A bicycle odometer (which measures distance traveled) is attachedk f:-k;dq -(w sxjk-r0luk2az near the wheel hub and is designed for 27-inch wheels. What ha -a0ks-2 (k:uk jdfq zr-;wklxppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular0 v,*omiz2pmf velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radiampo 20v,*z fiml 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 of tha7nz.hkfqhp.js h1 +(pxqzk* a8 *zx* a6sug7e angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fosuk4un5g :6gn 2fn k/krce? 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 yd9c1;do:ikihm2 +-*o :b givvbe f v7uour hands behind your head than when your arms are stret9h o gvmkiobbd fiu7civ-12*:dv:;+eched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thl* bxm6yh62owree at the pedal cranks. In which gear is i *o6mb62l yxhwt harder 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 being able to run fast have slender lowim- is2*4t kjmer legs with flesh and muscle concent*4ijs-t mkm2 irated 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.bw)+u1rgx )ao 7vzd+a 8–35) carry a long, narrow beam?

If the net force on a sm crt+ lzxsw,2, hs/n-ystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? hlrc-nt zx2/,s, +mws

Two inclines have the same height but mjf40s bm3av y8 livv,5ake different angles with the horizontal. The same steel ball is rolled dow8is v, 3v5vab mfy4jl0n each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an indttkx8k4v i.3qq pn(7 cline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?tq(8vnkp.74x i3 tk qd

A sphere and a cylinder have the same radius and the same mass. They staracwsevbd0): i / jg:kg0+hfk .t from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the g fg0dwk:baehs0 c/vi g:+.)jkreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most /4zmwgb*qx il w )sp7)moving or rotatinp7) l/mzq)b gxw*w i4sg objects eventually slow down and stop. Explain.

If there were a great migration of eextx a-c0 bh0*8iz(bow3w 2f9qi 7 hupeople toward the Earth’s equator, how would this affect the length of the2utw ee9h(a0qxbi*zi x 08fbw hoc-73 day?

Can the diver of Fig. 8–29 do a somersault without havingdhjw3-rpn h/ x0p kvi(.so/w1 any initial rotation when she leaves the (0jrds/phv i wh wno1./3x-kp board?

The moment of inertia of a rotating solid disk about an axisgab0g8x1(7 x(ff r 4xeobiio2 through its center of mass g20oiox1fe rxgb((4 bx7f 8a iis $\frac{1}{2}WR^2$ (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 holdz0ys)), m( ,tlus-p lbb; myjiing a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will ys-0 ),s(tzymlyi,b )p mu j;lbour angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havwmd7ak ,tx7 xs2l 6br6e the same mass. However, one is hollow and the other is solid. Describe xst6,mlrkd72xwb6 7a an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizonta i :fihxvz2hh+x5o1 +kl axle points west. In what direction is the linear velocity of a o k 1iix:2hhf+v+5 zhxpoint 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 decreasing?

Suppose you are standing on the edge of a large freely rotating turnj 3v qdm8o op*4:03wx9dto qh-c agky+table. What happens ih 39pqqcj *v4dom t k-xw8+03o:odya gf you walk toward the center?

A shortstop may leap into the air to catch a babpbe0.ih;ack h,kv7io6, 0 8ya 0ekahll and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wypkk.kb ;0iabaei ,ha ,v80co6e7 hh0 ill 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,uvg6j7m q2e3ie; 7nnb discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelluen7 gmj3 ni27;b6ve qer can operate to keep the helicopter stable.

  Express the following angles inw kh:ahw+,hp-/o gnt2 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 coincidencea8nr lhb(8**1uw zmm ul;fir 3. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earthlfm**;ma8ru (n813lhiw uz b r.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0dxkp++atey;s(0prm pazkb. sr +1f(;00 km from Earth. The beam diverges at an angl ka+s;.z(sr(x yerfa+ t1 pmp0dk +p;be $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is e4esh l*hc 01j,4fnt gfkw) n2turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelerationen 4 4e)t1k20n cf,lfjwhhgs * as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothewbz56:c:oyp3 h eb)ner child. If the ball makes 15.0 revolutions, what is its diameter?nbhzo3ye:p) 5b6cew:    m

  A bicycle with tires 68 cm in diametge54 bu3sqkk5er travels 8.0 km. How many revolutions do the wheels 4ks 5ebguq53kmake?    $rev$

  (a) A grinding wheel 0.35 2e4 s1qqm9,r o:vzkkj4mf .ybm in diameter rotates at 2500 rpm. Calculate its angular velocit.k,2mkfmyjs:qbe r14 o vq 94zy 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.uf-;bql h)wi)+,m 2 bhxcx9fv 8–38). (a) What )f;qlhw cu ib+h2bxfmx , )-9vis the linear 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 /n+om *r9gc-ck hn;pyits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pt+o-f wkw;c/ coint
(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 frowfnsgi -h) n70m the axis of rotation is to experience s ghn-n70wi)fan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to .1hwem-eu s9 h :ouchgfn2y9 ;280 rpm in 4.0 s.yh 9muef:s . -9huo;2nhg1cw e Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius ikor u 5nnt1*t ,v+vp5$R_1$ 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 rm lqgy.oh;z 7 2jwv1+//admhApollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m;mz1w/hqj o+hla /y.2dr vm7g. 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 resty,-u/ dbovxo :: zigfc6ok)*m,9 amme to 15,000 rpm in 220 s. Through how many revolutions did it turn in thisf,g* :ve6bdc9auy)mo ,ki- o:o /zm xm time?    $rev$

  An automobile engine slows dowb1hcbzbk ( j+7n from 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 f4u39g nhx06vpj2qti5 d2*rh q q) zjxzlying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finalqn 2h0d6iuqh)3 2jxrpt9qx 5 4zz*gjv 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 unifor76)() f 3kbwuplgvlkemly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the u (bf)g)k pk3vw7ell6 wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min I;3q.v w(ett ym ag04+my5isk rt turns 1500 revolutions before it comes gw y+0tkr smtq a4(5.;meyiv3 to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speezg(7ziqb, igo ud +12h+j0pifd uniformly from 95km/h to 45km/h The tires have a diameter of 0.80g i,+0i+bf d uo(ji2 7zg1pzqh 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 revolutiig6 tv0vj65cwuc0 b;ew(n jb4ons as the car reduces its speed uniformly from 95km/h to 45km/h The tires jbt0 vcbne4i6cw06u;w 5v(gjhave 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 ai 8l*y4 l6joeweight on each pedal when climbing a hill. The pedals rotate8lo *l4 jyeia6 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. What is thfu(7- 31hv gd9kg+qrin- kuzok 7jr4 xe magnitude of the torque if the force is exertfk uv ui+r o1-4h -kz7xkn7d g93rgjq(ed
(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. zm.d *-8z e,x yh+djsa4jiv;f8–39. Assume that id sy mvdf*z;a,4xe.j8j+ h-za friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,1zphu(hr,-- euqvk ro h*+su7 are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. ( hvur zqo e1h*7+s,hu-r kp-uCalculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder he qgkse1 xf* xwvj/):d9 pcd23yad of an engine require tightening to a torque of 38we :9p* 1s d/vj2yxcdf xgk3)q $m \cdot N$ 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 inertsl iv.b)u w5;nia of a 10.8-kg sphere of radius 0.648 m when the axisn ).vs w;lb5ui of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm o: ah,(c hniu)in diameter. The rim and tire have a combu ,n(oia: )chhined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotate( , jmozg y3ds1 7:onh-hhjg9md in a horizontal circle of radius 1.2 m. Calculago7, h1 s3jz jg:dmyn(-hom9hte
(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 e(y8df)kau o+ wheel rotating at constant angular speed (Fig. 8–42). The friction force between hyae )fuo8 +(kder hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig./sxz uwv p./y9 8–43 abou/9v s .w/zxyupt
(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 c0,o2g;y6el i xfz )ebof 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, unifojk-zq.2 mpe 1k-5k0o2afnyj frm cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite imen220ko-fk.-j fqpy1zajk 5s to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a( ndg1foo6 +k4 yflt-n mass of 0.580 kg. Calculate+to6f df l4 n(-nogky1
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3 k6iule1- ;goomp,qy:) 8rs ze$rev/s$ 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 smaly4 gwfvb0ax.7ddq1 +b l hand-driven merry-go-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 f aq10 xd+gywbvdb47. 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 offxeh) .p67g *1huu ptna and is eventually brought uniformly * t)h.a ph176guepnxuto rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 acctwd f )cl3va7)elerates 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 solellqm)c4zrm62 lt9t 9vnq7 )ucpy by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–426zclt9crpqqm9tn )vl )4 um75. 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 considered a lojmhrn9bd)loxreo tz9q/ :(+ + ng thin rod, as shown in Fig. 8–4 (zlm 9/ dojne)9rbt+o:+qxhr 6.
(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 7w nr7brb:rmsm 2l8(o 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 (dqf* t9b: q+koku0 yg8revolutions) and releases itfq8g0:9 +koq ktu*d yb 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 inertia o*qh y d fd:pai40x65oof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developsc0z 5 wo3ppsch4a)a)m a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls witho9lb )c mxlv)2ts1/q amut slipping down a lanvlm2)m1a q)9cx/s lt be at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the7t 3 x2 g)fhyg*(rrqiy Earth with respect to the Sun as the sum of two terms,3txg2(ih g rr7qy *)yf
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net w7p fstn(bsev v p;9h4,ork is required to accelerate it fs(4 sb;pft,h97pv venrom 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 ma/ 5g9enk h*xnha fn 5msr:t10gav*f1xp mx,z3 ss 1.80 kg starts from rest and rolls without slipping down hh:0f5 *n1*a mgma 3nf,1xxg 5rexzv9sknt p/ 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.f2sjew 75 5*mb-zxiqy It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation 2fi5 q * -zmjxe57byswof energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en4wko (y07k;k 0 kaoxwtd of a thin string in a circle0kay7kx; (0okw4 o kwt of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mnme ,vrf :. 4 cah,, (d pf,e0tduli*bq(wpl:iomentum of a 2.8-kg uniform cylindrical grinding wheel of radius::m4q,ldhw 0(d a f,e(i, uirp e. vc*,pbftnl 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform y0e yhkj0,l b+09ljhwthat is rotating at a rate of 1.3rev/s If he h l+ yjhj0k9yw, el00braises his arms to a 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 hub. 9yrgvql c;)u8y3w5a q-k /k ::ir5azhpuFig. 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 s3blrk/zqq;:: 5-5yiu9h au raw8.pvhgy )kuc he 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 i2zoujkbh us-zr.2 9 3xnitial rate of 1.0 rev every 2.x9.ubk2s -zrozj2u 3 h0 s 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 vertical axis through its center at a freyb 5 rqzj3f6 xde/7nq0quency 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 clay sticks to itx7jq 6 qr/5y ebz3dfn0?    $rev/s$

  (a) What is the angular momentum of a figure skater rl0y/8 trl:as 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 momentum of the E7f:8/muoww w alv w+4parth
(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 mofu * 1*jpllbo(ment of inertia I is dropped onto an identical disk rotating at angulab*u1f lo(*pljr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 c 4ggnyh4:u/x9 6loor $rev/s$ 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 7.w b 2 xi:vgr rfjpr7ig9c4d7of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 97:2xjgf.97gpc rd i rwr7b iv420 $kg \cdot m^2$ 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 r h *+kf-1hmlx93 mj3s j)bx/pg 1dzhois rotating freely with an angular velocity of 0.8 hjg r3xf9k-o3/s h m1lpbhj)z+1x *dm$rad/s$ 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 dw+cvw nnld;kn0 ) ytw(san ,*p.arf, 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 momentum, what would the Sun’s new rotation rate be?(round to the nearest intyl0nv*; +wn(ps.,kdtn) nwaceger)$\approx$    $rad/s$ (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 (s.k8nyl7y5ueo*b1m * isdjv 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 )fw jtj-eup92la 8wb;$ 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, thnj6c y4u )z)zmat can rotate freely without friction. The moment of inertia of the personc j nz)muzy64) 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 diameter merr 7-us+qcue04p pxuk a3 ev7q*iy-go-round turntable that s-u4v c0a7pekqux+iqpu7 * e3is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground wi: vsdf88r*+qbl iz 0eyth 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 do 0z8qr8lve*bds :yf i+es the spool’s center of mass move?

  The Moon orbits the Earth suq*+jt )n.dwenxp ou4q2- 5pfgch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (abow tdoq)pn .52xp-*ug+ j4 feqnut 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 rat;: 8mnzs+av9:1k sx)tgozud ne of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of aqa5 aka 1)z+wy uniform cylinder of radius 0.20 m acquires a rotational rate of from rest o kqaya5wz )1+aver 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 0.050 kg 9ez nxl.g e102:b ixfqr2mm 2yand diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation eqx0n1l mmgf2b:i 9z2rex2y.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 speed of thewxg+ kgj qy7o8+5fh rc .(sv*cph0;ij(i7aoo 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 cz xr/we /*ji9Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarter ezw*cx9j/r/i s of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is c6 k zqr7cwkvf29mn(4 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 f rq6wfk (c47nmk 29vczlywheel 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 illustrate3*iqcnyyn efk5 c y,+;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 (hoop) is roll, (ma0pduw 6kxing 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 freelnd 76 nmy,2lyzl;ks9mezsi5xl6 l1d*y (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. Amln5n* x2kzes6i6yl;1s7dly9mz l ,dt 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 stans5qa(i, jz*quxp )g;oding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb ,x5o gs ia;j*u)qp zq(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 mw375a-bz saklaking a turn with a radius The forces acting on the cyclist and cyclekw aszba 3-57l 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 ;2cf9o g;v4u lg nof+m1.5 m and begins whirling the rock in a nearly horizontal circle above his oc f;+2n gfo;v9ug4lmhead, 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 thi v7hho0x)6i-du jd wjswfm7o(l4*/dxn rods, making reasonable estimates for the dimensions. Then calculate the r*odsjflw/)u h7 wdd(60 7ixxo-jm v4hatio 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 cylindri:hsvqwv7zo p qa,*)u .cal plates, of mass uv:hps .wz7ov )a,q*q$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough tra(y2omde lxl/ 1ck of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to rex/ ml2od l(y1each the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but ) 1)uw zh:anqqdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car f,,g34)zz o gqh/lviireduces its speed uniformly from 90km/h to 60km/h Thezgg,i q3oh4/ ,fl) zvi 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