A bicycle odometer (which m aed.qe.qd g3klj62-q g o(qdw+r;fu;easures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use;due g +awj( ogq..l3kdqd-q6qf; 2re it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocitodgy5+jw s5/ jy. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of lins j/5+jwgo5dy ear acceleration change?

Could a nonrigid body be described by a single value60q+z8sf h 5z6glqinwzlh3 f+ of the angular velocity $\omega$ Explain.

Can a small force ever exert a gra-7o s5sv y7s)mk43l3jo vzgj eater torque than a larger force? Explain.

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

Why is it more diffizomc -,id kw121gbxdejez8l nr89w6 /9b0gjrcult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may heli/g10cw x w zre ,djb29861 -lrm9nbdgo8ejzkp you to answer this.

A 21-speed bicycle has seven sprockets at the rear w* w ;eggji/8kzhvgo 04n7wr7kheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? W*74wgnoeiw7 g8/rgh0;z kjv khy? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower leg1hwxq 2/0dtdu ( ;alkns with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why / lu( wka1ddxt ;0nq2hthis distribution of mass is advantageous.

Why do tightrope walkg,kjs u-):flaj 2/p/ka wkq3o ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system5y(j sj(q oc3u is zero, is the net torque also zero? If the net torque on a system is zero,(yj3ouc5( js q is the net force zero?

Two inclines have the same height but make different angles with+ ppfejpb)t 2. the horizontal. The same steel ball is rolled down each incline. On which incliepfp)2+tj p .bne will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously s5gdk3lk-2ki4va 88 2l( oktwyv.ce;efk 1llktart rolling (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 gradk2vk( k2 w;v. l8f4o1 klle i5lty-kc3keg8 eater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius : a6jq /d+i sqkj9dc.sand the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has q9 : +csdi/jsq6akd .jthe 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 cazw (ks1v)ui(onserved. Yet most moving or rotating objects eventually slow down and st()(sk vwza 1uiop. Explain.

If there were a great migration of people toward the Earth’s equator, how wod zymx mv+b2fdh)j1( ida, 2f9g et;9suld this affect the length of the(m)g 2dd 9sx+a h1 vytj;2zb fm9fei,d day?

Can the diver of Fig. 8–29 do a somersault wi-c 9xh9;uoosae3: pw ithout having any initial rotation when she-9eipo hu;so3 9cxw:a leaves the board?

The moment of inertia of a rotating solid disk about an axis through itts*6fyyd) ezh c+e ;(es center f(sy e)zce6y d+ the;*of mass 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 on a rotating sto*joj xac k:o88( 9oob2mnzwe6-szlw.ol holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increawx:o. wzb 8jkae(ojozo-lc s*n6m98 2 se, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow andq *t7y/et5xi ed cz:3(czc8pt the other is solid. Describe an experimentpet cze7 (d/yz8c*:xci3t5 qt to determine which is which.

The angular velocity of a wheel rieez yb/xr2:2otating on a horizontal axle points west. In what direction is the linear velocity of a point on the:2/ibre z2x ye 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 oolwc0 nl4.ty ) mf.:jtf a large freely rotating turntable. What happens if yo .lwn jfy.co4:t0)tml u walk toward the center?

A shortstop may leap t)qosn;/9 a0x(cku 32. lvwk/f jqv xsinto the air 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 the oppos0qk cnf uv93l2v s/j(xk.;xos/w t qa)ite direction (Fig. 8–36). Explain.

On the basis of the law of conservam;luo6wd+:l ation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss ou;+o6dw m all:ne or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angle :5hxe: lvf8 aui.o)69orve(am m, vjbs in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth becausemt)vx//1jebz of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) oft)v/mj1 /xze b the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moom 9i/btk ixz2;n, 380,000 km from Earth. The beam diverges at an ai kz9/i2m;bxt ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blendei13; 0 7n 4htgldsqvhzr rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to restztnh3i 0s 7lh4qv ;dg1 in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on0ot v3o*g x/51 -.loixjduyt e a level floor 3.5 m to another child. If the ball makes 15.0 revolution 3 x.-ji o*tte0govox5l1y/du s, what is its diameter?    m

  A bicycle with tires 68 cm in diametev gsgzr)u3 o4mo/w7g 6r travels 8.0 km. How many revolutions do the wheels4wo3o 6 m7g s)/gzuvgr make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Caxmg:m nw0tbn +0ve8i0lculate its angular velocit+b0n : xvw0n8tgmem0iy 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 re y/ 7drax+5)a pltjaq4volution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from ta 4j/px7l+) dt 5rqayahe 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 itsdnsf-py)e 0p7s .xyv 2+bj4 bx orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poil(bt3 rl0; exsnt
(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 i 9es sk k6vy.vlp+2o;d( uv:pcm from the axis of rotation is to :sy+uplo(svd ; 62.vekv ipk9 experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly xhx m.)spq; ,i;oiepfesu1 eo2j /1 *wabout its center from 130 rpm to 280 rpm in 4.0 s. Detfe ;hjiq1. xo e,;/*ism 12 e)xpuopwsermine
(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 w)ci;v-o znc i )onkyj4o1*z/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, astrona4hq8dqa;4eo8b,gwh m 8g f.jruts aboard the Apollo 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 revolutiog.8g8q bjfmw4ho8d ;h4r , aqen 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 220 s. Thr;r3p ig7ai mf-r a1/h*o;.gek- wpggz ough how many revolutions did it turn in/-grmpgwz7 gp.o-h;rf g3; e k*aii1a this time?    $rev$

  An automobile engine slows down fro-n1 0gnwi abs-1uldd,m 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 tt,ss0 rj ozx- 4kqk4z/he stresses of flying highspeed jets in a whirling “huts/0kqrzs-oz4jk x4, man centrifuge,” 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 uniformly from 240 rpm to 36,vnxuz d h6a1morm6c5*nlcvca-9 v660 rpm in 6.5 s. How far will a poin*az acnc voc xhul, nvrv6mm-96615d6t on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running a wblerr6*2:wg5 uc+ pqt 850rev/min It turns 1500 revolutions before it comes qrr gw2wbe5: lu6+*cpto 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 revolu li,o.og cc*-op;u(yqtions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ofco uy (i c-go;opq.,*l 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car redug1k+ jh *w2em 0 -tfkdmm)fpa+ces its speed uniformly from 95km/h to 45km/h The tifkjtd km f01g2mmw+ )-*+h peares 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 -u+e248kuw arz gta)eputs all her weight on each pedal when climbing a hill. Th+zrtu2w 4)kue ea ga-8e 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 fotk(ef5c fd d45nwvu6y -;p9rqb;;j b5ouxl7frce of 55 N on the end of a door 74 cm wide. What is the magnitude of f479t knx ;;pfuwo bddfq;lj-e bc(vy u 6r555the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–3z2ih k:o4 dvn79. Assume that a friction to7v:z4kdni o2 hrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of masc:a( m rjjs4l)6 ob7dgz+ hgpy0vx d--s m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held 0sd )vg lrc( bzx+ojpagyd-7 j6h-:m4in 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 requivi9 mw- k v)5*bpp;ksyre tightening to a torque of 38 *i v ;vmpk-9py kbw5)s$m \cdot N$ 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.n54 dn32ya o89melxn o8-kg sphere of radius 0.648 m when the axis of rotation is through its centelm 8n34e2n9xon ody a5r.    $kg \cdot m^2$

  Calculate the moment of inerxm126xb 7nh yjq6m*aotia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a comq*6njhxaby7 6 m x1mo2bined 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 r4yn,)et q:nw z:fab 1x vhdy5;otated in a horizontal circle of rb) qe ,wyn:5ah yvft:4d zn1x;adius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s0qsv5/o/ehuj, d2ph eh z lo62 wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clayeshlq2ze62ov /u 5o h,/0d phj 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. 8–4 q0td5+et1skqo kf35dn wd:l: 3 abl 01d+qo3 sktkw:5ddf n5qe:tout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass / rees:o nf6s: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 correct rate7f1j4famge:rkkmgc b5s v vs24d3d5 ,, engineers fire four tange4 aksc2e bkfgf,rsv553j g :dv71dmm4ntial rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of u5)) (vufytsf ja+qb98.50 cm and a mass of 0.580 kg.(qvyab+5su) )j f9tfu Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from restn5 +v rm22gxkud+ kgyu(lc9ahgf3q 33 to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivenh r0gt.pu/+s8asc a0v/n l4o f 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 +g c atvr a/0 psuo4/.8s0lfnhthe 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 +iz dpm( cn4cmch4:85ii xxp -rpm is shut off and is eventually brought uniformly to rest by a frictional torque o4izn: 4cimp+(hx8 x-id cmpc5 f 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 3q6la01knrn h- fsp;2hzw*4ao.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the actiona.piq+k ke:wsi32-g4 :xhzu f of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest tse2kiukx:pw f 3:i h.g+- 4qazo 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 - x8z ccfb:3.zguq,j d4buf1rconsidered a long thin rod, as shown in Fig. 8–4c:d- cq 1.z 4ffx8g3,zjb ruub6.
(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 masdo fll bx26epg ii9;r8f )z-t5ses, $m_1$ 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*lz 7z h/ v).nzee/xk82bike v (revolutions) and releases it at )ve 7nv e/zx/ei2z .hkkb8*lza 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 ;0twjf+ q;anf0 ehe3vof inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develu9.icnrybw7/ee,a g / ops 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 withoutp*oahqu3dtwuq 8)95 zq0+h vb slipping down a lane at 3.3hbu t 859 zw*pqva0dho+) qqu3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the payxt. w(, , 4c0ecigunc/4 ldSun as the sum of ti a(ct/x,4wd0ulne 4pcgc,y. wo terms,
(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 kg4w7kf 4lauc e5 and 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 a solid cw calf5e uk744ylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and ro(w2hpvempz 4/h/c tj 7lls without sjp mp/hv7 z( he/2w4ctlipping 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 verticalli)6i 6- .srcfod6 jawsy 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: ic -fwd osr)i66a6s.j Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of (1fy hl *vblv)a thin string in a circle of radius 1.vllb*yfhv) 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 cylindricn)7vsmq 6;epoq6em 9x8i )xf zotu-j :al grinding wheel of8f66o9t))mu 7qo:n qxep- jsxme; izv radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate -5h5i(i:g m :qt.xf)wwadhg e of 1.3rev/s If he raises his arms to a horizontal position, Fig.: 5 tmh(dh.a ) xgfiw-qgi5w:e 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 Fig. 8– q fb v/9lj.h6ip88bwr29) can reduce her moment of inertia by a factor of about 3.5 when changing pv8/h 6fr9.jqwi blb8 from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1g 0fmi0)uf0f fb.ynp+u0o5 el.0 rev every 2.0 s to p+fi0m.n0fubuf 5l ogf 0)0ye 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 axisol3 hu/.b7le u through its center at a frequency of 1.5rev/s Tho7llh eu/ 3.bue 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 it?    $rev/s$

  (a) What is the angular momentrpi djiz*cq, t em y6:9ih+,i9um 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 mommhm0386 rtujl1de. f dq.hg*p *hvh2n entum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk ofk o jv+ps 9+vn9ayoq1* moment of inertia I is dropped onto an identical disk rotatino*o9q1p+nav svk j9+yg at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a uh):(f(nq bipt 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 - ;f 5x cv)lxhxxkuo0g2l ;ayrj9 *r,hthe center of a rotating merry-go-round platform of radius 3.0 m and mo; kloxhr ux 0vj9r5xh gx-yclf,); 2*ament 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 rotq;jlxhw.;cpp c* g*o7 d--z coating freely with an angular velocity of 0.hl p p-7j-*; zw.cc coq;d*ogx8 $rad/s$ 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 collu :cpex)z ye)ai p;e win, -o37i0yxz0apses into a white dwarf, losing about half its mas7x:wiy 30-0)eo,ipzci eu)p ;aezyxn s 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 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 m-.t/:bqqujvcrb. c+winds 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 4cwd 0/ xmull*y .,ap1jk ;kvk$ 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, initu0w 3w3px2oq1rah7xo ially at rest, that can rotate freely without friction. The moment of7w231oqou xrpxawh03 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/)vlcvh7 t.b t of a 6.5-m diameter merry-go-round turntablt7cbhtlvv . /)e that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on tk.5;r9 b-yzdv1r c-uefbqs z0 he 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 5b;sdqzf 0ze k-uv1 cr.bry-9 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 +y.mf69 dk sjcc rhe f,6qp0w.such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle ow.,66f 9.mcpy kqsf rc0 deh+jrbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 18hjex4l2pjw7i, t 1 s 7wrjlk. 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 a uniform cylinder of radius 0.20 m acquire87 uwbaflsp d6 b-)3ims a rotational rate of from rest over a 6.0-s i3 wsu8p)a76bfib -dmlnterval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disjoyzxml/-vu70 wve i n890z*jks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energzomy0 z-ev7vn90j8 ixj uw/l*y 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 anguc7ocdx; eb(v1/qhi e;lar 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 withxj6o 0bp8kg* u,q(pie 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-quarters of its mass in the process, what would its rotation speed be? Assume that the star is ae(pugob8kqp*6j x, 0 i 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 for it to use energy stored446ujbtb4j ya in a heavy rotating flywheab b644jytj4 uel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able 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 illustratf5s7xwr9yhv9 es 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 (hoowzd m5h9hq -5rp) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictj a4rxg;b;ulp:r a 6f)ga,+g ewx./ltion) 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 4,jplbwua r;fgag+;:g6)axrxt ./ el ) 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 standing vertically on the floor, and a9kb1w q-xx/arhsiwq ( 3j*i2 we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (a qwii2s (1*- 3bw/ qx9xjhrakFig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a tur-hdhsi6,-m3yz kzh7 sn with a radius The forces actingy k-zs7zh h,-m6sdhi 3 on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length g pz.4 dvt/fy9h)l3 baq9tk 8s1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 sq3 b yh).89at gks4dz 9vtfpl/. 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 e7z fw:v6bj:kand 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, anf76kv: :wbzejd with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricmomn-b:v hmcpbn3 :2+ al plates, of mass3vbmpnom- mb:2+c :n h $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ron+m9xmsrw lwi,*ka2 *ugh 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 the highest point of the loop without leaving the track? Assume* anrm ww9lsi +k,*2mx $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not;jyr 707ec/ nrkoy. bc assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a;cdix1aty h323m0c o cv6t5f ls the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m.3c521 ihl3x6 fodac ;t cm0tvy (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