A bicycle odometer (which measures distance traveled) i 1 btr3 ij. wlrv zp2q+vct,pl,i.d91ts attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wit,b1l,2 9tvrl t .v 3p cjrz1dq.w+iptih 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a 3x5pb-wq;usj mi 7vykvv 8;4kpoint on the rim have radial and/or tangential acceleration? If the disk’s angular veloc; x38pkvv47jw- vq5imyuk ;bsity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ofbv)aks51sf e:8p2;mr;nz t fj the angular velocity $\omega$ Explain.

Can a small force ever exert a t,yep5rg+.st 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 yogos/3 t13jws1/yipqon :ws3our hands behind your head than when your arms are stretched out in front of you? A diagram may help you to ansow1ois3g/s3:3woq s/t 1 jpnywer this.

A 21-speed bicycle has seven sprockets at the rear wheel p950yuh yq /tmx sse - vcen.7h9:dic8and three at the pedal cranks. In which gear is it harder to 0 -dpqvs8s7uy5mx9cyh.9 :nti ehce /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 being able to ru hsi3- nwb)epyrwxm)f 2r-+l.x2xvu,odr7 9wn fast have slender lower legs with flesh and muscle conceduswfn .-x)+39r,ip oyb2re7vmrwh2) l xx- wntrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a louz)zmode/ 3mk; vxp. /ng, narrow beam?

If the net force on a system is zero, is the ne-d+gu ;y6zf aouf7z/jx x7*.zbnou;4 2 htgkc t torque also zero? If the net torque on a system is zero, is the net2xfuodf/u* ;ny 6. g4cgojb7zaxk zt+huz-;7 force zero?

Two inclines have the same height but make different avp()p nz0 w2js97iq ts pqz4 6b8uucb9ngles with the horizontal. The same steel ball is rolled d q(q z9p wts6zb487pu n9bjsi)pv02ucown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneohuwcjv4wh8: 0c vy(,kea1 7mausly start rolling (from rest) down an incline. One sphere has twice the radius and twic(aa40h18y:v7mcu kj eww cv,he 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 saqg rhj9k((0rlc s 2pa/me radius and the same mass. They start from rest at the top of an incline. Which reaches the botjl q(9(rasrph ck g0/2tom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or rzcyk q7r8z() gotating objects eventually slow down and stop. cz(y)q7k8rzg Explain.

If there were a great migration x dqwf*-zua(;of people toward the Earth’s equator, how would this affect the length of th- ;dx(*ufz qawe day?

Can the diver of Fig. 8–29mgxk, 2kgi 1q: do a somersault without having any initial rotation when she leaves , g21qg xim:kkthe board?

The moment of inertia of a rotating solid disk about an axis throuxf grvg8r4ob2qn7a ..gh its center of mass is.abo g.7r 8qfnrvx42 g $\frac{1}{2}WR^2$ (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 ik yl1 ts7xl:t w*y5 gohr0xybdw//i+5 n each outstretched hand. If you suddenly drop the masses, will your angular v:/sl ykxly *i1o7t wb5g/ +5hyr0xtd welocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is */ao :oug otwjxh n8db1l6(azl *0ih(hollow and the other is sol*t(o06lb8/x luo:*aod j1n(h wgh ziaid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal l/ nytgyc o:z9 1n v6(zxjwj*7axle 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 wheel. Is the angular speed increasing or decreasingnngljzx 1y/ v9:wzotc*6( j7y?

Suppose you are standing on the edge of a large freely rotating turntapszkw6 q2f 34dble. What hap2kzq4f dps3 w6pens if you walk toward the center?

A shortstop may leap into the air to cajzj.00q f ae3v.,+0:scxwvwm.tbdkntch 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 +cv0d .j0j fkwxvq0sam.t:new b3,z.hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angull u5u)o-cxxn4 ar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellec)5uxx-o l u4nr can operate to keep the helicopter stable.

  Express the following angles in radianfm)pvwd k)7 *ss: (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 op pyn8r9gm1 p;f an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamet9 p pp18gmr;nyers (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges v*ak+(xr; . s j (32/ibjnukjrjaleh: at an an3vbrs.ix h2njk(ja;r/ *:a+l j (kuejgle $\theta$ (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.60j1j(ayuz omo 4u,th When the motor is turned off during operation, the blades slow to rest in 30juzjot 4auy ,ho16m (.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 mfeiq mw; xzdf f8.d;qt7r. 8-x to another child. If the ball makes 15.0 revolutions, zdf 88ex.. f7q;- fqixrmwt;dwhat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do dmq;5dw*g 7qdtgqwd;q7d*5m dhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C;twgomd.-) jis2)w(n o yor2falculate its angular veloc(tm2)sf2w;j. w gioy o-)r ondity 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 rjdxt7co zd32)el j296h4how 4eeybv8evolution in 4.0 s (Fig. 8–38). (a) What is the linear4od2h279 ejyvxl4d b8c 6)hw3etz e oj 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) in its orbit aroun/9jd5vxz(a usd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p3mzrsc 3dp2wqz14f9+nfnx ,r oint
(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 cen*3zcbwd t2t 2so p bm16c6x(yitrifuge rotate if a particle 7.0 cm from the axis of rotation is to experiw16pzt o2632x*biymb d ts (ccence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acg3absbu k 8)f/celerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Deteb s)8g3fkua/brmine
(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 radiusi7u 50-9 b i*pxir,jrq urwi9v $R_1$ 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 spacecraft put them-j6jpjtu)y.fvt, o 0eselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip,vjt0 u6-.)ojyepftj , 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 15,000 rpm in 2 qz-zp t-4oc73uq d5mx20 s. Through how many revm 4zqtdzp3qx o-5 c-7uolutions did it turn in this time?    $rev$

  An automobile engine slows down froc9pr*/gc+d,fv-b5 / tqugg 6rpn u(v hm 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 inwk i;):k bl 27a1 ; o/uon,feaem8vkfg+uz9jc a whirling “human centrifuge,” which taok e7u: ju;kn89abe fl1ai,2;k/fvcwomz )g +kes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly :qo(myrgyr +3 from 240 rpm to 360 rpm in 6.5 s. Hoy o+3(rg y:rmqw far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 r6hsh 4x)ut3s.o4mfz-kb ykxg f zu ;90evolutions before iu3 z m094gfyu fhso.k;t-zb4xh)kxs6 t 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 caribin 5r(dx,s e*ap*b 2 reduces its speed uniformly from 95km/h to 45km/h The tires have dp br5(sae,xbii2* *n 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 unife*7m*:0rtq hiq6 h ; e(qzzuyoormly from 95km/h to 45km/h The tires haveq(:y0o;qzquem6i7*th er * zh 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 climbing a hu g6na9ux*ie+,v; g9 9pee lzxill. The pedals rotate in a circle ofugex6 p9 x*ize v+,9g9n;auel radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the e e prv7/v44 cmkqb2/rv,qpyk :y (*fyfnd of a door 74 cm wide. What is the mag4*/yb:q2 ryevy 7vvqm(f,fcp k p/k4rnitude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the ah25ymg3(e(qu : /jec myj2tazxle of the wheel shown in Fig. 8–39. Assume that a fr mu/t m j2e3czy jghy:aqe5(2(iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod4omx/k) i/2h8bl frqy which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then release8/b)rox f2ky4i qlh/m d. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requiaz37 b4*(fb pjfg: gbhre tightening to a torque of 38 hb z7ba*g:3 jpbfg4(f$m \cdot N$ 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 o8wa*(d/ui3 k-lgjw/e4myh21u gvso xf a 10.8-kg sphere of radius 0.648 m when the axis of rotation1m yj(i e-2s/8g*4olu kgh d/axvwwu 3 is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm inko (eq;1wv0 9hvwm/lobrp5 u) diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignoredho el/bqv p1v(wworu;0m5)9k (why?).    $kg \cdot m^2$

  A small 650-gram baltq. ove*wzmm m70h )2nl on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. mmoz7 n0th 2e*vw)m.qCalculate
(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 pv8ytvbc4dzx5 /;:lhbqy3 )wtotter’s wheel rotating at constant angular speed (Fig. 8–42). The frid:t)4 hb3y5c8 xytl/qb vz ;wvction 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 poo:ycy h3a15oxsjlms 9ivpd .1 )cc/t+gc -,mqint objects shown in Fig. 8–43 a.yi-y/ot 3+d1 mc o cmqcl1)ps:aj v 5hxg,sc9bout
(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 ofln/;wb n( 6fnf0 yvih) two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correctj,().h mkium/jypef ) j -ujv57phz8 n rate, engineers fire mkfp/ j)ijh7m. 5ve,8hp(uuzn-)yjj 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 satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder x+bq x5nvc1 ec 8*0hx-e*ibf/-n tkom with a radius of 8.50 cm and a mass of 0.580 kg. fb ixhxn e*1mvbx0kn*/tq8e- +c- c5oCalculate
(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, hs3m;ke 5m+lv accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small 6nsm144wn/4 hwwr qlghand-driven merry-go-round and is able to accelerate it from rest to a frequency ofr4w q/n m4wn1 hgsl6w4 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 i/hb(7 cii7gu pv 2vsu;s eventually brought uniformly to r gbp;7ucu7 2i vsvi(/hest 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. 806wp l bl/rw+,e;q7wy di.f l)ny q8kn–45 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 action of 2+qly9 p:ju v;sqbb rg(*elh -the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Th hq;e *q(b2r :9lbpug-s+ yvlje 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 considered a long thin rod, as shown in lru )tef)nj0tn 5oi(8wwgk 7+Fig. 8–46n5erg u(jno)iwtf7 )k 0+ lwt8.
(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 mass4c, u-q*vpfrq 19-xdzzki hw7es, $m_1$ 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 accele(v n: awguzif,s r.qqjeg)1:-rates the hammer from rest within four full turns (revolutions) and releases zqfwunig:( a),.:gjv-rs1eq 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 iggzc(rb 2n1(6huk-v w nertia 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 tor nw2pp3bd3rb/- etk 2wque 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 slippin,u vhbam-0)izmu :, png down a lane at 3.uha ,p)mz m,-n vbui:03 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of tvrn0w1tgct 4 f9/yw ,r1fcnx, he Earth with respect to the Sun as the sum of two terms1vtrfnwg,049xtcf cr1y, n /w ,
(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 zt/cm(n dwf7bz:)-)c9 u ocrdtii 2g+ of 7.50 m. How much net work dnobgm c-c (7t)rzuzc/i: d i+t9wf)2is required 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 and rolls wt zkdle )j)7ph82u b4jithout slipping down a 30.0 zljdt7k hb j )e)842up$^{\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 balanv (7vfd3of b(eced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the ve37d((vbf ofpole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m o2if8(t,+aue+hqort 2 )vbd f2n,dlwomentum of a 0.210-kg ball rotating on the end of a thin string in a circlv 2orf8+2et, obdld+in u)ha(qwt2f,e of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.fzeubk7h8sni-:l0 ga0 q7i r48-kg uniform cylindrical grinding wheel of radi ki0rgih478ez0fs lu- qnba:7us 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 ate1e c vdy dk9k)u:cx*; his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a9ed:euk * kx;cvd)yc 1 horizontal position, 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 Fig0,c8;o( it 6y(tn.gl0 ktf1 ninfq gnd. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is hc n( ttgfi n0gf d1(.n,ot8 q;06nikyler 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 1.0 aym,bv(:np:ek :jl5cbq ) g3crev every 2.0 s:( b:cm)c , k apej5ygbnlq:3v to 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 vertcpw-i p6akybu1-3bi * ical axis through its center at a frequency of 1.5rev/s The wheel can be con1u-p*3 wbc ibp-6iykasidered 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 momea:n vy 6cdb/b* obl,-wntum 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 momentukyyfg-df-dr:6 p g km6aw,)r .m 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 d x x/f, 5x3qyyz(s**cf j/6tt txbuw;nisk of moment of inertia I is dropped onto an identical dt/tw*,t b/fzf(qus jyxy6 5;n x*cxx3 isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 22nc 8mlpttj 0(og5ix9.4 $rev/s$ 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-ro*xj8 py;l ujo2),r*k4w ho(gxg a1ubqund platform of r g);hxbgu *j,ow8*pok 2xrj4u(ql1ayadius 3.0 m and 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-go-round is ro(tdonyx7p3hzm 2y2:::p(h b,d ldf 80tv wvqgtating freely with an angular velocity of 0by8hqfnvg2 7lzy( m , 0 :p2t::d3pvtw(dd hox.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half alzm xp g kk/,-p:ml nca1a5+6its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carri-1 ,na m5cgxmplaa :zp6l k/+kes 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 chpea.s,pj3), pa*vxn3ul z9winds 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 nz4* kdo5 mkm7gviq5 nq8z7t)$ 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,ohc13v8* jmtd(uu( jy that can rotate freely without friction. The moment of inertt*81y(( cduvhm3jj uoia 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 dadjlni r97*su.dc 64diameter merry-go-round turntable that is mounted on frictionlessid cls 79rjda*4 .6ndu 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 ropedb33s a - z:fwte3hx8w rolls on 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 8xha-sz fe:w3 b33d twthe person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eartnjhmaj*4v3 (gs l+;uxx zu ik,cu5e5,cp.h 5sh. Determi3ecz,j s vxgm(il nhpk5a5xsu, 5;uj.uch+* 4ne 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 the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/(hl/ a7 ranxm2$s^2$ 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 radius 0a355-efyacm e)i x e(q.20 m acquires a rotational rate 5c)xe eaam(5i- y qf3eof from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.0ri tf2 a s1;ut7y u)f*s-xao5j50 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0x tu;1a 7srof*jf5a)t-si u2y -m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheeliczomgxtno22c1, +v 1 ($\omega_R$) 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 grx6xg in1(:cj1 ue4p akeat, 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 spherxgxc(1 4pkna6j:ie1u e at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automojez.i k,)rmk/oi5 l1jbile 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 cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able k5miikr/,l j.1 ezoj) 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 illustratei0 ck8hf+ k5+l)cm,mlcq,xoqs 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 5tx+exmitt)fapm8i8,gpd 8 ,) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we igno0 xp8+bd o q2;cor4sgkz ,nbw;ya:ls8re 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, g2bb+ w:qds;0 r8zs;oylxn4ko pc8,a as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has rv +6hecm q(r4cadius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a srm(qh +v cc4e6tep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with s,n*lvsuz4oo r8a7g6 npeed v=4.2m/s on a flat road is making a turn with a radi a oz7s l,*g4rn8uno6vus The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m ylz( c o;6:jes(0k mzr/0lwhoand begins whirling the rock in a nearly horizoo;rlk0w6 z : 0lm(c/yjhe(ozsntal 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 cy +fkftz 6dbh ;8elx-am:xe,b (linder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arm (b;6t+:xfmb8ed azxfk- lh,es, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrica 20/bdpjd4gq xn/nm vzq*um2)l plates, ofnzvj2n 2*d0xq4mp /db/)uq g m mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro b n)um qy4swe-xjso35gp4q)9lls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble mustwy-mx) qb3sgo4q5nsu9 )ejp 4 drop if it 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 djpq r,1 ws7eo-o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cfd.p*juh:bj+-pny; 0 ws6g dp ar reduces its speed uniformly from 90km/hbdupn :wjdp* g0py- j.shf;6+ 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