A bicycle odometer (which measures distance traveled) is attached db-z v11lm+fav2p,1mxil +xm near the wheel hub and is designed for 27-inch wzm1bam,m dxi+v -1l p+v1lf2x heels. 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 ruh mjgkdq, aa3q0fp1y5+ :4slim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangksuq+m,0 d apqlaj h5f4g:y13 ential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular j+wea 3 aokz;lyq,+d6velocity $\omega$ Explain.

Can a small force ever exert a greater torque than ah fw(wippaua)e 7gu 9vm i3;,+ 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 headvu)*0 d+ su;fglki/bm than when your arms are stretched out in f+sv)gud 0klubi / f*;mront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocd2a759 aj:/p rt n*g6dpre xhjkets at the rear wheel and three at the pedal cranks. In which gear is it harr56da2:9/ xp g7e*th djarjnpder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beingmcny1,s )t)nb pi c9)z 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 dynamics, explain why this distribution of ma)y9 cinp )s1 z)mnb,tcss is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long1-oiomfl.k:hhe 8m 7 z, narrow beam?

If the net force on a system is zero, is the net torquefd4a*4n j5 wp.lt1jyu zkp ,(o) 4ndnt also zero? If the net torque on a system i 5pu pj)a4yz1 j4t.ktd*nd,wf(lnon 4s zero, is the net force zero?

Two inclines have the same height but make different angles with the horizonta*wgwr.y u.dk +l. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explainww .k* yd+ur.g.

Two solid spheres simultaneously s cm ((rykxq)52khf v)htart rolling (from rest) down an incline. One sphere has twice the radius and twice5q2f ( )y(kcvmhrx k)h the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They stae ou)ldm1bw/:2l/9ed* sx kltrt from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the g1e u el//wd9k:oxt2d*mbls) lreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that moment/61wla:1s1wsfjyy 0 4hotkznum and angular momentum are conserved. Yet most moving or rotating objects eventually slow down a4 11af1sts: z nly/y6k wjwho0nd stop. Explain.

If there were a great migration of people toward the Earth’s equator, hohnk g 895pi/spw would this affect the lengh/sg5in p9k8pth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initic xq 1ba9qxsoqm-0+4h z) *(e. lxoei+ldat (eal rotation when e9 o+ab)e 0(qzh4eqatq lids*m lx o.+ -x1(xcshe leaves the board?

The moment of inertia of a rotating solid disk about an axis throou;g65+idy k a;0rnw jugh its center of masnak y;o+u j6;wg5d0i rs 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 outstrt 9kjnt.lp,gf-* gt h d*zy,u5etched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay thep *u,kf. 5g t* lg,dyzntjt-9h same? Explain.

Two spheres look identical and have t,0 *r /lqyw wuylt)+.7gkdpe hhe same mass. However, one is hollow and the ot0euh *7p) wqy/tr,yk.l+ldwg her is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel 5d*8e* i gg gplnv.9hwx+ ab /(fpvaw*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 acceleration points east, describe the tangential linear acceleration of this point at the top of the.9nlv8b/ pgea**p vhgdfg+*(5xa wiw wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Whatkdy/ l tll3x.4 az(:ka x dk/.:yla3ltz4a kl (happens if you walk toward the center?

A shortstop may leap intolk;e 5va fzefw+swfw066v nn5k -wv 5/ the air to catch a ball 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 rotate in the opposite direction (Fik nfe w/vvk5-z;5eww6f l s5+n6f0avwg. 8–36). Explain.

On the basis of the law of conservation of an:okga7iep5 (di qh +7cgular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more way5i dc: eip+ah7qo(gk7 s the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiaw. 5rbk( z,zorns: (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. Caduj2(8: dsnl glculate, using the information inside the Front Cover, the angular diameters (in radians)(:dlgs2d8nj u of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon nq0*iq -*n bj)ll7k/mru0akq , 380,000 km from Earth. The beam diverges at an anglilj mqk l7 qranqb/)-nu*k00*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 ra3:aetf ;b1t9 ca) pbfrte of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the anga)tr 19f c:tbpbfa;3 eular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to aspb fef12v884c 7xauf w:qw1c 3k1wxwnother child. If the ball makes 15.0 revolutions, what is its diava ecw71:k8wxfupfwqx1 cbf32184 swmeter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ha ; q3jl03xx*ybveik/ow many revolutions do th;eq3bvx xjly/i a0k*3e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in l p:z 6,h79oe1ot7zbmv e6wsndiameter rotates at 2500 rpm. Calculate its angular veloe 1,6tovh m7z79 obz :w6slenpcity 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.q(znmy6, 0ree0 s (Fig. 8–38). (a) What is the linear speed(6mze,yn er0q 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) in its orcoo (/o(/ fcadpps yp;f9,gm; (dv cf9: jm/ombit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point q ;t9y1epufo5
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm i4h cg+1jv t;e n)wsi agdtw249/ v)icfrom the axis of rotation is to experience an acceleration of ;1vtsac+)j/ e4 )viigtgwn2 wh9d i4c100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 20 an(( 0edjnkrn4ddi, 80 rpm in 4.0 s. De0k r(jne,d4n0a(dind termine
(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 s:gjw c2 9+/s/x-1eybuufuak $R_1$ 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 abo0kyjrdx3(s1j3axpbmk/rb4 p9 /8n/c sw0b aoard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of thwjo9bs (r /43p1pajbx/kx arbmkc00y/ 8n3sdeir trip, they accelerated 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 1a7qi63tmho vy /g tf(,5,000 rpm in 220 s. Through how many revolutiontmoyi(3 gtvh fa,6 /q7s did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpmdc5 32lmr c-pm 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 j:d8s :n3ibof flying highspeed jets in a whirling “human centrifuge,” which takes bo:j:fs8n d3 i1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 8uq/0(cam(ak:ovuxq 360 rpm in 6.5 s. How far will a point on the edge / uq8q 0(:k(v amoaxucof the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at uh2-mn:pvg*yb4nl( ; ym30yo 6 lniqrflw 03l850rev/min It turns 1500 revol g wbn3hp6( *0lyo:vlqyf2nirnlu l0-3;mm4 y utions before 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 nm dgjiw4)l* qmkf+) 1jw0vx1the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diametewkgnl*dfmj )0im w)jvx1 1 q4+r 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 u,rqas vkwlyp l91b7 g m;pz(bo0(e/+vniformly from 95km/qz7v /0evp+ ;sogrlpbm(9lyk1 b w,a( h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weigouwggz x)6t 2axo;n 2)ht on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 x6txgoa2n2gz)u );wocm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Wha8) tbupnmr83qkc;j 4 8xtx(yb ,ude3rt is the magnitude of the torque ipdm,k)y8cu3 tr b ;xb 848jxqu3 ter(nf 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 aboutv3/:25xuv gd3ikb6rto o0dmb the axle of the wheel shown in Fig. 8–39. Assume that a frictio5d6oo3vtd 2 0ux: kivrbg3b /mn 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 ma9wld0 )ibozz1a4l.g t8xrd+ ljhpm 8; ssless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of t8rl. ab jtihx)1;w dl+ 84z9po lm0dzghe net torque on this system.

  The bolts on the cylinder head of an engz7v35enprjj 4 ine require tightening to a torque of 38 5e zr4pj3v7 nj$m \cdot N$ 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 scgvaxe) * qo3m;yh-oz ;)hi lh,n.br7phere of radius 0.648 m when the axis of rotation is through itscgoevih ;* baqr-n yhlm37z h.;),)xo center.    $kg \cdot m^2$

  Calculate the moment of inertia of h(qp jn q8.)0hgcclp5p0n zw j9(w(d xa bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 qg(8.pnj h0(0 (p)hpl dzc cnqxw5w9jkg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rky +nrd 01pa0ecnrg; ulf*l1 eb q:75hotated in a horizontal circle of radius 1.20bkn *c+hn;e1 :f7erldg r1 p0lyauq5 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 b cy tc 52bu3j(sc1kj,eowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clayjcuc2 , k5jtyb(3cs1e 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 g t+c2kw8)xq npoint objects shown in Fig. 8–43 abx k +t)8g2wnqcout
(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 t p7o nyu;yzu*0s/xja v 67t.wbwo 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 spinning at the correct rat pw02d4qkhji* e, engineers fire four tangential rockets as shown in Fihp42q* jki wd0g. 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 radi mxo ,oy )opmz,j1p*mw+n9uf:us of 8.50 cm and a mass of 0.580 kg. Calculpmny9*z1j , :wopmo o ,xfum+)ate
(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 xh) vpr./y )cvvfb/-)yfrka 1w5rnm* to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round an bf(7;+ 6az svvowps4sd is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.sob7s+ faw ( vs6z;pv45 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 ro ku,b0 u,jq;abk8v+ ketating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a friction0+ ;a,8 vubkj,qe ukbkal 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 ac*+il/ chf w1,jca1 zlxcelerates 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 action of the forearm, whi-gdwu+)7yew 1 i5mhy9 1ogopvch rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acdohio+79g1gp v) wwuye1y -m5celerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–t9t 7fb0+gpba5 -u+u kcqhld)46pldbf5ut)b70uk + 9gchta q-+.
(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 consistt*s gaq9mj.g8 4 0nsmcs of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer j-ys32i 2hqjx from rest within four full turns (revolutions) and releases it at a spes2 qh3y2 jjxi-ed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of in;hwc/g / (gmoxd-cmb7ertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ulw0igch kv66hh -(7faxjq 7 8of 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 -g5w cnt8uef3c m 54gr;2h fgjqkf)x5cm rolls without slipping down a lane at 3.g4ck -hnc5 gwfqg5m5)ue8 3;r jxf f2t3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th gyl42:vyk*02l1 ky.v iamkrhxi mm64e Earth with respect to the Sun as the sum of two 0:1ykk h4m mkgvi6mil*l a2ryxv.y 2 4terms,
(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.- mxcrqyc0hs cde2ni445/ zav3y,, fb84bb yc50 m. How much net work is r5-xs /ccyv4 imeq,2b nr b0ay yb4 c4fzhdc3,8equired 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 asi )6b ;k)rgqfbo:k88o8v nfbk1c 3b knd rolls without slipping down qn:s vokr1fk)c8;f b6ki b)8 g8bok3ba 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 verticall5av0bi ;smm (qy on its tip. It starts to fall and its lower end does not i b(m5vsm q0;aslip. 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 momys0kwl (4y utc w) )(* yngmsxu67;ol*fpx,i oentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of ()ysn7)* lm6yx40o(c wulp,ity fo*sxgw u;k 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular7zrc8tut )h9c momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wh rtcu c9t7)zh8en 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 o9 by-vx0 3ilfp :h*qfhis side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, thpf*yifo0qb3h9x vl -:e 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 of iner+ux. 1 v blgazowx7)d.o/b dk1tia by a factor of about 3.v + b.)d7wb1./ ulzoag d1xokx5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0donc4 )()+a ykcoxf o 20u(ox;onhvz- rev every 2.0 s to a final ratezoyncn2;(vxc ox f0 )-) kd+(h4oouao 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 wm,y2h7txz q zta0+t4gcz 2k4vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the whe+tzgxkw7cmz2t2aqy ,4 zth04el after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a1l l.rp26-y b,u fg 6v8aoeaist 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of54g,0m70zl1 q blqgn n o8py5adtc (dw 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)6 ve n 5l-lsq c0ewj)upht;y7r3-fws of moment of inertia I is dropped onto an identical disuwf- ;pt- lesjc5ln yev0q7hr)63ws )k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a/ r; ejki3rm4zt 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plateh1)- vzo0hre form of radius 3.0 m andorez0) ev h1h- moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-h;l9 *drfr o-,fnar0sgo-round is rotating freely with an angular velocity of 0;90arf, n frhors-d l*.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 event4o83qhx- 2c+ d rcgwixually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carw-cxgh +o3 xq4id 8cr2ries 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 ;v9t38(0pdrmg hbi tswx, sm,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 vc*vy ;5eofyd) o15 lv$ 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, initiw 2gb/kez9y;ually at rest, that can rotate freely without friction. The momz9e/u g2y ;bkwent 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 diamcv-grr 4p 9yptl 4rq;hb ih1/b0+ y(nketer merry-go-round turntable that is mounted on frictionl gr vkh4n(/1qt90lyyp-4 bhpr;ic+ bress 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 w2jwxe; zdku.se8n -5z oj 3:ythe rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the swd jzoj-:5 z32sen.ywku;8xepool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Ear cgm;9kh4k,jtth. Determine the rtgk 4h cm,;k9jatio 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 the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 c26qqlf+xh6.tm: 0xc 1vbu gdm/$s^2$ 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 pj+m tiun)uqzu,8y xpwy1 q4 c; (jm5:radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motpxyuit + u:(uz)cq1mq wm 5n ,p4jy8;jor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindri7dgvo3:t8s * mx *yemrcal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 mm *v tdorg3 es*xm8:y7. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed9rwm om8z2u 2w of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as p w 0xn2, lxfw(at; :kck q2cgib(t66xgreat, 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, kx60 qgctwa; n, xfc2il:px(b2t(6 kw what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutioezzb,c-g1/.1yo+ xa apm zuv- n automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrixgz,e/o czza 1b y+.-pmv-au1cal 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 illustrauooc8)mqr aq3efq0 s43h8o 5:;ts(t z bx(hcptes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface akd 1 , y,.5de0shbhkqcb(uib9 t 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 u42i k*zi4o akp(:owi.us lxsj 1nz4a,n +ie6 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 horio+kiiwjk6z.,a4 li 2(: os*i uxns4 a 1peuzn4zontally 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 v-rqw4a f n8f(nertically on the floor, and we want to exeraqffw8n n 4r(-t 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 v=4.2m/s on a flat road is making a turn with 9w,ab6s 6-n2*txma rryu 1lgja radius The forces rs1*x6a ,m 6yjrulgn92 -btwaacting 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-t,q g+a(x miv into a sling of length 1.5 m and begins whirling the rock in a near+i a-t vqm(gx,ly 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 achieve 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 rods, maki)3x6w ekt t.cong reawo.t k)6xe3 ctsonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plkt)/ v w*;*wybl kkdu:ow6kw- ates, of mass u/w:k ykd 6l*ww*vk )o-; bkwt$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 5iwz6wz- n7b9 a7mcn)m-ov,gyxkp 2a r rolls along the looped 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 the loopgoz5k9,- ax6c2 ivwwn- bm77)mpy azn without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, boxjg sn *bo 30mp3h98k-a:znby3 a*vkut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutioj*/o2kv7 s8y6cpg clhr uus 72ns as the car reduces its speed uniformly from 90kms7c22y8 pgh/ cr6k*ovju7 sl u/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s