A bicycle odometer (which measures distv2.y coa3(0kqxf1 uj qance traveled) is attached near the wheel hub and is designed for 27-inch jq.xq3y 1(cfa20 v okuwheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotatf vzl6l n4.ei)ud j1v,es at constant angular velocity. Does a point on the rim have radial and/or tangential accevjz64d,n).i l1 vufleleration? 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 thxxn lt+(i,6y ue angular velocity $\omega$ Explain.

Can a small force ever exert a greater t d-r8zh,38z rtt:oja((5th7vj qu offorque 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 difficult to do a sit-up with your hands behind kdr6.v4ob3np *bv;2pydft l7your head than when your arms are stretched out in fp 2b37vf*pltno.;4r 6v kdybd ront of you? A diagram may help you to answer this.

A 21-speed bicycle ha, -o bn-ibpc(ds seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small reab co-ibdn(, p-r 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 sl t;* ()gjvpwyvl(ir;o -qbhl0ender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thipvj*lt oqiwrv g -0;)hy (lb;(s distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow hi 4oz vhk;.rg2e.r-zg6h.uv beam?

If the net force on a system is zero, is the net torque also zero? If the net to rctild4,z(h+ le l)b(rque on a system is zero, is t ihze(4l,d )lclt+(rbhe net force zero?

Two inclines have the sameh5e 8wga7e- dx height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball w7gx8eh5ed-a at the bottom be greater? Explain.

Two solid spheres simultaneously fw mvmsy1p0hhdq (01: start rolling (from rest) down an incline. One sphere has twice the radius q0dh0h1( mf 1w:vspmy and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start9mgnnznhe0 dw0 (si3 wq67 9 ywlgr:.k from rest at the top of an incline. Which reaches the bottom first? Which has the greater splyw9 nngew09hgk(w zsdn q3mir.6:07eed 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 conserved. Yet mv 8 ;mh7oz,qv /bz)agcost moving or rotating objects eventu c zvq8;m)o /,zvgah7bally slow down and stop. Explain.

If there were a great migration of people toward the Earth’v kbtq51+v9gq na39ho hf38 ehs equator, how would this affect the ke9qg+3qaof5h19h hn t8vvb3 length of the day?

Can the diver of Fig. 8–29 do a somersault without havingmtq 0)cvs9/ra any initial rotation when she m r) /stv90cqaleaves the board?

The moment of inertia of a rotating solid disk d bz9l6f:ot4e about an axis through its center of mass iltfd: 6bo49e zs $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass i5xjpuvsm7)-84 hr,wt ps d 7mtn each outstretched hand. If you suddenly drop the masses, wdw r57x87ut,vhp) mpjs-mt 4 sill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identig wsr z 6iyu8 u7hq8m4s450qdical and have the same mass. However, one is hollow and the7u58w4sr sq mhi8gyizu d 406q other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel x w/6z )wubn+8c6plxu rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceln ul u6xpw +cb8w6/zx)eration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotativb.s.f4 *hj7h igufx m;-trkg5s r0m3ng turntable. What happens if you uj mkfh-4rf3 mhb r.v;7 .isxg*s 5t0gwalk toward the center?

A shortstop may leap into the air to catchzdfa9c8)br8ulc+a:a/f;)uf x8 i)m hy zj:rp a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If yfh m/: p8 auczu bxf9)fyaji;)d8c+rr )l a:z8ou look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angu: 7 9w1b. 0l hqpv)am7ccvsvfklar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopteabmfl7sk):9c h0v p cv1w.vq7r stable.

  Express the following anglp*n1o -(k;th rji 35cd03 tu x5obyrstes in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of 3fzn xcw.vs,-q7zz1;xb-qo ugow 8 y8an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun andx;yxqc wg-wo7, 1sqo 8nvuzz - 3fz8b. the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eartaq u dkk975:coa;mh +nh. The beam diverges a n h7a:;q+mokcd95kua t 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 a4udzn-zdrtw3iqn/ 6 : rate of 6500 rpm. When the motor is turned off during operation, the blades sl:dwn6u rt z -dizn/4q3ow 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. Ihg+,(z/1x3gr 7 xtwjhzh; e u*es*ptyf the ball makes 15.0 revolutions, what is its diametertxg ez/z37+twj hr h;(xue,*s1 g h*py?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionjrujwg), /usx/8l+ u ws do the wheels8 /jjrgws,xulu) u/w+ make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate i*to,s c;je yw.+0-l8 lk9ne5 tnwqjjr ts angular velocie, j q+tjj;.no t5w0se l*wcl9rk-y8 nty 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 .l ,z5q xbnq j jd;y9m0ai9lq/complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated. ,i 9;n5 9bmaqlxjl/qj ydqz0 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) 8lak55y e39yh z,7s2xhoasqh in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe w)o/ u)cpvfxang/)7 ved of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis gobc.f24 .yt:dvk 6zxof rotation is to experience an acceleration of 100,026f.y ovg:z 4.kxbtdc00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centegr7911 vf+q.:lhhahgbn8 a uc r from 130 rpm to 28ufqa:1h19gb.a+7 cr gv nhlh80 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusdm 1lw/a 4:t.ei+ ce3x .elzx vij7/ji $R_1$ 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 spacecraft put thems7q. 3un*x hfwzcadvg v2 ,-hlsdyo 1*/elves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min,n3sz7h. 1gvw f v u2*x oal/*dqc-hyd 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 resrb+7tr*zt a v7ldn. u5t to 15,000 rpm in 220 s. Through how many revolutions did it tur bt5v.+*dr7z a7nrtlu n in this time?    $rev$

  An automobile engine slows down from 4500 rpm t55i ;*6a5i chpv:rslxc:jmc 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*(9xjwp yp ao2(5cz kghspeed jets in a whirling “human centrifuge,” which takes 1.0 9koy(p(pw 5j2*xcz gamin 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 rpmaka6 1ibmkawd 7qtv-5aa1e ,4 to 360 rpm in 6.5 s. How far will a poin,1 aa adt6q4ekb715avm a-wki t on the edge of the wheel have traveled in this time?    m

  A cooling fan is turnc)wpdt ,y+vd96n ng/y ed off when it is running at 850rev/min It turns 1500 revoludyc,)pwvg nt/y +6n9dtions before 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 sp ib -r7 w3(vmz7yzch(jeed uniformly from 95k m-(iw b(rv zy7jz3ch7m/h to 45km/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 revolutsue mmouo 885s(; nqs7z ip6cowh;b16.hz d 0sions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameterdw5z ;o1.m 8(uczbphi7eh68u;somsnq6s0 so 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 weight on each pedal when climbing 1(8) wapjub lsf9m(dr.xdw,nu(p z0pa hill. The pedals rotate in a circle uwpuar .)(zl 8(9pdpxs(, 1dfbjmw0nof 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 mofv5i9 hb4:qhoev-ru8p o ;+ wide. What is the magnitude of the torqvvh um8o5i-; f+o4 eh borqp9:ue 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 t9f 7q:ykm:nsi he axle of the wheel shown in Fig. 8–39. Assume that a friction torque of:9y sf knmi:7q 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless 4w 5exxy9ds-k;y iy 1.nfmdc3 rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the mxiydc fys43 ;ewydk.n mx91-5agnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 3e z10u+avoz 6khk7sv 1801ka7 v6 zoz+h s1ukve $m \cdot N$ 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-kgp9fk l( xopn)3 sphere of radius 0.648 m when the axis of rotation ifx9op(p3)lk ns through its center.    $kg \cdot m^2$

  Calculate the moment of iocy9 zqib/ f//nertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combi c o9yz/f/qi/bned 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 ho6b f7 dab2w*rph-q+ ,2thpa thin, light rod is rotated in a horizontal citd2hh aabwof*p , q7h+pr2-6brcle 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 potter’s whe)mdnufkjx c6-- +4jbq el rotating at constant angular speed (Fig. 8–42). The friction force between f +k-c j4xd)-qjm n6ubher 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 +3fpbq.*n4/7r bx1f rfalc/k q 4towsof inertia of the array of point objects shown in Fig. 8–43 abou4fqlxr .wb/c4b 1+fp3sf7kna/rt oq* t
(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 e6flsp/:ijuz:cv5oro7 ir 32 two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical sateenqj5 z 3( ;:ab9u1ovai rvfpf7+nk-n llite 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 forceionap(f3+; f ra7j- n kvbu5n19 vzqe: 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./jsn6 ckbedp+p1 . :z*ekif/q50 cm and a mass of 0.580ipp*e fk6/nj:kcs +1b ze/q .d kg. 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 ba1v kvbte2x99 ,avs0 qit, 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 on a small hand-driven merry-go-round and3:bs gtzvuql:*5u)v v is abletvbul ) qzv3u: g:v*5s to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, 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 eventu /,d-il b.tj(yszn)lu*p , jsyp,hu56t x b1yfally brought uniformly to rest by a frictional torque .b/x *j1hls)ijy t d-b6,5uzupt,y fy s(pn,lof 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-k1,w.qatgupw7( lhf0 yg 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 action of the forep:9z uww l1x (bo38dskarm, which rotates about the elbow jokl uwd( wsp 1b3:x9o8zint 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 canj3o;tygd91wcbowt6/ 5hw5le: p +xbc be considered a long thin rod, as shown in Fig. 8–4:5lcwb b6 d wytept/9ox31j; +woh cg56.
(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 masses, dc7* -pixx0a 6d8gm b:tva4tv$m_1$ 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 f-uox:ov ;rn4j vbjg+ )ull turns (revolutions) and releases v;xjo j:+ gv-4 ourn)bit 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 ajjn*eos8nblka + :-x ; moment of 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 dev- m+2vyrh:oyna/meu9elops 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 9.09r)bu0 vk9*boexeh k, cm rolls without slipping down a lane at 3.3bkk9*o h e)vrx90ue,b $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the su41kq;s,k. we( jkc7xjz/r jus m of two term;c qjwzj r( k 7xk41u,ssk/je.s,
(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 aqucwt) z:ub(x2uwiww r 5;q39u 8;qh rnd 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? Assqh );;ubw rqw(: 5iz 8r239uxucqu wwtume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from re7s5ew/7+mv vh ga* g22eoiupa st and rolls without sle m+/7pa*s gvv52whoe2ug7 aiipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on it-+-td* xjyp rns tip. It starts to fall and its lowyx n-+-rp*j tder 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 energy.]    $m/s$

  What is the angular pa6pa1f + .ihwv/icogan2 co 5k:4(eqmomentum of a 0.210-kg ball rotating on the end of a thin sta ecih :ovaa5qig n(1p4pc+2/wk .o6f ring in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindfnyb1y ;z(d+ac5cv8 qing wheel of radius 18 cm when rotating at 1500 rpm? c+5yv(yfazq 8d c;nb1    $kg \cdot m^2$
(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 rotati l4u z4h6h-wgt:k atm2ng at a rate of 1.3rev/s If he raises his arms to a horizl tk-4:muhzwt4 ag h62ontal 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 Fig. 8–29ss )rdkcug78 )gn0+*xxxl g )j yqw8f,) can reduce her moment of inertia by a factor of about 3.5 7d)gjsqg)f yr nxk0cwl*xs8u8 g+x) , when changing 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 rat:9m3p39n2uu o;samq z6 qqnlsd/ egr5e from an initial rate of 1.0 rev every 2.0 s to a final r5 s:;e/ o3d3p n6 l2m9 zarqqumu9qgnsate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through itw39l lgd4afr1s 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 cla4df3 9glra w1ly, 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 momentum of a figure skater spinning atf4xv*o1x9ks9 1efggvg e(cx 0 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 qbzj k xqxo58-ui6xf6kg8k +6the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is droppm/yt j0ox(+sv ,h; ggoe(rk;6p lz 1x b6azx.med onto an identical disk rotating at anguso+6vz,em(0ply1xa/(x j mk zx; o;h gtg6r b.lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at8qdf +t fp(n (*enby;,3yqy rl 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 rotati1)lr gk m5: bbygsq52nng merry-go-round platform of radius 3.0 m and moment of ily2 rqg1:n g5 smkb)5bnertia 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 ) uc*v7(sejhky9ijp/1d+q wu freely with an angular velocity of 0.8i dp(+qy /c *h eswu17uk9j)jv $rad/s$ 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 even7npmeuf c8iwqw5:j 0.tually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no w few:7nc.08 ijpmq5u 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 winds in excessf hm wzrpo-,kn34e 1zq66(iya 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 ucygd syj :32q1 i*2xi$ 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 resty,zse9 jit1l1, that can rotate freely without friction. The momyit1j lz,e1s 9ent 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 stc9dhrpv o d4.87o:27tfu3d sqj2txf d ands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mome 4ur xp:j82h9df7qftv3d od7st. c2do nt 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 role 5ob z1ueyn2.ls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwib 2z1eyn5oe .unds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the sameiw ,y) wt0wmf, side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to fwmi )tw, y0w,its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from res9 agqeay5fk3pp8 p3xh9v1: s qt 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 of a uniform cylinder )lrve g,w.i(s of radius 0.20 m acquires a rotational rate of from rest over ) (ws rie,glv.a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindr8 nk jl.nqu( 0ewdt1.sical disks, each of mass 0.050 kg and diameter 0.075 m, joidn 80w l1 q.e.tns(kujned 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 angulv clhfj:jst6db.n/;yh;+t:u) /,upb l ot9 ihar 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 -sq my9nyk b7ak(, p3h-hlhl, times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitationalmq-,,lyb- 9a7n3hy(kspkl hh 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 automobile is for it to use energy st65lrg; mxuc dk73t3edored in a heavy rotating flywheel. Su6udg x3 ;ckmdr7 t5e3lppose 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 illustrates4/. jxbps(/-vzbv nf31 ,n*pblngeqy jg, l4 m 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 rolling on a horizontal surface at speed v=3.3; 7s/fncez2p;b2-p)jb dy hgnj25 obz $m/s$ 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 frictiod*hulc. pmje /u u+(wop )3.zvn) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig( h/3mozpp uc+ud.l.v u)*wej. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, m 2ygofn4tned 9h 4s):and we want to exert a horizontal force 44thdmfn2e: y9nosg) 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=h-0nakexasm*wu 1gy8c;v:z : 4.2m/s on a flat road is making a turn with a radius The forces acting on the cyc:k*cn; eam:-1us z yawxh0 gv8list 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 0.5ap efgw+2 2.wkrnrg+e( anr320-kg rock into a sling of length 1.5 m and begins whirling the rock inggarf(k.neew 3r +rw2p + 2a2n 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 achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, m1/wlisik s3n 6aking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched i3 l/s6n1iws karms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyli o-l23 sw:bzsvndrical plates, of mass 2ws3vzbs o :l-$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t;f:6 6a u o4n 4xojrqy4*y nsmxu+h n8fu*+obrhe 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 the highest point of the nru4our4uy8o4*;b + n6f6x +*ymja fq o:nxshloop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not j,ytcv 7l ,9o ke)n*ngassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniez(,lf3 hdq8ptc95z iformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. zf , c iz9etp58q(3ldh(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