -the 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 the greater rotationalzmous4k6bw-) 8 m,2 zyqlc5dl KE?

We claim that momentum and angulj g0 kt9p/2k 1o8glsnqv:q;spzt+2u s ar momentum are conserved. Yet most moving or rotating objeps2ultps:k8s q;zjkn 9t 0gv o+/g12qcts eventually slow down and stop. Explain.

If there were a great migration of people to/w1h ,hlq66t z +oua jsf2vmx6quz 2t1ward the Earth’s equator, how would this af/ht 2,1 qxmvu1uawo 6l2fs zj 66+qthzfect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatio9q; iiia 3na,gz mp i9rw-,+pnn when she leaves the boarri i q,9zpna3aw-; 9m+gii,pnd?

The moment of inertia of a rotating solid disk abou,t )xhtfa1y3 ot an axis through its center of mas3 ,)ayfo1tht xs 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 in each outoqj +su3wd 1c/rdq1j3 stretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Ed1sdqj /owcq +33ru 1jxplain.

Two spheres look identicalucj jon ;hz2k6wj4*60wzw c;y and have the same mass. However, one is hollow and the other is solid. Describenw j; z;wju0hocc62zy j6w* 4k an experiment to determine which is which.

The angular velocity ofe.u 8o vrre fi0b/rsa( -82ucz a wheel rotating 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 ae80oz(us- 8e.rc rav2i bu f/rcceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standiv*ohptxqoo7 w,(dyfb,i ; :9xng on the edge of a large freely rotating turntable. What happens if you walk t(hxyo*twobp d:oq,9f;x7v i, oward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As rb5. t /cfxx)uhe throws the ball, the upper part ou x5/ctr)b.f xf his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a hjfb )t.5wkf1buisk,y(z( j k ug,3u e)elicopter must have more than onb, tkubu.sf g1jj)ki ,e (uf z)w53ky(e 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)l9h;5d4yd 492xqf7j9j1zxhzi i vcxl 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 coincida8t h,a:q6bp1*xul7yj 2 *kaeptn z)y.tnp.bence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun anu.ajty,qbk) t*8 a .t xzne*ypn lp16ha2b p7:d the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Th+t n 7;o- kn3bx6rwdlde beam diverges at an anw b+l6 nrxddn73to;k- gle $\theta$ (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 6qt4s s9ldlod7r2. q/ v500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. sd 4t2v7.srql o dl/q9What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball hihzw oc* b:7 68wkjm9f:cnl:makes 15.07l n fwij:mc6c: bh*8 kohz9w: revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameeob *6rt6kimo.0oh7+zeh my8 )z,dqt ter travels 8.0 km. How many revolutions do the wheels maktzoe) 6k0t rqi8.ym+oomebdh,*h 76 z e?    $rev$

  (a) A grinding wheel 0.35 m in dia homwdje j4ix-8:e/ lud-yj)2 meter rotates at 2500 rpm. Calculate its angular)e2h/jyo djlj8-mi: -xeudw 4 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 wxi llo hn6/r8;x +4ggcomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centlxw6g4x8lhng ;/ ori+er?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity opvv y8w9 ov.w6f the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poqj:uw3o: 9fcwak6t2q3 ,fc ak int
(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 part5.ityt o ia kvoj::p):icle 7.0 cm from the axis of rotatioao5vi)y:tp:k.j i: otn is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centegv(wswh,+h ow15n-t w r from 130 rpm to 280 n vgww,t1 +hhs5ww(-orpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius d lo7e5fn09 av$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecrafo -z8z hy7pyusnr ;6;tt put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The sp6 ;zsyt7p;8-uor yzh nacecraft 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 acceleratep21o 5 fcc4jzes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutif4 opc 152zjceons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to :pnjv7nnsf7q ip)/)hbrww8 81200 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 flytgf) ogkw;7smk7 . c(jl+:qm,:pwklj ing highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to:k. 7g(s mgmj)op+jtl;lkw7k:,qcf w 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 acceler76d2; h.2 ks zqbkb tr+j)nntxates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wi) 2k6.2bd tzqrxnb jt;s7khn+ll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is e p1sau-nd 12drunning at 850rev/min It turns 1500 revolutions be pus2e ad1-d1nfore 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 mak8g)pnwf5 jgm5 e 65 revolutions as the car reduces its speed uniformly from 95km/h to 4pn )5g8 jg5fmw5km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fro;ioea)7p8vc:3pp u tzc a7z7z;+sd 5rj 8k afom 95km/h to 45km/h The tires hav8cd 7f):a8rz75;ak js7p oz3z vipeco+tua;pe 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 ea li0yr8lhp2 v;ch pedal when climbing a h8pl vr0l; 2yhiill. 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.np2 b*wtsms-k e(uq)b/n8m r on the end of a door 74 cm wide. What is the magnitude of the t.pt sb 2)smnwn qm-*/eb8ku( rorque 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 tor+ ;yn2m3hxbas que about the axle of the wheel shown in Fig. 8–39. Assume that a friction torquynh2 3 ma+x;sbe of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m ,7ugeuf9 gep6y99w+tpd xmp/vex4o., are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then r4x 96/ogpyp9d m.t ef px9uue,gv w7+eeleased. 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 torque yxt21:x4fe sn4psp:a w*a5yb of 38afp4:ts 2b4xapy51wxs* e:n y $m \cdot N$ 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 ron8r we(fhrk;itk*+s48 r .qa adius 0.648 m when the axis of rotation is througha sqhkfre(;rwtk8 irn* +. o48 its center.    $kg \cdot m^2$

  Calculate the moment of inertia of x5 -7tvvhfd8g a bicycle wheel 66.7 cm in diameter. The rim and tire have a com-vf8xv5 7dtgh bined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end o/ce:f 8z r4krqf a thin, light rod is rotated in a horizontal circle of radius 1.28kq/ 4rrcfz e: 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 a bowl on a potter’s whe0n3 ket- ;kaegx .*mxkel rotating at constant angular speed (Fig. 8–42). The fricgmank*0; t 3x- eex.kktion 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 objects shown in Fig.zp,0x m8ldw:t 8–43 8w:mzp xdtl,0about
(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 total97n,1ira b 61i8mkdyxpj kw*y*pop r. mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning yy)(a /xz6gg gkmi29r*wv ef02oa6hcat the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellhaf *y9z ow6r6kx(cggav)22 eym/g0iite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform iq8 i2 kl w14n w::fce3k:zk.ohhcha4cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculateoikn44zqwcfihc2lk e18 ::hhw ka 3: .
(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, acceldqg uebu a58q7ybs*20fjf-o .+t y1p lerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-g ck (xx; vz3ayf70u8bgo-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 k7 3fx8c(;uagxybv0z 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 1z (ondty2+; fmx p7sx7s;xem80,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 yos fdt;z8x( nm;77mp+ xe2xs $m \cdot N$ 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. 8–4ag)sh bpe+w9*7vy l0a5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the act7 ln2a8yz2;iq bjj)y bion of the forearm, which rotates about the elbow joint under thzal8jq ib2 ;7bj2)yy ne action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be consider8ty u*.plia6n ed a long thin rod, as shown in Fig. 8–46pul6yta .*8n i.
(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 ouq7m m2 jwh8d*d7j..5dmbbsvf two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four 9d :d0uas3ax v-f6qg)fjjy1m full turns (revolu9 10 d jsfvxaum3 )jg-qd6fay:tions) and 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 inert1* jc.i/b -q f 1qs4e-ah(x f.mnvgyhkia 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 de(f;so 3dpaj 1 n0vz9sn0- gtlzvelops 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 rad* c 7x k,jr +g4pd8egtykw9+kbius 9.0 cm rolls without slipping down a lane a+j+pwkx *9g84rke gk7d ,cbytt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with 7*uyap 6egrkbyb5/ozyws s9v()i7+k respect to the Sun as the sum of two ty9uwo+6s 7* ysakbgei(y)zbp75 v kr/ erms,
(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 mac, g*(txk nzdexx;wsi308 m4w ss of 1640 kg and a radius of 7.50 m. How much net work is reqt;kmdcx4083,ws e * x(wiz xgnuired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest ank +q u ,3wdzm44s7:,hvoi9yt7rel fizd rolls without slipping down a 30+zr isi9u,ym3 hok4,7 e4dfvq7 tzlw:.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. Ito a6kk/y2 ktoni 7vq;9)(hp 8tb8uru4n*zzya starts to fall and its lower end dopi yha8ovz94k ty) /uo *zbq2;k8(rn kut7a6 nes 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 the end of a thin +p1bnzg/ju.k usg/x- string in a circle of radius. /un xpj-g1bz+ ksug/ 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of1,daw c2z 5risl(2cqa l 46nlf a 2.8-kg uniform cylindrical grinding wheel of ral2 asc6 a ld2qin rfc(zl4,5w1dius 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 platform thatn 3hei*w 3z(j/s:dlwf is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, lz wj3n(fh3ws/ e*di :Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mcnx6x;dooqt* -ty/w6oment of inertia by a factor of about 3.5 wqo;/tocd w* -y6 xnxt6hen changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin root1qzg5+ k3 mjtation rate from an initial rate of 1.0 rev every 2.0 s tom+ kzg51qo3j t a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis tj.(/ sepkmc4 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 fla. mskpt4(j/ ect 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 angufr9 v zu42xx:6xs4 1r7 yuetzblar momentum 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 moment /fxi- aoytn/(um of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an i0im w l249f5sk oxy)(;ykboaidentical disk rotating at angula0ix (wm5fa iybo)k;492l kyosr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 37*vi :m9mmydqr1svx(w; ds e2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-ve7 l0n1 nlslp ,r9*fcround platform of radius 3.0 m and moment of inn09l 1n7e *v lpl,frscertia 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 y4i xndic8c c095xc :ovelocity of 0.8 80ci:4cy o x9dccix5n $rad/s$ 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 white dwarf, losin cd:ufr1-;zgv5r+f ( ) grirmrg about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round cu5 : zri(rg-vfmr )f+;1g rrdto 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 ocw*+2hywk v t pnhg017f 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 7xsf u4xi,lzk(1 g wqcrf82b *$ 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 platformw z bckae5dbij *885)k.q7 qqd, initially at rest, that can rotate freely without fric )qd8bqz5q w.e8ik*5ka7bdj ction. The moment of inertia of 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 diameterc*bsx, b. 7) vlzr8ttd merry-go-round turntable thz. l7)*t ,btrc dbvsx8at 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 o(::sxv vth*,uhgwj )q .ss/ dtn the ground with the end of 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 with: hw) *ssjt gvvxsdt:,/q u.h(out slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side :m6:u0fi ut1i*0wgzsp8t pvk always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital g0t6pi8u zw: :1k vutpif* m0sangular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest i605 3plj*ainphrb.s.r/ d x zqan( a*at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder d)yu2m 8vuc 9gof radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constav)m 9c8 gyuu2dnt 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 mass 0.05 a3bo oo :w.,gxylz-y70 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-alz:ob73. gx oyo w-y,yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the jew.hr;ass1yv f .(w2rear 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 St.bc uushfzx5iw 41 vn ;ol)rl-7 40xkun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would xs 4u 7r lvcwf kxt4.)bn150oi;-zulh its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energf iqzu+y50 u-1gb5o f jascw;8y stored in a heavy rotating flywheel. Suppose such a cary5fu+15 is faq;o cubg8j-wz0 has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an wijr8 tm)jx mppzni-6 g47pw.b .3ji4$H_2O$ 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 surfaceu8-vy h lp lqsc456w6b 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 al bf.8v9 my2,lff/r ww 1jky+qnd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of 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 show.r9 bfk/ l21+ yfjyl ,qf8wmwvn. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standinfc+:y c1dvv0-s16olx *n xi hig vertically on the floor, and we want to exert a horizontal force F at its axle so that it wiy6foc:vdv* i lc 10-xx1s n+ihll climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roadjg5bngbfg 1qy3v+g5-a22 n q l is making a turn with a radius Tj bgqaf2l5q +ng-yg n 3g125vbhe 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.c a5ptfu4u 5 ulo )qo3w0d6x9 6tvqs-a50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it fromqv9 ow 5qc35 6tad6tu fsxu4a-0oplu ) 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 onui a5xs u6j)6jn2.g 68 vculcylinder and her arms as thin rods, making reasonable estimates for the dimensions. T6auiluj .sc5xvn2 ) 8ojnu6g 6hen 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 assemblys xz(q3 df9k68f 6q8rcmkv 7bz which consists of two cylindrical plates, of 6 dczs3k8 ff v7qrz(kmbqx869mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alono;hh3 .z4d3k,z q3 ,wtdhlox ig the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if iti4 3dw; hktq3d,o3z,lxhz o.h is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumeo ld/7mbz) fml /w3d6-x pybd5 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ph p(:32zgfw gha2dg 5car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelep: gz2gga5 hdh3fpw(2ration 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