A bicycle odometer (which measures distance traveled) is attached near the wheelpblnq6qk z0- 9 hub and is designed for 27-inch wheels. What hnqpkz-0lb q69 appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim h)xsxet.ks e2 id26)jx ( lryk.ave radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accelerxj) 6lx s.sety(xr2)e.kk di2ation change?

Could a nonrigid body be described by a sing b;l/ojvnkaqj )zfw7h7qv9 09sy +t4ale value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?vddc- 7re)2 sexp :v3+o,d-0bv vzmco 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:t2 ktrofi3,k do a sit-up with your hands behind your head than when your arms are streto,rik tt3k2: fched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear ww xsb9)aa4)v e cl*aa:heel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? vs4:c)a )9aawe*abxl 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 hav v)3zyy- frz f(cl:murzm-k98e slender lower legs with flesh and muscle concentrated hiz-zv:cmryul- (fz 93frm )yk8gh, 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–w5e-k9 4rgo7ki-t/ fthq3t fl35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque ra-;jk y2.mcni)tw8galso zero? If the net torque on a system is -twjgr2cykm. )i a;n8zero, is the net force zero?

Two inclines have th;*; 5rlu setgog5: ayce same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of ya:;r*g5esu;5tl o gcthe ball at the bottom be greater? Explain.

Two solid spheres simulta dl a59x3psi,mneously start rolling (from rest) down an incline. One sphere has twice the radius and twice x5, a9idm pl3sthe mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same e ci jb6qirzy:xh*ga;ei0is.sb 2: 23radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Wirzies3 sb0. ;yhi: :xgb62je 2a*q ichich 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 conseiccukro ; 60f/rved. Yet most moving or rotating objects eventually slow dc cor/0uf6;k iown and stop. Explain.

If there were a great migration of people toward the Earth’s equatoae2guh /81 x nz,pfz0wr, how would this affect the l0xune 8 gp2wh1, z/afzength of the day?

Can the diver of Fig. 8–29 do a somersault6unnn0wrk ;(x (a9 jbtvb-r1 m without having any initial rotation when sjnt va m9 nubnr(0;6xbw1k(-r he leaves the board?

The moment of inertia of a rotating solid disk abmq681h gy:zzy 5zxh. nout an axis through its center of mass zny1:z h5y.xqm6h 8gzis $\frac{1}{2}WR^2$ (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 oj18ce+ b ee0a;js/bc/fn,dedp u) d7*gt5 jgcn a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity inc etdgng1ds c e)a/jjp87,+*/f c b;c ud5jeeb0rease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, i2 0utbznp92s one is hollow and thetp bni0zs 2u92 other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horiz20rxl tjy3+htn4( sjvontal 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 acc(ht0vnls4jy2 xtr3 j +eleration 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 turnt n o )wm5 xfcy8;jht59pu)8rxiable. What happens if yot58o9xm)x)irh cjw;y8ufn p5u walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As zy 7ht0e6w5gu-gilo( y,l*p rhe throws the ball, th, i*op 65w0glluz-gyet7yr(he 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)l7z* vaihybzszh+ ,)5bh y/c conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more wb)vsy hz/z5 ha+c) ybzl7h i,*ays the second propeller can operate to keep the helicopter stable.

  Express the following angles in / 3dg y+r)0mj dhtoj 5scl,( a(bbk7sdradians: (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 amazin7o+uz7x fm6xb qen 0*zg coincidence. Calculate, using the information inside the qu+6*fm7x0e nzz b7oxFront Cover, the angular diameters (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 fromr l.u5puv+0ctb tb(kstu,/ y 2 Earth. The beam diverges at 2svt0.ry5ut/u c+lt,(kubpb 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 a rate of 6500 rpm. When the mn gqc0*qcu296*krs vjotor is turned off during operation, the blades slow to rest in 3.0 s. What is th06j c**qqnk2usvc9 rg e angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child6 9 l4zmthwu0v. If the ball makes 15.0 revolutions, whhm v04tuzwl69 at is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km2,-daizb hf;6 oka(rae z( dq7. How many revolutions do the wheels mak a; 2(obre (z,fz a-ik6d7dqhae?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its an.1 (d-odj gw twbouzk7cpl681gular velocity jokd 8-w6o1g( .zuplbdct7 1 win $rad/s$ $\omega$ =    $rad/sec$
(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 ekk9mk/ qkwlrscj / 80-l- 0fvcomplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from t/kc/9kq0ew j8vf 0lr-k kml-she center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of j) 8nfk q9zt2lthe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poif;w+hmm:5*9 i hmax mo3taj5snt
(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 from the axis om x8z,f (mvb c8opgf13f vm38fb 8cx1f, mg zo(protation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its c zc+- bg)b9e4cxkay4 d c7n+yoenter from 130 rpm to 280 rpm in eckda+y4z97 y)4ocbx+b cn-g 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 radiu+j)haxz x 1lwc 32/ilns $R_1$ 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 aboa6wnsl8.qfq2tvk-v: qk: y,o srd the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of t l,fqsqokwy68t :ks.:- nvqv2heir trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in s;9dx vdzal3j,g cctdq 6(n)9220 s. Through how many revolutions did it turn in tqnvzs,xjad( 9 tl 6gc3 9;c)ddhis time?    $rev$

  An automobile engine slows downdqoqps46v0p ) 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 flying highspeed jets in a whirling “hut0,pilj+i 1d5 tu6eg abe(/ihman centrifuge,” which taeiijbt1p/e(0 5ih6, g+aldt ukes 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 from5 pjcd9a )a 4c8az6nux 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thuc5xj6az8dnpa9c) a4is time?    m

  A cooling fan is turned o)6tbo i191vbvh bu(rvzd h7u4ff when it is running at 850rev/min It turns 1500 revolutions beforeoz vub14r6d7 th1u9 b(v)ibhv it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uwdl 1+qsotht/ een +:3fr2.n uniformly from 95km/h to 45km/h The tires have a diameter oewtuhdr 1 +./n2 o f+s:nl3teqf 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 cas9m j ei11n*)7 lf9gmt nsba,cr reduces its speed uniformly from 95km/h to 4mfmb 1,7ni9g*nse9l)j 1 ca st5km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weightcfvj, 9-.tl,na, o av 5mq3rml on each pedal when climbing a hill. The cm qa,l,r.vn,am jtvo3 f5-9l pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force oaxs nyv9h4x.6 f 55 N on the end of a door 74 cm wide. What is the magnitude of the a4 nh6xsxyv9.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 bgvfhykgs 0dq,xj)e 9. +a-4 rfor84mn pf k02axle of the wheel shown in Fig. 8–39. Assume that a friction bk egv -.grdfk f s4+jo8a0,2 4 xh90nrfqmpy)torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arhkfdp p- /(1xbwd tq+;e attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially thwh;(qpf-b1+p/tdk xde 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 tightenin* s)gw,h jca1f xi)ui*2rmmd7g to a torque of 38*2u7wac) irfj mmhd1*g ,i)sx $m \cdot N$ 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 6qod2coynd0)c +k0vnb9h, xa 10.8-kg sphere of radius 0.648 m when the axis o c0c,x2kqo)6+vndaodb0h y9 nf rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametc uwqu nxgg;i) d 7n1irn(h,w0ty;w29er. The rim and tire have a combined mass of 1.25 kg. Th; ncnxdu 7;guwihywr1 9,( 0t)nwgq i2e mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end coq (gzl0itn,u i4bi8h /( 5xbof a thin, light rod is rotated in a horizontal circle of radius 1.2 i,o 5l 0ctg(u(4zxi/nh8 ibqb 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 bo sdu*fuk8+-l xwl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hdx+uuf* -8sl kands 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 p 9j d5/94xs*bkwi7jlsjpiz ; soint objects shown in Fig. 8–43 alwk;*j7ji d49s sbxs/5 j zip9bout
(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 atohusu7u(a (j c1ms 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 s(ve,+p/l hqtwzt/ mo)ri 9t2oatellite 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 fohw(ipqzmo +ot)e/tv,2 rlt9/ rce 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 and a m* qbm kvps4g(,jicv29ass of 0.580 kg. Calcul9sp*b vci,v2m k(gq j4ate
(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 g3 i6(tk ldn eor.r1e9.k(smwrest 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 small6gl 6v:.tyozy-2n8td) v9 *m;nbr cy m4g vgko 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 kgn;n6g 8v:tmbl4tyk)or9y2g gyo.v z -c mv6* d) 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 is eventually brought k1nxs9n p-.n.mcz g,. o (k9fmmhs*xyuniformly to rest by a frp fm . ho.k(1ng-cz ks9xsx,ym*9mn.nictional 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 accelerates a 3.6-kg ball at 7 r7pafh e5w dy3y0i yk4b++qn+ $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which vwop: 8pzxy;m a+4b:zp;h8 -iyjw ) bazh;8)my8vp:wwo p x ;a zapb+ :yib-j4rotates 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 con; hiich v2:ne.sidered a long thin rod, as shown in Fig. 8–46.nech2i:vi; .h
(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 consistsr z-(,mwq+ qno 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 accgvc 1bfzl7/5f kt2 4khelerates the hammer from rest within four full turns (revolutz2gf kl57c/41htkv bf ions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of in ;ut7-6uee hepertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o 7dt) mo+3pa b13iv6j smcodr2f 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass7guen rn;:(jw5j 307wfs * )coqlzvsu 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 0s quc7)ge3 jvj :n(solfwwzun;5*r7 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the bc0nuf7k5 12 fbj mly /-/toursum of two termsnm- kol0rtf 5 1bcbuj// uf7y2,
(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 av*t9;wyj 2ehrev35/ ix-l ognet work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid g l3-evar/ih5jx92e * tov; wycylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg sdtznl734:st r1xn(- v swn p)vpe4zm 2tarts from rest and rolls without slipping tsxw zt:ev)r7 m vz4spn 3p1-2n(dl4ndown 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 balankdf *4iht/ zwqo r.4u;af12kx ced 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 pol1ak;x ti4 wq.4ru/dh*zff k2oe just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of tbks. zn85+i7d(+h wd4qtu 5fp jgc1ta 0.210-kg ball rotating on the end of a thin string in a circle o tk+c 1d5sthinwgj z4bf8+ .5u7 tpq(df 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 cylindrical grinding whe-a5jb3x0 bp vf ,mo,nu b)lw2oy2,ryvel of radius 18 cm when rotating at 1500 ,ru2 jbb5n bvp ),wvfyaylo203,xmo- 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 that is rot )b wq4 7/q1+sjg /xd*rfz0lwxkzm8n fating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreffg *xwl8k 1sq4j +)/w /zx70ndmbrqzases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moxy2; suf7/oazw;rv- dment of inertia by a factor of about 3.5 when changing from the straight position to the tucfxwyzr ; /d27-ovuas;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 her spin rota9 eg86letir4dcq(8u ftion rate from an initial rate of 1.0 rev everyu89 6tecg48lq fed i(r 2.0 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 rotatigdtjyv d4p.3t5*3bl ing around a vertical axis through its center at a frequency of 1.5rev/s The whee.id3vbp5 ltt*y g4jd3 l 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 it?    $rev/s$

  (a) What is the angular fzr*;ox4 p2yo momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth01 5gs j.pwjex
(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 droppe2 ebmpam*+0ejqe fdf ;(g hk19d onto an identical disk rotating at angular sbge*eq d9f( ;m+2eh10 f kapmjpeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns pawwvz/u-2q1ua z-rvopn y3 v:/ 4ez.at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plalr2tui73vc k 2tform of radius 3.0 m and movc2 k7 rl3u2timent 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 rotating a0 gdp-1j xl4szb9t )rfreely with an angular velocity of 0.8 rlgst1b 9jxpz4)-d0a$rad/s$ 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, losingz:fl7 )295s mkw yb,psi0vyjt 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 newyly 52 jtwzk,pb:vm9i0s 7sf) 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 e p+s j4/yf0w-pfp8j ltxcess 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 i fmm/v0,sxg4$ 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, that can rotate +jjwzgx8p5xz5 dh*j ye6jw:u.51fnn freely without friction. The moment of inertia 565g:fywj+ j ejx nz8w.5*z n1xduhjpof 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 diamete-we d,d.p k8rtr merry-go-round turntable that is mounted t d,.prk -wed8on 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 th ,bk;9kh*wdwo w/up+m e 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 o9wb;d/+ k*mou,p wwhknto it, Fig. 8–50. The spool rolls behind the 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 trf5)gvpd cd13up *9bhhe Earth. Determine the ratio of tp9d 3prgu hvb1c)5*dfhe 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/olzs7k p2;m oudg6b/3 $s^2$ 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 tm 11/zp ;,kubezs)4as( djakphe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0kzdu,4zs a1 p) pes;makj(/b1-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made d0 xqp)s.wx06c0 tnkiof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solip0x0 st .c6niwdk0q)xd 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 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, howw1 bgo e5u89he is the angular speed 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: 8vi/o is,armna)w1 v, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitaiv /vwr: ,8ans)oa1im tional 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 sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobileruy/ nf)4jmr l 2m4v2x 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 masv42fxr ml4) nrymj2u/s 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 illustratef/v9m+z6ofeny 7soji ) 9se9ds 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 rsow .- bl1zjna9*ri +folling 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 b2wcfykz (*4oe5 +i 1o9f9l kl qy9ey,d;nzwq(i.e., we ignore friction) about a hine,f *919 e blo k2qlkyfz(yn o;w5c4y+qzid 9wge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is n sg4m/:j ,hgo sv.a1fstanding vertically on the floor, and we wan4g.s:m gvfhs o an1/j,t to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is ma7q 6(:. kiuhgjwt;wi1m qb,wcking a turn with a radius The forces acting jw:1wuqth,qgi7bk6;(.wmc ion 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 and begins whir c7edlv.q,x xcpno(43 ling 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 required to achix,p qcc 4 3lox(.n7deveve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and he*ixqdbfz ; jr51lok47 o1ni. qr arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, 7q;rxzii1 qb*f. d1l 5k4nooj and with arms held tightly against her body.   

  You are designing a clutch assembly which consists . r1cjl:g z-lsof two cylindrical plates, of mass lg:1czjl-s. r $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roll-ua je3tr 9a9gs along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach ujt a9g39aer-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 ng i qp4i34pum+ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uni03v7m,ikpanrl4 (, of hqpw 8pformly from r,ah7,4 0p(qw 3p p8fvmlnk io90km/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