A bicycle odometer (whe68p vdbvnos4nn + 3u0ich measures distance traveled) is attached near the wheel hub and is designed fo8 3po0vednb+sv4 6nnur 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cons5c ub hrfj6jy:l)n0e- tant angular velocity. Does a point on the rim have radial and/or tn-jc5f lr b0:6 h)ujeyangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by axgw:;8 / wivzb single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a great b5.jwk39ou vler 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 do a sit-up with your handso02k8n.bu6q9ovptcl behind your head than when your arms are stretched out in front of you? A diagram may help you to 2t.pvcknb u8 o06q9loanswer this.

A 21-speed bicycle has seven spr+nj*u7k8o . )qegazjenqf q4 m,-n6igockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a 6q ujg8*k 4n) e.z,-mjqoeq7+fnan gi small rear 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;ohe7jdar - -s to run fast have slender lower legs with flesh and muscle conc-j r-hs;7 deoaentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walf b95 u ze852i isoqnci9ctk-sa0l7e 4-sd0htkers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zeh7snuf*p ((uxro? If the net torque on a system is zero, is the net fuxpfuh(( *s7 norce zero?

Two inclines have the same height but make different angles witv up gt4na+2ejz2 1s)lh the horizontal. The same steel ball is rolled down each incline. l vg e)utp+21jzas24nOn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start ),t )nc8p*ps h12irvk hd(ikwrolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed t 8hd knc))r (w2pi*hpvstk1, ihere? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sal:nw .mi:w 2glme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Whii: w ng:w.m2llch has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular 4q6pf57toan 4 jhm-w rmomentum are conserved. Yet most moving or rotating objects eventually slow 4q7n 6p m5jrtwohf-a4down and stop. Explain.

If there were a great migration of people toward t 0(au )f(nr*ni5 sieywhe Earth’s equator, how would this affect t n5ayu( irwf*sien(0)he length of the day?

Can the diver of Fig. 8jyi rqk3 bgp8ajea6n ;ix3h;2 lqx 14,da;(tv–29 do a somersault without having any initial rotation when she leag2khp(n a y 6arb;i;d8a,vxjq;tql 3 jx e13i4ves the board?

The moment of inertia of a rotating siwel, 4uto /0zy/m tu-gu)lr3gxii/ m:d f61molid disk about an axis through its center of m/mdl1 izt:l w ueifi -goy x)/ u3um064g,/trmass 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 stool holding a)adytd 2 l /zvp1juvwym3.+d + 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stya/v3w) dyd+vtd .lm+uz2pj1ay the same? Explain.

Two spheres look identical d u;o,e1xb dl2zy x7f5and have the same mass. However, one is hollow and the other is solid. Describe an x;do572u lye zf1x,bdexperiment to determine which is which.

The angular velocity of a wheel rotating on a horizonta4hp0x +1/6 w n 3s.iul pjekv5(l qh9a:wvidzkl axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleratiolhv5/6s0(j xun hzw k 3q:1ii+ppeawld.k 4v9n 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 frei0jbiaq v /hy-,q /:kfely rotating turntable. What happens if you walk toward the center 0bh: /kyjf,iqi -aq/v?

A shortstop may leap into the air to catch a ball and throw it quickly. As.gb0*p dtje y1 he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hi yj*0bdtp1ge.ps and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular*vw +lvxqsam:*n7zo* momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways q s*n wx7:ao+zvmlv**the second propeller can operate to keep the helicopter stable.

  Express the following angles in faxc(.dzk gh-80xn9u 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 coincidenc6 l(/am3ljwxrmelyz f 2-pwir v,l;,8e. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen olfwr, l3ilm 2 p6 ;j(rm v8l/-awyex,zn Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dv,-yxe etn*1el ;z+ x. b7/8ksbpl1oqwz7 dyliverges ak8;e+qeltx s- n/.w* zvb1b, 1ydlxzy7o pl7e 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 a rate of 6500 rpm. When the motoha*a gedu,jf3c9b (*g r is turned off during operation, the blades slow to rest in 3.0 s. What is the ang*cfga3 d(* bghau9 ,ejular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball k4)y74mrcet h.j; o fqon a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter? .ht)o4 yfj qmc7;ke4r    m

  A bicycle with tires 68 cm in diamet eql c*fsf577wer travels 8.0 km. How many revolutions do the wheels make? ff c7e*lw 7q5s   $rev$

  (a) A grinding wheel 0.35 m in diameter rota .b c2mtdh;0 3r7khntztes at 2500 rpm. Calculate its angular velocity irz20h b ; dntc7.tk3hmn $rad/s$ $\omega$ =    $rad/sec$
(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 revolution in 4.0 s (Fig. 8–38:sf7 i/upir5t). (a) What is the linear speed of apst r5i :/if7u child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in 8wtjlf*2w8 gh9d .irpits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point.jjhai11c v y*k qex49c/ mfp-
(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 pa iqcvtq b9i7:8rticle 7.0 cm from the axis of rotation is to e:v9iq7b8tq icxperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fromnt/ av22 );gw(lw suxx 130 rpm to 280 rpm gxxw t(u )sa;/lnw22vin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu e*xvg/hhw: 8cs $R_1$ 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 spacecr91 - j zku+mqvr)3yoyxaft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, r-xyj)o3 quym z1v9k+they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000. e91z;tiid d gel8ieu/;pxjcs1o-tu a*s88i rpm in 220 s. Through how many revolutions did it turn in this time? duzt8dc t9.uipslo ;ise i-j ;1xe8a/8 1gi*e   $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula brbnsv;aawx;/;4 k(rte
(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 uxe3ugfn)y4xgaw5w4ye d + 85nitn8+ jets in a whirling “human centrifuge,” which takes 1.0ynw5 guxi t4 n+ea5w+e8 43)ndx 8fyug 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 rpyqwn mb0e6hh78, sv; lm to 360 rpm in 6.5 s. How far will a point on the edge of t0 ;8l,qy6hewbmh svn7he wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 ,,gb:i*-vwnscqj+,n n6k hgirevolutions before itic+nn q, -ng*w i jh,k6,s:bvg 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 spfqcikc:i v 6+w6-yfga jf2sk*m4 7e1reed uniformly fromqjaf* kfm6e27r fw i1 k+-c4v6gic:ys 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 rev w6etfd2t;6h8 vmugl6olutions as the car reduces its speed uniformly from 95km/h to 45km/h e6v6t l6mh wug tf2;8dThe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on e5gces6 s3o+ eiach pedal when climbing a hill. The pedas5eg3ec6i o+ sls rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm widpsa8j acb9h43v25l+j jj )v5 ucadn;de. What is the magnitude vj; jvn38upaj al49s+bc5da2)5dj hc of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about ,c(za 7xemh aa9)jor5the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of haea5x j r(oamz,)9 c70.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslesdhfx8 +bnf 3;24mws(zpxt2l/y doi s.s rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position andy.bimxx8 d;fd/fsw l+4os 3 (hz 2tnp2 then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine r/0p,qov:eybq equire tightening to a torque of 38 yq:v be,0poq/ $m \cdot N$ 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 1uh)jg md+y(u; 0.8-kg sphere of radius 0.648 m when the axis of rotation)(jhy;d +umg u is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. nxx4(** cftikumc)cjb h*:r 0The rim and tire have a combined mass of 1.ki*4x)mr* xjcu(cb *:ch0f nt25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end jw2+9 sk5j* p+t;,qe r qlunwaof a thin, light rod is rotated in a horizontal circle of radius 1.2*k2s9wu+5tpj;,rej a l+ nqqw 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 wheel rotating at constant angu*oq1m9hcx ; upha:5bblar speed (Fig. 8–42). The friction force between her hands and uo ;cph9* b1a :xqbh5mthe clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–4 f2zpquu prl5lc942c+uesp5( 3cs 2zp4pu u(5uecr9p5 qfl2 l+ about
(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 consis7m7 d)/. np*pzlwujwjts 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 cylina;*3m ysal tuwv84jlyjvy fi4sq + 9jd2*r:;edrical satellite spinning at the correct rate, engineers fire four tangential ej*8qdyytv si2: ;v; a 9+lmjsr auflj*y4w34rockets 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 8.50 cm and a mass of 9hcpf9sb9q i+8fj*dq 0.58c qp9q fdsh9 +*9fjbi80 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,3rw;x ps: az5g 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 tan);ln)j zu iukun;y +a4gentially on a small hand-driven merry-go-round and is able to accelerate iu ;z+nn uj) kya)uil;4t 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 eventually4 zb:t,of,i:+6,chvi0 wrywz eo2twr brought uniformly to rest by a frictional toeitow4z,:h,2rc6zi o:f y0 , wt+bvrw rque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball:tx +uophy6/nl6/tct 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 thqo6/bn nr-:weej b28xf;meq f6 g4y.qrown solely by the action 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 tof6be6y jqge4xf en 2 r:q;b wmq8-no./ 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 beqxw(9p 5h lgomek:fun (/ +de9 considered a long thin rod, as shown in Fig. 8–46.w 9m /p(x:9ue(nlke hfgq+o d5
(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;ib5 re1q9ddutvz 3,b, $m_1$ 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 ebs8n;p fcq4dc,9z1 mfrom rest within four full turns (revolutionbmn1szp;e 4dc,8f9 cqs) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertiasjt *m 52u ajks.5au-p of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqm u:o y/ar*tlsj87h(v;pp dd3ue 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 witho, .8*( z vv-e;4py+w nmf1nrj k9 burbqs9lnncut slipping down a lane at cky-lz v;,+j (. 8qp4nfn*rr9mnwbsen ub19v 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Ear 15-l od;y9edo utqu wq0t57zxth with respect to the Sun as the sum of two e5tw90t 5 7-d; zluoxuqod1 yqterms,
(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. Hb/d - u5vz0ebzjl,60k4lpca i *n2ghzow much net work is required to accelerate it from rest to a rlu,ec p6j0knivhl z*zg5z- /b d4ab02otation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wit(,m c - aednu/iu*)ydzhout slipd(d u *yea,-zcm)/ uniping 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 its tip. It starts to fall ac,y)07)+xzw zcacq(z+zn+ kbw k8hxw nd its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]+ )cb,)c+hzaqkwc y0 xznxkz7+(wzw8    $m/s$

  What is the angular mj v;vj)os;bwknwbq/7cb 0.u;omentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 1vq;ju;kw cvb7/ jwbobn 0.;s)0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of ajc)cqw3+pv4k, r c4e7uc rrz( 2.8-kg uniform cylindrical grinding wheel of radius 1c ecrc43 cr+jwq(7p kz,v4r)u 8 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 rotatin c.b-bh-;yu 4v,csplbl: ig( fg at a rate of 1lb-, .up:c b s yvg-;(iblfc4h.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 reduw ( +fq*xn i8g05iaihdce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If sin +dgx8h( 0qfiwi *a5he 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 initgxi +n0gae/fd;h h .z*m,o+ ftial rate of 1.0 rgf0/int hh+ ; gz,+*odf .xeamev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throu/i9rg2wxkokc q( /rqlwm y.j:dc6z1 )gh its center at a frequency of 1.5rev/s The wheel can be considered a unc/ z g1mr jdywc.o k/)ri: w6q(2lxk9qiform 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 momentum*0.c nntphi2.4 m zllo 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 momentuf*h cf ul a 1kj:t;g62ft):qmxm 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 mom)b*skab6k j+ -ms1cexent of inertia I is dropped onto an identical disk ro+1bx*aemkjk6s -cs) btating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns mie cx(;x.+g 2, fogzqat 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 tvfbhk7:2* un) lq; bqifu:r7hhe center of a rotating merry-go-round platform o2l: hi:rn)7;qhuu vf q7fbbk*f radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is r-wx cp53ytjnb4o e:v,f tp a*2otating freely with an angular velocity of 0.84e tt v ppwb 2a-3*,f5xcj:ony $rad/s$ 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 eventpqjbek3 i wj9 )v::ohgcl6q aq;.jp8-ually 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 angular momentum, whk-po6 qp : qj8lwv.acj;e j:9i3 hbgq)at 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 involvez eks40ps qxqul41h.o( v9b.z 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 wi ;8qscfd ut+nr.bip:12mp2 $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely witaw1lv7 n 9lcj9hout friction. The moment of i9 w9lc lvn1j7anertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6. h-6cs rx*u* gjco, yl,l2:iuet+i nf65-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of,f6+o xuy6ljrceg hi:lu2,c n*is*t- 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 wit 1ginmr;us3d8 -qf-vg4 t/xcsh the end of the rope lying on the top edge84/ fts1-vmg- x3dc sri;gnqu 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 center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Dete 8dk6xqh 4d zo/w0egg1rmine the ratio of the Moo g 8z /xwk4hgdo10qde6n’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resnatdaynb1 1uhfc3 oh ;c 8v9(yb 0qf8 t8t(l+jt 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 of radius u;3 nx. 1*yxk zzpg 9c-xd,fub0.20 m acquires a rotational rate of from rest over a 6.0-s interck3p xzfx ; gd -.1ynb9,xuz*uval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 o/3m.,yk/w)g-- ,a n emyvd izmy xse5kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 5)madv/mxn- z,gmo y e3,y/k yews.-i0.0050 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 angular speed of the b xsxl l:pxd7e;0r0 bn+jx3p6rear 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 sqj0p j*59a( qwj jihm/ize of our Sun, but with 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 spmhjaji qw(0 5*/9jjpqeed 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 stored in,hvpxqz0z 6 ey8c41v c a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a 1 0zy8vexv,czqc 64ph 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 a8 :jtu vlrwd-u) ig1xspp6:l 2n $H_2O$ 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 oeg i).l)xj14mg-gps z n 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 fricti 6 io,/a t)/lte xueedgl5u 6fjtrl187on) 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 gravityi1ttu7e) 8ldr/ujge6/o 5f xa 6ellt, 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 vba wkgc:g l9q7j5vd- n;vau/7ertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). Twbgcl:9aa5jv d7; vkqgn/-7 uhe step has height h, where h

  A bicyclist traveling wit:d;mq0ja0 rsqh speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist an0 srmqj:0;daqd 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 p pu2nlgx*i+p kaerc kk3ys( 9,1)i 9tinto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 r,pkkpc( )una +xel1pgsi*yk293t9r ipm 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 -7efidh;b ,ihbody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms he ihhe-,db;7 ifld tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricalvd y u qy*wi/*6 ldxg6ch:0vz- plates, of hl*ugycv-iqy w 0v d:6*/xz6dmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 akkxdh/z zac1sg3i90(t n )*wg long the looped rough track of Fig. 8–58. What is the minimum value of the */9gg0antish)kzz c xwd3k1( vertical height h that the marble 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 gr i wp047w*g a,zm47t tx8 a8jk*bfiodo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolu/wig5al.h5hklh+i3j tions as the car reduces its speed uniformly from 90km/h to 60kmhi5k/ .i+ah lglw5h j3/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s