A bicycle odometer (which measures distance traveled) is attached near then .*mdyhub8+ :qeq ,rnylw(v8 wheel huw* rqd vq 8b:.nenlym,uh+y8(b and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Dzjq2sy:n r- ukt w4::;mqxn z:oes 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 whny ;wr:k-zjuqsqz:m2 4n t::xich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descri /hu)h1so yi a6n3syg*bed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?xsig o ab47 k*+l(3aur 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+fg7u:dbr wywse08ee1)l .lgb s 3j+w your head than whe js)weeydbwbf08+ 7 . wrg+3eslgl:1un your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the reldk3fr tu f*70ar wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? ltkff*r0 du37 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 run fast have slender lower legs with fles7yh73 j:wgwkh )7nssx h and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distr g 3ssyhw7n:7)jwh7xkibution of mass is advantageous.

Why do tightrope walkers (Fig. 8–3530ev f/*zy-.dbu y7qqqemc;m ) carry a long, narrow beam?

If the net force on a system is zero, is w9:iy4e 3ntx vtg p 26dkkph3pu,gb07the net torque also zero? If the net torque on a w4k:pxp b 3d3g6vuhik7 t0,p eg2y nt9system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hmdsas5y 1b,. horizontal. The same steel ball is rolled down each incline.am,hsbd5 .1s y On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (w2oo xt ntk,d 14lt9t:n4 xiv6from rest) down an incline. One sphere has twice the radius and twice the mass of ttxn4xon2wdo,kilv9ttt: 614 he 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 rwed,r pjv*gud1x e+*epgb xo, wys 46;j:m 52kadius and the same mass. They start from rest at the top of an incline. Which reace*dejb:wo4j mw5*;1y rdges ,v6k+ p,xxg pu 2hes 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 rotational KE?

We claim that momentum aq qdjs7ed80tj6 /d 2zwnd angular momentum are conserved. Yet most moving or rot6/7tqjdsedd2z qwj 80ating objects eventually slow down and stop. Explain.

If there were a grean67a(gu d6jhjgk szlu8y9a pcew; ;01 t migration of people toward the Earth’s equator, how would this affect the length of e ;7;lzs 6cug6(gaujdkhnj10 8ya9 wpthe day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatioit.brjgdy) gsku+,)+ n when s.ytj),i u)rdggk+s+ bhe leaves the board?

The moment of inertia of a rotating solid disk a8l5.. ymy1r ,;ws vljjnokgpo +9s5jvbout an axis through its center of mass 5jy lvs lm 9wrp. ooj+vj1.;5sn,g8kyis $\frac{1}{2}WR^2$ (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 rotatinv +r 688k6sjudp:4j(ko:rzk zmg(vl6bp izn;g stool holding a 2-kg mass in each outstretched hand. If yjkop 6sk p+g (4 u(8j8z:r kznir:lvdvb6;6mzou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However n o7(km57fx v.s ml.1ns8esqm, one is hollow and the ot5nf(mnmxok1l .7vss m8qse7.her is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontalr+7czb, zkb im),/dfg 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 atgf,,izmzr +b )bc 7d/k the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standinqgrz8b1-eirs ; /j5kgg on the edge of a large freely rotating turntable. What happens if you waliekrqj5 z/g1 gbrs;8-k toward the center?

A shortstop may leap into the air to catch a bw+dnfc)o*6 3 qt ee4gh4 y3aw7pwjz.t all and throw it quickly. As he throws the ba ejztgp6q+ w3acye f3tn*. 7)w4wod h4ll, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dik13 ryb 4ohd4pev(pv p0q2i+qscuss why a helicopter must have more than one rotor (or pqk0 3 p r(p4hbv1i+yd pev42qoropeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingq bjz4 (75 whht4a6fc(w lwxl* angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate,*n(e8ox ltrxz)1 6 hmh using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moo (8mzh)*ethn61r xl xon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direc h)jc fe5gl9, b rl+opyf+ umvc;-v,(bted at the Moon, 380,000 km from Earth. The beam diverges at-h) l b ;9upcc+rmg+lf5 fo(v,vby,ej an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at p) 4cd8*x3 sli g+dzgha rate of 6500 rpm. When the motor is turned off during operation, the blades dx8s i lpdh4zg* g+)3cslow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on 6sv9co5fz+3d)t jfy5npn ua) a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is i5 d+ vt9fjn)p3sf)56u anzc yots diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolul.e.4v u 5sr7xe iuykmn9 (8hations do the wel7muv r.u.i 89e(nhyx 4ak5sheels make?    $rev$

  (a) A grinding wheel 0.35 m in diamt+omm/:,c ojpeter rotates at 2500 rpm. Calculate its angular velocitm/p ,c+mjot :oy in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8– s:4d(:hb.tg imsm7u u38). (a) What is the linea4i.m:bg :(sd uhs t7mur 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 velofi(- :p n8xnkzspb)h9city of 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 speex.poznsojh,boptq ;(q.g)eregf 1*4+ 02s lg d of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle n. d*zw -c.lxfet7mg p ,(.ayo7.0 cm from the axis of rotation is to experience an x.t.wglan (-.dfeo7p c,m*z yacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 12igr,wn vl m*qts4,q630 rpm to 280 i 2gn q,l6qrmt,*vsw 4rpm 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 radiusv ud9(0e+hl9noqpmqvh k;w5 ( 2 hf8fq $R_1$ 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 ab5-phlgoy:t h +-nuiwb)as vm1,ha4 ;woard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start -)nly;aw pva hwh g4o 5+:t us1hb,mi-of their 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 unifor 8/mnorjkhml,pw 9 j:ud:n/7dmly from rest to 15,000 rpm in 220 s. Through how man7:r/w ,nnk9/8p du :lmjohdmjy revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 j q izb9hodb)1ncnh838)l r/ irpm 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 highsp wxs- uvfh 61-+2l)mkk;dcv wqeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching ilf+w hmxsdk-6 uw-21cv )qv;k ts 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 unifo 8 ybuvym9.lstl:1s/ lw+:rq prmly from 240 rpm to 360 rpm in 6.5 s. How far wivs:98/ ytw+p. ll 1bq:yul rmsll 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;fxhyu2z*: 7 phco 1ux 850rev/min It turns 1500 revolutions before it comes to*hufxxo7:1 ypch2uz; a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unif1mm,qx+dj l 30)9+pg/dp7zehsni dxv ormly from 95km/h to 45km/h The tires have a dim9 i dd xj vs7++,30)zqe/ pdnpxlh1mgameter 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 m 0esoe *z)i/ndake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diam) ie no0d/*sezeter 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 weigri:y8.jb x2apht on each pedal when climbing a hill. The ped.y brjx:2ai 8pals 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 on the end of a door 74 cm wide. fcc0zd(9ygwoq (ii 87 What is the magnitude zq(gf cc ywi9dio(780 of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Asspq t+1.hbt,hm ume that a friction torque of 01httb. +ph m,q.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,m+8p7jn 7m-ucq y t3ftb2fj e3 are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initiall3c+yte f jq2j7t3bu7f mmpn8 -y the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a tq9q/ vzu e2w8iak0k+forque of 38 i8 kk0vq/q+zu a92wfe $m \cdot N$ 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,cu:75p3iupcy4f4 kc e7 w hqs of radius 0.648 m when the axis of rotation is thr q4c7h4u uyp3kpf:w5s, 7ei ccough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 wfzi eezx0sy/9 (tt9.f5y 7o ucm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored7.e0fuwo9z y fxtiye s /(z59t (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, lightce :s0ya+k gno,/ d+ gl*bsvlyk 64yp; rod is rotated in a horizontal circle of;l dknag++v4/ poykb,cls sgyy 0e6*: radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating a9a/ywrv/*1c lk yqb3k t constant angular speed (Fig. 8–42)r9w3abl1yc/ *kkyv q/. The friction 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 array5exu2v;we9 0g aae,n pbd mw7; of point objects shown in Fig. 8–43 aboute2gdnu5bem e9 7p;awx;,w v 0a
(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 t)3: l *o1rxp 1zc qvps(tuq8tmotal 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 fzgy 87 pq:4e88hg d1*icb puhsatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kh1qp 8zuci84 y7pd g:h*b 8fegg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm a)vsd*i0-d o: w a3-p2kkmsruind a mass of 0.580 kg. d*2upsmis0-a kdo w:k3i-vr )Calculate
(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 froml6ds1irwf zz+qb. k. 6 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-gk wwmj4:/b w-dl d1mpbd)(7udf q80uxo-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 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, neglect8fwuuwjkd7)4 d-0p( x lwqbm/:1b mdd ing frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is evensrhqvg;v:863rn +vmatg0y cd w2 9ss,tually brought uniformly to rest by a fric 8hvtw,sgdv0 vq+9 m ga3nyr;:ssr2c6 tional 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 acceley m:n3fcrqzt 8tso1gk(.1m0p d6fh0crates 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 thex(l5 jzgqvn;c 3u/owtu)0df r,0qk : y action of the forearm, which rotayz qrxq(3jl0ng5 : ukc/ wfo,vd0tu ;)tes about the elbow joint under the 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 con0.u8e)i k-q/qa anpgca 6z 1dlsidered a long thin rod, as shown in Fig. 8–46.qq)dalc081gn/u. - za6pi ea k
(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 j/91vkch0+q*ursuhl 5pq2e sconsists 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 from rest within four full turns tpwn0u7 () wsr(revolutions) and releases it wuw7(r pn)t s0at 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 oe.rw;+ rk b9uqf inertia 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 dewu ttl 2up84-xvelops 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 andnm8a+j oc 7jp( radius 9.0 cm rolls without slipping down a lanejnj(ao7c mp8 + at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earc u 2tsn;z:rn9qz 58hoth with respect to the Sun as the sum of two t:ho9ctr ;58uzsnn z2q 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 mass of 1640 kg and a radius of 7.50 m.g: 9ebg-boj,6f oz ua/ How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is aa:/,bggbfjz-6oeo 9 u solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest +mu5e4 abr31m xmnfdaswj2:9 and rolls without slim42 f1:rjb+we xnam3s du 9am5pping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It stawgs./xy g-ste6tz . vxv +n) bt11tww:rts to fall and its lower end does not slip. What will be the speed of the upper end of the pole bngw1 )+s 6-g1zv/ wtex:xy.tvt sw.tjust before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball t/ z6s4ym. 6 p4(3qmfdzmzwlxrotating on the end of a thin strinmzfwlzq4 s/myx3z6p6t 4. d(mg in a circle 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 2.8-kg uniform cylindrica35wr /9hwafq *dpqk v0l grinding wheel of0q59kvp hf w*3d/wraq radius 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 po/ d/)(if on +exnsq yp, :kyj+(fdfz6latform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal posi :ndi6f q e/+,px)jf kyo(f(yon/d z+stion, 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 thej0c zila t qjke+vnj4m). /lg)yk3 l:yx1f 7q: one shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuc:c) e .jmvi)yl+l:g3qx 7n z yjltkq k0a1fj4/k 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 jdpjvn3ig.;o ixx.d4i g1gvbh,5( o0 her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of ox 3 vgd j. p(.j015i og;hivgd4bixn,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 through its cw3k 5lcp9wr b-::nvw nenter 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 wheel after the :wbl95 wcwvknp-3r n: clay sticks to it?    $rev/s$

  (a) What is the angular mr06 f ,j4r bshe) qx4:ddsh(3j3e qtpb 0j5uphomentum 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 mom xddvg-u3pb*; zn+ wd*entum 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 onto1i6go.rmhn) h22l plr an identical disk rotating at ai2 )pr.1lhrn6 lmog2h ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.;nzwy6 esb ;c lcw79ujm 8y;94jhhq q34 $rev/s$ 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 g8;vgs. :m sfkwb3 gx)stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of i v8sg:gw 3mxbfsg.)k ;nertia 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 vel 2u*96yf8d l t7odt)wu/rpwizocity of 0.8fi p6u/r z2w) *dwu78od9tylt $rad/s$ 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 ag(wt6 nus 1ic2 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 carries away no angular momentu(twg s1n26icum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess os-i k17f)ynzxdd xw :8f 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 12su+q3h j ,nnst g)qtrn 8k*k$ 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, initiabj6u 24j)ccg n (vi+c ,cjnr8ully at rest, that can rotate freely without friction. The moment of inertia of the person plus the platforubjj)i(, 2vn+gun6c j8c4 crc m 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 diameter merry-go-round t: rs v jiu,.kdkm*qy23urntabjq ki:r.kd my,3 *uv2sle that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls ue rju+b21n6 ae6erv2on 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 without slipping. What length of rope unwinds from the spool? How far does the a1 uer ev+2j6u6b2 ernspool’s center of mass move?

  The Moon orbits the Earth such xpl3gb9kiy yd 7s*3,xthat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.),y 3i 7l9*bksd3ygx px    $\times10^{ -6}$

  A cyclist accelerates from rest at a hk/jiua( )wc3rate 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 3k4og meno z+.shape of a uniform cylinder of 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 mooz4 nog+ 3.kemtor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mas*ym hal-q*va 5s 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of en*qy m5aalvh-* ergy 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 spephe,-m(2;aadoqv, 1cn1 ph dled 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 great, were (a4t*gf lhmvd0eo8fm :b /vq(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 pd8ebfmmtv v: l0qoa( 4(/fg*h rocess, 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-pollution autpwbgrpb ;. crwt,1+v)ym6 )e lomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 16 yr )wt1+ ppwbe.rv)cmgb, ;l400 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 illustratei - ,isgmijj4 uzj8c.*s 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 (hodw5v+cn 96doh op) 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 lengt +wf fjzkwu+b td4+ 2uwmq)9bsu4j2 x+h 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, b2+qk fm2 twju+wsud4)+f9+4w bzuxjas shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. )hun+zss d 6qpu8*h/pum7q l6 It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that n lp7qhuup*6zqh)6/ du s +sm8it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spe5bwxwh86 lx*mc 6b*fyed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cy*b wfw 58cxhy*6b6mlxcle 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 int24n1bcd(xjvj:tt7h t o a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requi1c: dvxt7jth4jb2t (nred 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 a2shc8hw j i e- tk3zqm75hyz87dd3 vb2s thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of th8d qkdy 3st 28zi 72-mchjev7z b5hwh3e 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 cylin :3m4-t v.u(dlayk zrydrical plates, of massdz .muav t(y3 -l:ry4k $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a 1c/5(ik 9: xsoeieuvfs.mm)rlong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble m5c)k/ e.ousxev is i9:m(fr1must 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 do nma8qc nu )46iqot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tmxkn f*p2yp s) h/gx);he car reduces its speed uniformly frohmkgf )y x;sp/xn*)p2 m 90km/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