A bicycle odometer (which mhdy8n*,g uh1my-zox 5easures distance traveled) is attached near the wheel hub ny*yg d1oh, h 8zm-x5uand is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vel(ww4a kp m 8gn4fubfp;06j3 (lnq;ajk ocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would (j (aanwwp6km4p8f 3j klfg qb ;0n4;uthe magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valuetw+ox w* xz)1z of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lcz ywb e 6.38a7xc(cs:4zyim qarger 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 yov; ihm+4e g 4dih5*96ehw,tsfc 1zeqour hands behind your head than when your arms are stretched out in front of you? A diagram wm 44 sedgz heh5*1v6oc+ htfqiei ;,9may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedak0niiyc 0m: sua4 dlu02 kfy7e53 5fvbl cranks. In which gear is it harder to pedal, a small rear sprocke l uknf2f45suv7d30cb eyii0k0:am5 yt 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 slender lower legs )s6 rxndl(/f79ug,yt3 a siy gwith flesh and muscle concentrated high, close to the b3gag(s x dnytf /r)uiy ,sl679ody (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35)wbymxo3wiz1 ; /3we()qfuuj1 carry a long, narrow beam?

If the net force on a system is zero, is the net lz/72i ots.gu torque also zero? If the net torqg/tzs2 .7iul oue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hiycthb wyh 36)0 qep)6orizontal. The same steel ball is rolled down each incline. On which incline will the bi0)pq)6hy6ewych 3t speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollin/u2fgypop /0w g (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reachey2upp 0og/fw/ s 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. 7ml+lzb;-l) pf* b4yiuhahoa3+ w/cxThey start from rest at the top of an incliu4mlw/cxfahay l+ bh3-oblp)+ i*;z7ne. 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 are conserved. Yet mo),*4p) r /hd3 ezjtwdzwjku l:st moving or rota ur,wzpw/*jzj):4ded3t lhk)ting objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s eq/fs58.qx/t 1:h zjww8lo fhwauator, how would this affewa8tjh:5 l18 o/zfh/w fq wxs.ct the length of the day?

Can the diver of Fig. j46zhp;o9 sukx f25wv3b6 zrr 8–29 do a somersault without having any initial rotation when she leaves the b 64k pvuh6z2brwox53s; jrf9zoard?

The moment of inertia of a rotask1 .htf unlo;.rr(k8 ting solid disk about an axis through its center of mass k;rokr stf1 .h8.u(lnis $\frac{1}{2}WR^2$ (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 in each outstren*lsu p)09k9qqym s( ctched hand. If ss c (n y9*pmkq0qu9)lyou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same s-pf*665 o5h xcpadwy mass. However, one is hollow and the other is solid. Des-p*wh6ya s d5ocx65fpcribe an experiment to determine which is which.

The angular velocity ofysb 3iem:3mc q9d5m*u a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of thmi 9e*du35bcsm 3my:q e wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotate.eao ; bwm5 j,;/ic /8bpvamding turntable. What happens if you wal/,i;mbvdmb ewao. / jae58c;pk toward the center?

A shortstop may leap into the air to catch a ball and throww69b,n,s74hao rcz f e it quickly. As he throws the ball, the u crba sh7,9o6fze,4 wnpper 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 conservat dws19 7xheqn ,x5isffdr67- yion of angular momentum, discuss why a helicopter must have more than one s1q7sd fry-i5xf, e nw 7x96hdrotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r9zkdhmbld;ot ,7 8lgfh-9e6c adians: (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 :hzzcmo ue8*+/ gbp8nan amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) ofou: gp8n8cb+m/h z*e z the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed a+ u6 6ezzetq9dt the Moon, 380,000 km from Earth. The beam diverges at an antde qz6 9+zue6gle $\theta$ (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 650055h21(1yaxyi t xz gcw rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angul1tiyyh25xxca1w5gz ( ar acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5l* gvodkx7ov:dsz +;7 m to another child. If the ball makes 15.0 revolutions, what x* vgd kvz+;d 7:o7lsois its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoluti* s3q9 2fefcbmons do the wheelmqsbe3fc*2 9f s make?    $rev$

  (a) A grinding wheel 0.35 m in diamete*n-o txr,c;ogr rotates at 2500 rpm. Calculate its angular veloci, g-;tonrx *coty 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 n.m n;vi 2tsd5complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear simn d;5vn2. stpeed of a 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 igf+b h+qa6f* +i7cs0ql *rxo lts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee ebdd:. w (pfi*t:4hvjd 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 z 1a;udy( vh5u7.0 cm from the axis of rotation is to experience an acceleration of 100,000 v(y ;1aud u5zh$g’s$?    $rpm$

  A 70-cm-diameter wheel acceleragzx5i3 4m+-7wnyj cpgtes uniformly about its center from 130 rpm to 280-i7gz wpmxc3n5 g 4y+j 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 radiuhy)ty1)h y( wcs $R_1$ 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 them:d:jt lcut2xaj2.m c;selves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no roc:d: jl ;uxa2 .tjc2tmtation 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 rpm in 220 srl)*74r*oclxel a5ds . Through how many revol* xco) e*lald5r4 ls7rutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal-x8z7 yfrz6xoi6nzwxf-mjv. l+7 x z 4culate
(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 hi69 r7lhy-6cvmxm7g b fghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 com6h6g ym-bx m vcf7rl97plete 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 uni fkz5* :iwq/gz,b. fjsformly from 240 rpm to 360 rpm in 6.5 s. How far will *fk:g, jfq.zz i/bw 5sa point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 150h q9y y)fp1ep)0 revolutions before it comesy)y ) pp9fh1qe 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 the1pknu.r sekvo9297 pl car reduces its speed uniformly from 95km/h to 45km/h The tirp19pk.7usl eov n r29kes 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 redus4jwoxl: zs5,fxg ;z/ces its speed uniformly from 95km/h to 45km/h The tires have a diamw szof 5jxz,sxg4;: /leter 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 climb4e )noevetwl(v /;xk 3ing a hill. The pedals rotate in a circle of radius 17 cm. ;v(v4o3)/ xe wel kten
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force old g )db*buy3w-/3j lof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forcegyb) bdld w 3-/l3*ouj 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 cyl z0 /e excp7/j8:deFig. 8–39. Assume that a friction torque of 0.4 lecxe/ / 08zd p7:ecyj$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atjw9w-88z,roq5w -)ebtyu ve ,lp qox4tached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position wlvoz u-4wqxe,8oq tjp5-ey b,w)r 89 and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a s)rifh6+l(*jm/ojfgtorque of 38 jsflf g6(o +rhji)/* m$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when 1az7yq r6)wasthe axis of rotation is )z qa6ywr s71athrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of izbyk31vq:sz , n;n5i a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of then,bv51yz ii;sq k 3zn: hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end oxl bwf o,*ubc8 u:20g qx:yu-0kifp 0uf a thin, light rod is rotated in a horizontal circle of radius 1.2 m0:8:kc,*p-q i02logfyu fuwxubu 0bx. 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 angulaas 1r *jmw xe80a8bk)or speed0sawko*mre8 1 )aj8xb (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of po9m0ncu:b xh9h int objects shown in Fig. 8–43 about:bhnh0m9 uc9x
(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 of9mavu1gtq. yjv(, 9zr 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 satellite spinning at the css-hwd ; ),jq,nw+fyf1 yk(kjorrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If t1 )-;nf ( ,wk +hqjjkw,yyfdsshe 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 uniformq.tk5l-j+d y1w 8 t ay (wtu4kbs7wxz0*v +ikv cylinder with a radius of 8.50 cm and a mass of 0.5ayy.1lk-k ux*s(kw0 tt8 jw7+b vd qzvi +4tw580 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 it from u 0 e75wxpr9cjrest 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 smu 6 a,dqy1p ctdst30r8all hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume tcqu16 d8, stay0dtpr3he 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 ; 5vyvtf -tf5vyu1q z)at 10,300 rpm is shut off and is eventually brought uniformly to res-5 vyy utzf1 f;5vvq)tt by 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 accelerates a 3.6-kg ball7 z:vvbaa2l0e/z ,uve 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 thry.i q b45/, 8 q73hjf:zlhrd ycphosf:own solely by the action of the forearm, which rotates about the elbow joint under h7. jr:y8fho/ 3q4:c,ls dqi5pf hyz bthe 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 ca /bfm8:*fybop v si.k,n be considered a long thin rod, as shown in Fig. 8–4fmp o.vk*fi8:s byb/ ,6.
(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 tw;1dkk ajcx qb, bxb )y(p(sv.2o 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 full turns (ren;e0bvk e o8z4volutions) ao e0be n;k84zvnd 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 mome pfn,b/2nkr-)e.3x:pzwhuront 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 develops0: djik: ia;02oxlrd l 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 slipping down eap9q3jw3;)f ya+yam a lane at aw3yaa ey jmf39;p +q)3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect,o4wt3w kxpg80 k,itf to the Sun as the sum of two tetx4gtokw , ,8fkp30wirms,
(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 radbhj,mv-v*tm1 : ci9hl ius of 7.50 m. How much net work is required to mthb:im j,v*lc1-h9v accelerate it from rest to a rotation 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 fra-9gdv(+ ga236*)u1c1hg9gjhn uhc svqtm i som rest and rolls without slipping down a 30.0a sqmh2ugv-g( s31utcn)dcjvh *6a1g hg9i +9 $^{\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 -: g*cfnlm 9p2f ixk1etip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just :ke -mgxf pc1inlf29*before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentu q(dti7 *;nnf,qdu l9m6gq1j0 s k+odsm of a 0.210-kg ball rotating on the end of a thin stringlniojk+ sqt(;dm9q16g7 duns fdq0*, 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 u9:s o866fi jscei-lzwniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm?:zwii- e cls89j66os f    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating atz) w em)wz/ot8 a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, twmo e/)z8zwt) he 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 reduce her moment of inertjv,m)stvvivt 5 fq.oa.5q)5n cll4.tkc ;7nvia 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 h)5lv7,5j nvv;)i qtt . slvat cf om4.nv.5kcqer angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase htqwhj5 sg-w1 a*3-4cpzkiqq8,js tb3er spin rotation rate from an initial rate of 1.0 rev every 2.0 s hib-,qqs33 kqtwpw 1jz8j5c -* gst4ato 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 arozx( yof5f 9+mfv5ghe7 und a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chuxf59e (hf o5ym7fgvz+ nk 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 skater spinnidc6)z j2 8qlpsng 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 m9(aatezc 6cq9omentum 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 of inertia I is dropp /g/kuaxe 8go.t2 ;uoxed onto an identicx xogt/ue o2k8;.ag/ual disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at6ysm fma 5p d4a+n65fq 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-ro zpi8c1bs)n;t und platform of radius 3.0 m and moment of inertcsnb1zpt; )8 iia 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 wjh3k-5hgt hp* nq+y:u ith an angular velocity of 0.8 hp h+jq-:k* 5 tuygh3n$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about h 5 sl sf(f,(tpc6c/9qzcn24mr,ulc lk alf its mass in the 5tcc,( fuk6ln ,(mfzl/2cl c9qspsr4 process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ofmihqta;4r /+i ) eplei,a3po7 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 vluuyy99zb0 ;d5db fgox)i j bt3o; 5+$ 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 ca *1df-:/dbuwoe0:sg 8nb0dsi k l t7yln rotate freely without friction. The moment of inertia of the person plus 0nbi:/k 7o-wysd dused1: 0tbg*8fll 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 osy;. b rwmwm-l-6xmh:f a 6.5-m diameter merry-go-round turntable that is mounted on frictionlw-bmsm.m rl- y;6xw :hess bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope ro 6q2;2o4 dlethukeb2d2 nkcv1lls 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 unwinds from the spool? How far does the spool’s centerbecqkvd l e242n2 to2 u;k6h1d of mass move?

  The Moon orbits the Earth such that the same ytdkcp (mx-. lsq/ .0fside 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/xp.m0ks.fd y(-tq cl the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fw58i iyuo:rhmp.;;f h 0qzao4rom rest 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 0.20 m acquirlssv;x;iy m.0 ea8n:des a rotdmyne. a0;i l;s s8:xvational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of )h6 uh mw 7bifl05n(pbmass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diame0f b whm5h)nui lpb(67ter 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 rear why p1)x8yby 6mreel ($\omega_R$) 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 greu0(m, ,gtc5ow2zi l rvat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star ,c0,5vtmr2wogzu ( li 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 lowro9fzj1ws,u3 sc m );:pa a4sc-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Sup3s,wazurma1cs;j c:9p o)4fspose 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 illustrates 2c*cw n5fks; pan $H_2O$ 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.vhxf64sm 8pt-uo zf7q1, cdmd5 d3ho23 $m/s$ 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 itom +0/;( jnq*uwuxnM and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a waut;/ q (n+0o xjiwmu*nll, 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. 8–21g.]

A wheel of mass M has radius R. It is sta pksoi k4c*q+jy *tzm 0.5g5vznding vertically on the floor, and we want to exert a horizontal force F at k*+ 5vsy.t5 mz4oz *0pjickqgits 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=4.2mz /y ktrhhr7:b;zrf ;*/s on a flat road is making a turn with a radius The forcbzkrrh ;y;:h 7tr*/zfes acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengv313 twd a;ba0rcse+tth 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 rpm after 5.0 s. cwa 33;b0trsd1tv +ea 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, makk4l-xi(w - *lr8h hg.rfmh2gking reasonable estimates forl* l -8-hgig rwr f4k.h2xmk(h the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two c*8anzgzy 2d .iylindrical plates, of masz gz8.2nydia* s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track -kikifer 3+m ; 5zct6uof Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest3 ;k-iurmz+ ie5kc f6t point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumecmd.hk3 ln-k p*(,gpfa+q vq2 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformcfp mhx/hk9i za; 421dly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eachha/cz291;hipmdx k4 f 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