A bicycle odometer (which measures distancelp .qe-ql8,nvqz,0jd traveled) is attached near the wheel hub and is designed for 27-inch wheels. What lqp8-n ,.0 l,ezqqj vdhappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularps.r te/d8 16gfyd4p2yr z0jo velocity. Does a point on the rim have radial and/or tangential accetj1oprpys/e 6d.r8z 2d 4yf 0gleration? 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 acceleration change?

Could a nonrigid body be described by a single value of the angular vpyzxd7u.a lyz h)t2./ elocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largerhfbvg gzmy3:3o:e)fpb f u/48 force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficulf9o7xsgwd :j 4t to do a sit-up with your hands behind your head than when your arms are stretched out in front of you9f4xws:7djgo ? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalks wd.3k;ua(a cranks. In which gear is it harder to pedal, a small r;(.ku kda3sw aear 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 iy ahbhjl90k suw 73w,of-)9b slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis9u jwkb3 ,0y79) liaf-o bhwhs of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, n8vg+ore8sz 6 sgs*v w4arrow beam?

If the net force on a system is zero, is the netusa0- n7 6vn 4rg(hio:gv1gji torque also zero? If the net torque on a system is zero, is the net fos7ggir0a in 1 v-6uonh:gj(4vrce zero?

Two inclines have the same height but make different angles with -o0bs9l0 nyre the horizontal. The same steel ball is rolled down each incline. On whicb0- 09yosr elnh incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollivey;q3nd,b u )nb1ti0ng (from rest) down an incline. 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? W3yb0iennut vq)1 ;,db hich has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same)qyuv iykuaa6 8 0*:ku radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greau *uv:60uqy)yk aki 8ater 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 -ht d/od l3c)lfdt 06wconserved. Yet most moving or rotating objects eventually slow down and stop. Et0cofh wdlt6d3d )/ l-xplain.

If there were a great migration of people toward the Eii 1vxl3cnp vgn(qwa0hpw 20cj ,k +:*arth’s equator, how would this affect the length of the i0,lhx:vi* nj nq k(p1wcpw+ca3g2v0day?

Can the diver of Fig. 8–29 do a somersault without having anys5jxkoyqeou+zvz4 t((adc -y ;; /v m+ initial rotation when she leaves theqavs ;e(mjy -zzy k4otd+v(u c5 +x/;o board?

The moment of inertia of a :o;x: nyopga -u. k1sxrotating solid disk about an axis through its center of masx :.1 -oonkpgaysux; :s is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting dfm8l5*d2k tm - t9sruon a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, t9l r*5dk-2mf 8sdtu mor stay the same? Explain.

Two spheres look identical and have the same mxw62su-32wgvy woz d7xyf;r1ass. However, one is hollow and the other is solid. Describe an exper vws x yg7-fdrw6z o3;y212uwximent to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points weixz7pogst.kgz. cy h8 z 7lrn:2 4dm.)st. 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 :8z.m2opk74s) n. d7tzzh.lg cg xyirlinear 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 ah4:d3s gu3irmwfm.3n large freely rotating turntable. What happens if you walk t 33gs fm43 drmnhwui:.oward the center?

A shortstop may leap into the aikgqj+ 6o: sk:y7b-to0k4db xe r to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in thek-6dko :o4+: b7q0 gybjtxks e opposite direction (Fig. 8–36). Explain.

On the basis of the lalur* /pqr2v l8w of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller canpl/v8rr u*lq2 operate to keep the helicopter stable.

  Express the following angles in radian8 luhu06vuc: gs: (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. Calculat l 8qm; f9 1jb6x2pg;fghwpan+e, using the information inside the Front Cover, the angular diameters (in radians) of the Sufpl+gxa gh2 ;8w nbq j6mf1;9pn and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directe+z)-ioj yjt- yqqo;/m d at the Moon, 380,000 km from Earth. The beam diverges at+oq-tymiq z;/ jo-jy) 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 xfmauq.l3,1k , qkez 4rotate at a rate of 6500 rpm. When the motor is turned off during operation, tleq q m3,ka4xk1.z,f uhe blades slow 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 a level floor 3.5 m to another child. If the ball makes:fd ioat 8-/:dcu;tjniqa-mv0o j i8 6 15.0 revolutions, what is o-o vt;:na886i a/ umqjdjti-cd0f i :its diameter?    m

  A bicycle with tires 68 cm in diametec471 aq6lzdtuf6aab :r travels 8.0 km. How many revolutions do the wheels utqfc4 b z:d716aa6la make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250c .l:)d* lebfm-wm( pf0 rpm. Calculate its angular f bl .(-p*fd :mwc)elmvelocity 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 revl1 onznafhd83ou 8r 14olution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chihzrn1 ud8aof8ln1o 34ld 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 Ear7 0aii:,iqb avth (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-pbm2 w+yuv h3 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 centrpf c-2bbb5.3ncfvo d 1ifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10ob.- c2 vbnc5pd b1ff30,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelprf19:njsul q8 ti0i*erates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ r si1:i0qjp*f u89tnline
(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 radiuevtg. )o)7 q-lgnrv6d s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraftp0nb)g2bm e ,- d pemrr u5.lvy)a/p9r put themselves into a slow rotation to distribute the Sun’s energy evenly. At /mpyneb,02e .ra5-p m g )9pb)rrlduvthe 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. 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 15,00082u1ld4kkw aanj- et(y c i7 :+akdgv5 rpm in 220 s. Through how many revolutio4uiwv21+acn-d ljkakty(ak 58: ge 7dns did it turn in this time?    $rev$

  An automobile engine slowswn3 nunuwz3(/ down 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 hig; n xkq1cormb 582a2-ml7phie hspeed jets in a whirling “human cemhk1l8 ir qo2 -axpm27;e n5bcntrifuge,” which takes 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 uniformd4;/ dvd2qq r 6cyx4 u28zvzkcly from 240 rpm to 360 rpm in 6.5 s. How far wx cqkdr4z/v8z4 2dy;uvq6 2c dill a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it afh26 4liveb+ is running at 850rev/min It turns 1500 revolutions before it comes to avflbh+ 4ai6 e2 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 unifor/j pby r0)q e2.1ppctvqo18 srmly from 95km/h to 4.1vp2rc8o)rspbqq jp1 e/0 ty5km/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

  The tires of a car make 65 revoluti,2j .(qpxvf+owq4hz gi 97o;jpt 9cd wons as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diametervjptd;9o w z92wgj .xfi + cp47(qqho, 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 a7ya fe2pql23r bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a cir p3rf2el yq7a2cle 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 *lz+gafa l07( y*8nr*bsvr ab q ,t:ejis the magn,z:v*8b (0rlqy* asrbt*aefn7+gla j itude 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 abouz x1dx2j3w tv7trqvs:p h hhh(.,af :.t the axle of the wheel shown in Fig. 8–39. Assume that a friction t.vfv:xs( hxttw,: 1qhpj rd27h.h3azorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,o*,bf6 u gjl;d 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. Calculate the magnitude and direction of the net torque on god*;ulbf,j6 this system.

  The bolts on the cylinder head of an engine lbc +u1m2mp7 ;vodv:prequire tightening to a torque of 38;2 u7 dbvpcp +:mlmv1o $m \cdot N$ 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 s p,o3lwk),av aphere of radius 0.648 m when the axis of rotatio )vwlkpoa,,a3 n is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. 9z9pd ;u0f+lv qnd;m,nr pfi3 qqut*4The rim and tire have a combined mass of 1.25 kg. The mass opd9v 4 niunzq rm,d ;+tf03u;*lq qp9ff the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the e: 0w1 koq.phhqnd of a thin, light rod is rotated in a horizontal cwkoh:q0 phq1. ircle of 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 potx 8d9cvw6m,ok4j z3amil 6q s/ dl(u3fter’s wheel rotating at constant angular speed c4d68, 9(/ aiuxd3 3lkwjvzfqlmm s6o(Fig. 8–42). 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 arr3m,h1t4lbb ehsp-hn -.lk3saay of point objects shown in Fig. 8–43 abhls3bn4mtl-bks-eha3 ,1 .hp out
(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;u33/mtq 7nd ,zpjezm atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the coral d;+0rv9 nlyrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a dyv0; + al9nrlradius 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 cylind dxa,;* hamhhe9mma 8vn/ .vkz15b9pzer with a radius of 8.50 cm and a mass of 0.580 kg. Calculamm.a h;d5xb9pveamh , 1/n9az z* h8vkte
(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 swino q3pf,zt8n8 p1p smj;gs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially d.ej +e8 vda(. y6gqx i*xlo6fon a small hand-driven merry-go-round and is able tov . ogx6e6(ie+*jdfl axdyq. 8 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, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotax uqv-tu 9q5pm/v-p6j ting at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictionapj9 m p5-uvt-uvq6 x/ql 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 acceleratey2s-yngv y7wwd;xa // s 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 b)8bgytz,m; f :pt w8fly the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45.)fzb;tlgpfy8wmt, 8: 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 long thin rod, as2 h)4ejzh(wt :;jfuy f shown in Fig. 8–46 y;uf:hz()4wt2 hjfje .
(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 co(tnzriw 4:g w-nsists 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 vuq2gp;zv:v k5o 9id)full turns (revolutions) and releases it at a vgqi5v;ok 29v )p: uzdspeed 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 inergwdtp,z)j 0w(unp7 t:wro21q tia 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 de +8l1; oeaicuyb4ph f-z+f 5npvelops 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 anr/o-pzei (kdf z+h,.ubg :,dpd radius 9.0 cm rolls without slipping down a la,ddz/-eb+ o ,h:f(p g.u zrpkine at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the vzvog2*vwr* /j1m v0(sb8jslebfx 3. Earth with respect to the Sun as the sum of two terms3ovm br.2bvjz s* 1ev80jgw/vs(lf*x ,
(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 andjc9 2 y7p: i/ruiqix:1 p+kbaq a radius of 7.50 m. 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 aq/rkpu7 :j9iiyq:c i2 ax+b p1 solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and r;o rc xmuy3x2f0/ k4g clq5ukr.c0h +nolls without slipp x2u3+ 4kgc uhf5/l0;o.cnk0cqymrxring 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 verticallj h3:hxwzr ,k6y 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 pole just before it hits the ground? [Hint: Use conservation of enerkr6x: zjw,h3 hgy.]    $m/s$

  What is the angular momentum o*-dnc td a)j1xf a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spea) cn1 t-jd*xded of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindriri 6ad;-u 5 j0hwohun4cal grinding wheel of radius 18 cm when rotathuwni- drh 6juo50;a 4ing 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 that7vhqw+tz1f)o is rotating at a rate of 1.3rev/s If he raises z fw 7qh+v1to)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 F/.*cp sw4n (w:kwz+jxn yx6rtig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position /w6*tcw yp:n .s4 +kjzw(n rxxto the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an ini 9y1efp eq16ped)v2yctial rate of 1.0 rev every 2.0 s to e12 61cp9eypq eyd f)va 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 througigi3 ti:v(c d-h its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rota:(tcii iv3g-dting wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spin+aqozy:-,pwf .nhs(nvj 0a. nning 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 mpykjl(l -orn521z f)comentum 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 ocbfah;( v 5w 9/m+ivzof moment of inertia I is dropped onto an identical disk rota;ihwmabfoc(+9 v 5/ vzting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at8 foiedx3y m7) 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-roungd(qt)we,*0xgc t3s ad platform of radius 3.0 m ta gtwd*3gx) ,qs( ce0and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely u by.j)5fr z+ 5+sdnixwith an angular velocity of 0.nfxryj + zd5 +5sib.u)8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwar*bs.7nagfpuw 8dc 9fzen x2 (v8fx:x4 f, 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?(rouxpzn87wsx4:un .v c xf (f*2af8gbed9nd 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 zk9a y-k hknxfq) ;v6m.zw s,)in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass :x)v/bmwj dqj.rll5 -$ 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 res)d:cqm us0*gli um2 bx64 ruw*t, that can rotate freely without slb0 6du4:qi*wxu rcg 2) mum*friction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter5umgi sbl/ m p.b*so+8p4cje: merry-go-round turntable that is mounted on frictionless bearim/pseoip4b5 gmjls8+ u :bc.* ngs and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the +x,)ziydxm*z 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 slippinm,)xixdz* +zy g. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thatuqmfwf +7+mf 4iq:* be the same side always faces the Earth. Determine the ratio of thew*f im umb 4q:7+eff+q 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 restsa3uqgbg7mgo)2 a 6m5 at a rate 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 ixm:hk:nc10 j of a uniform cylinder of radius 0.20 m acquires a rotational rate of fnji01:kxm c :hrom rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of t a )v4xjwp5eh*wo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 w54*eax phj)vm, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-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 adna:-c .4 uhkengular 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 v4 p-lg:i8gp ltimes 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-qua-lgl :p v8pig4rters 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 l5f*k* xpd ;hsgow-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a c ;hk*fd spx*g5ar has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able 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 illustrates ao9 b0j7b g+a+n 9jikykn $H_2O$ 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 roll0:y-fn eedg9ria n 1wxybu 2-,ing 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 e c;f o.dp xgri(n b(68ub z09bnzy*x8and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the lizo 8*bz(.ce ndigb x(9f8x6 n0yur;b pnear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing qtj5y0er* rw1 czdo ko65o.,zf +s3atvertically on the floor, and we want to exert a horizontal force F at its axle so th.a3o5,rst 0o*1yo 6w+zej ctfrq5kz dat 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 **s-nt-s qw nzqie9d(flat road is making a turn with a radius The forces acting sz -*nd9* wqit( nse-qon 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-kgb;t ++ n-chhv2lvey65c q gfr. rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his h er+- n+bcgtvcflv q2h6h;5 y.ead, 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 ac5h- yy5q hmx0s a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightl-hmxy55ch qy0y against her body.   

  You are designing a clutch assembly which consists of two cy-l ) 4dau*l7 : /ehpurwodksi1lindrical plates, of masa4 -d/7wkp: u1eu hlloidr)s*s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 an0yd:usfi6s 6u d radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must dro0isfd6usu 6:y p 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 3+rd6i lu8pecy)y cl;do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revp)e;q:3n4z51 wcu mx/ zapjrj olutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a rum/1:4zj nzx;qpac5) e 3jwp) 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