A bicycle odometer (wh8 fho9t botz/mi .x7c1ich measures distance traveled) is attached near the wheel hub and 9 /.o1 8h7tctfoi zxbmis designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cowv szn3 qav)hw0 wtual: ,f66v50)nipnstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accevl : z60,i sfu30n va)vhw)6pwqnw5taleration? 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 velj;ruic .q u0a9ocity $\omega$ Explain.

Can a small force ever exer++8y lexkklj a: x1y;dt a greater 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 difficult to ilko)8 mi(z; ldo a sit-up with your hands behind your head than when your arms are stretched out in front of) 8m ioi(lklz; you? A diagram may help you to answer this.

A 21-speed bicycle has p ajjdgg,g3m7grc-0. seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket?g3mgj0a,d.cgjgp- 7r Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend onmpe9 ,7/ ;lxy*qf.+ vesili pg being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain w+i*i/.fy9 p eex7l ml;,vpgqs hy this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l 3/suhdi3ulzde/1j q2ong, narrow beam?

If the net force on a system wgw kfmfmgw(27i5n2 t(. ;aic is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero?2w(m7i(.g f gfkt m2 5wciwan;

Two inclines have the ,4gr.wg q5ec xsame height but make different angles with the horizontal. The same steel ball is rolled dq5.wegx, r4cg own each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline.si6mfry dt)17 One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the boytm6 sf)7i 1drttom?

A sphere and a cylinder ha9hbye bg0t)+vve the same radius and the same mass. They start from rest at the top )9h0+t eg bbvyof an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ar *exp+rpq o p.65vfisv1cyx 7tlzk+n6vv+7)r e conserved. Yet most moving or rotating objects eventually slow down an*+c) vnxqvv e pi6f57+r1tp7k .6px+sylovzrd stop. Explain.

If there were a great w(alj4e(e p/g aw(fn)*ft4 samigration of people toward the Earth’s equator, how would te ga 4wnl4pa(()ew* at (f/sfjhis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial4chqufo,p7t,4 wzm 4r-9gjbx rotation when she leaves the board?t 4o47,c,q hb-u9gx4pm zfjrw

The moment of inertia of a rotating solid disk abou6dfo:8 la7 psu3kr;a dt an axis through its center of maspf l ska8au6o;d73:d rs 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 r6cevf:1pp2+seh6t t c+c 3:sidtcmo*otating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your pm v32itsdof:h*c c e 6s+c:+6 cep1ttangular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is vg71p1)p pjnh hollow and the other is solid. Describe an experiment to determine whph n)1v7j pp1gich is which.

The angular velocity of a wheel rotating on a horizontal ax8(hc. lp;0sjriww 6 kwt3 hl9ule points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Ishwwr6l wc0 lkt;jh u938ps(i. the angular speed increasing or decreasing?

Suppose you are standing oq yxl*9c4p z1 5zygz;gn the edge of a large freely rotating turntable. What happens if you walk toward the cz9 ylg p1qz 4zgy*;x5center?

A shortstop may leap into the air to catch a balxq.;*qxese+u4tgin;/uf 4w*sm9 q fc l and throw it quickly. As he throws the ball, the upp.m ;c/ns*+9gie f44q s;ufx *tuwqeqxer part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserdljqb5i suxge7qwo-u )/,a +6ria2y 0vation of angular 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 helicopt/da7sebuqal6r5gq -2w+i0)x i uy, ojer stable.

  Express the following angljl0e3gnrtyiinl. : 1, es 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 an amazing coincidence. Calculate,hk. 6a+cp y*jl.namr9 using the information inside the Front Cover, the angular diameters +ya *k.6hn.p9cl jarm(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges w1j0h65 oi5gpk gimacujq 0. 8athjao j p0.c6miiu 8105gq kg5w an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 65006-)o9r 5o so-a 6sum)dfmwqdi rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladefimu9-- )6wd oo maodsrq6)5ss slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m mc 1uekid) b. .iaj*p6to another child. If the ball makes 15.0 revolutions, what is its dij1)pbmu.k .iie cad* 6ameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. H 3fch2okhrk2k k69+dr,ye j)low many revolutions do the wheels kcrh6 r2j,2e+)kyflk kd93ho make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotan3-phxgtz d7*lal 81ntes at 2500 rpm. Calculate its angular zxd*ln7htl p-8 a 1n3gvelocity 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 exmyde 1ev;dd(i(4 fd1 r,ke*one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chyfe4dd1e(dv*1 rxe;,kmide( ild 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 t4z01p5 si nyd x01invnsg .c;hhe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a polfn/ apzi)yr)f jt.4.23q rxhint
(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 centrij)2i1pi pb9yp as6ji7fuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1006p bipj1ip)yisj92 7a ,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates ( - ;d+jy;bmkbeaz byv.0b9iguniformly about its center from 130 rpm to 280 rpm in 4vk 9ay;e bbgyji-z;0 d+.mb( b.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu-0sfq pxb6yx0 nup;6 axm x/;zs $R_1$ 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, astronautfyzx.w ck-q-c vn6b4l- v+o+ws 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 revolution every minute during a 12-min time interval. The spacecraft ca- -q+. kzwv6y-+wno4vblccxf n 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 so lq*mh r-u8qpe( y2av5*4pvqz:q)lh. Through how many revph )8v 5yo*h2 -q uqa(qpleq lmvrz*:4olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.xx 9bofyv gff+/8/0,bxv qtu45 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets inawr -i5ei )ff-biza397* clut*wl.ut sl6q1j a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final s7f.q-l )z5j atuise*li ui tfb19wc3aw lr6 -*peed.
(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 diame 0kfodh 5+4xh5tsmi)y4bvsb,ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a poinxi50s 5h)k ,fymtvsbdh 44bo+ t on the edge of the wheel have traveled in this time?    m

  A cooling fan is turne7 yx xvl*,6sp;e1f;h(mg bobb d off when it is running at 850rev/min It turns 1500 revolutions befo m b 76he1x obb*g;sxf(;p,lyvre it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uutr kep)4 eq 4gf,rm+4niformly frkeq,4et44mu rp+fg r)om 95km/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 revolutions as the car reduces its speed unifor9j+r6fg7:dbkzmrhxmy k g :o v+7 n.dr k+)kc;mly from 95km/h to 45km/h The tires have a diameter of 0.8kfr xk6 r : 9n go+ z7yh;vmc+kb7j+r)dg.m:dk0 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 z 0i+lba+ vw0fclimbing a hill. The pedals rotate in a circle of radi0 +zb+0awvfli us 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is):lmg nb4t4pd the magnitude of the torque if the p4g:tl4mndb) 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. 8–39. Aszxp9s v8nv0l(4jv -w8ipeyp-sume that a friction t s(pvvij - npx84wv08-lzpy9eorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to thsc:h*-laew rxlh t hq +c59s7/e ends of a massless rod which pivots as shown in Fig. 8–40. Initially t /t7+qsc- whhrc5*alxl: hs9ehe rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine reqaiko;k41 )at.wj il i,uire tightening to a torque of 384ai) ai.ot,;kljki1 w $m \cdot N$ 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 p7 s:,8muppb0 knmcr: of a 10.8-kg sphere of radius 0.648 m when the axis of :mnp 0pp bcmk, 7s8:rurotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Tbi-5b rwwyq 37 hrgzau249zjo 6*h1 eghe rim and tire have a combined mass of 1.25 kg. Th *za- j3boqw 2zguiwgh917hby4err5 6e mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a kquxr7, o83copw0q s ;thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcuxou,0oq7kw8cq;r s 3 plate
(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 wheel rotating at constant angular i;1fhy :ols n:h e/7upspeed (Fig. 8–42). The friction force between her hands and the cl1slohny/ ;e7u:ph:f i ay 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- ; ksdb2*hl nstsxa9i-yb9q- Fig. 8–43 abos hkb*iy-9lb-t ;x- 2nssqad9ut
(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 consgwj4 xrx d- y-bg/or69ists of 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 hju2a-r-c) t aik4otzn y+ epg d6417aiodh+7satellite spinning at the correct rate, engineers fire er y1++ n6i togaza ) dp--ckah4h47iujdo72tfour tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifhy 4j .cu8brm/1(sqr aorm cylinder with a radius of 8.50 cm and a mass of 0.58r abs h(/j 4y1m.cqu8r0 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 bat, accelerating t da*d- 0 cqwmxfecv ,fo*n2+ llu97r;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-roun)66b.(k;8ci k j;xdfclhwofo6 hk-f yd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torqueb-kh6di.j6oyhfkco; ( w ;ff 6k8l) xc 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 ir6o18 9xgvnsys eventually brought uniformly to rest by xyro 1gvs69n8 a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelera5;xy)n: sa y k66 ksymbv9kb-jtes 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 by the action of the forearm, wh 3z3 cqr8m5a2 9uwkyzzich rotates about the elbow joint under the action of the triceps muscle, 3zq 8wz y 92ra3zmuc5kFig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered n(fbok-r8tr+hq t9y6+sc f8 za long thin rod, as shown in Fig. 8–48ttksy 8 cq9f z +ho(b+6-rnrf6.
(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 cgo z/ 8rrz-icmg(,fxgiw,n 55onsists 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 syf3,g,h*2kxryxo uy:u l )b3full turns (revolutions) and releases it at g ,x3ly: skxo)yr yu*23 f,ubha 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+r:wi s31 izlt 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 dea+;yp3:0d *s9nfo wgymulta+velops 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 without sli0 nay wew7b0o;pping down a lane;y0w7 ebn0 woa at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resdf*n/ n; cpjlml -iz6.5hqcb2gj37 z kpect to the Sun as the sum of twonj/;cm7b23 zfig.* hk lcjq nz-dpl65 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 kg and a radius of 7.50 m. How much net 8z9nabij 2c 1hh 2r*;tul fe.4 rjmgl/work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assumenr4cru2fl le za9h2jhig/b* ;.tjm81 it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mmi3mowlfjw 4:3te 5.dass 1.80 kg starts from rest and rolls without slipping down 5jto3dwm34 wfiem . l: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 iaze ) y/tey5f0v7a,bbjd 6r7yts 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 grouvd /eyay7tf 7a65,br0ezybj )nd? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum 2cxlpz+d zo ys2uu8y1lb5 )m9 of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angularmzspyz ydb1+ u5lc98u2o xl)2 speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momen z.yqjq4hhm)p6 hl7 m:8ore/wtum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? mj.o) r4h8qe /m: zhhw yp67ql   $kg \cdot m^2$
(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 plapw-pw t3zuzj/ka3(* ohp-2ottform that is rotating at a rate of z-t*3upwo az2pjhp(-to k3/w1.3rev/s If he raises 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 Fig. 8–29) can pr8d/bewqd *.reduce her moment of inertia by a factor of about 3.5 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 s b*wdd 8rpqe./peed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an 7h ria4jbu5q (y g36ujdu3c2pinitial rate of 1.0 rev every 2.0 s to a fina iydj4 qjbp(uac6ugh72ru353l rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a rj8q0oy28 an 0fwc z.9we;r vvfrequency of 1.5rev/s The wheel can be considered a uniform disk vcrz08qfe j a.02wo9;y w8vrn 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 momentum of a figure s+.nj3/qze q u;wi s6d1iiay3qkater 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+- :gdc-eyfd:w0e2 kq 2ifuxu momentum 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 of moment of3 e86svwcn qp- inertia I is dropped onto an identical disk rotating at angular s p enqv6s38-wcpeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnstp tu(w1zv71f at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round pzbwg h46na a*4o4 u-sllatform of radius 3.0 m and moment of in4hoga alnszw*46- u4bertia 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 soq(du2gw3fk) :,pn0ib1wm l with an angular velocity of 0.8 oq0 1g w,(:3smfnpbd)2 kwi lu$rad/s$ 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 ys vkjh w52j)9ahag4+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 angular as)2v4 5khjhya+w jg9 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 wx /od:5pwdfyp 9p7 h/sinds 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 fo sfw b-)4rfq.*)cxa$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that ::j jwtm)e0nfg+nl-(b1khl dcan rotate freely without friction. The moh f+:b(wdknn gjt 0e-)ll:jm1ment 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 ox0vh,r7,y0i xl) n fwgfli ,)wf a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momxg,wh flvi,0in )7xf wl0y) r,ent 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 enqekus+)gk1ix2 bcnv: ze7g 7-d of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s kxzg+1iqcg):eeksv - n72b u7 center of mass move?

  The Moon orbits the Earth such dw/v(f ,wez8 b4vt*e nthat 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 particl/v wtv(84ne *zdfweb,e orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resth pr7- /w/oq.alv(,py5 vrtf k 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 un1h j: 3mivsr 7/-y. gvg7 tc1hxuts+dfiform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the cs-m:gvu1 j xs13r7fi vt +h.dht7y/gtorque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eal) 9pkw xx05qnipk7z;y ;ea a1ch of mass 0.050 kg and diameter 0.0759x 1p0 zwi nq)7;xayea kl5kp; m, 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 angul himhyt pm(j60(q (0tgw vl-q0ar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, wij8q8/h4/ e+xd lad nfere rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radn+8dlaqe/ 4i/ h8xjdf ius 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-pol,jrd9co7xt,gw (g iv;q6/ trplution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, dvpctojgqrt, /w 9(x 76gr,i ;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 ah6x q*rens1j* c84lyon $H_2O$ 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 hjk5:,0r(z kqr sw/ :wrlmft8h /it u7jorizontal 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 fricti( y0eou ha9q-f3(cktxz -hqy/ on) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the ox9kz h uq3(f-q-aeh0( cyty/ moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl 13ks/uf xlpoy+)m+ axoor, and we want to exert a horizontal force F at its axle spa x )fy++kxs lmuo31/o that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making ak:0+mw xyn ageicww*.sp21(i turn with a radius The forces acting on the cyclist and cycle are the nor *n+ ew1x:2ig.0s(c wkmip waymal 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/v+kre;y n8nw rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circnew+v /y8nrk;le 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 sotbyh 3 +ay7m1olid 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 ara +b7yo y1t3hmms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two k3dbeg6j 0i 9 n9 zu1g(8syml sby.qd7cylindrical plates, of mas euqd.ys1g i9b0 dg76z3jymkb sn8(9ls $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58- 1x o91d4y*z rnz5 qk.pe:mftj8nziv. What is the minimum value of the vertical height h that the ma81d1:nzp v e 54ry .moxnk-jqz*9zf tirble must drop 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 do not assudn1 1a6vrfu- xi vsc7n/byu13me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as k+,a .u:it j vdt/5dzqthe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wt 5 jdvtua:ki +,.qd/zhat 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