A bicycle odometer (which measures distance traveled) is attached near theej 6p*u91p:giqzy3 i37th uaxn:p fb3 wheel hub and i :npth: g6i7u 33 ap*9 u1xpefzyqbj3is 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 velocifg9 pwf)h:fhe7-)akxty. 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? f-9f:gfahw)) e k7pxhFor which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be de ,m5bdprwst(4scribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largzh 6f b y9y)1w0m3fxikw3:pjyer 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 difficuyd-4p uxywc- 61 46dn u(5ett6ergih dlt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A did t5h -u ewnei gpdt46r-u6614c(xydyagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and th(yopa. 5a o*airee at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a laraoyi*p.( 5aoage rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend gkh7j:yfb3 .non being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the :3g.hy 7jbnfk basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narroiujy(u fy q d5.b*;a;vw beam?

If the net force on a system is zero, ta *jbj-mo4.q is the net torque also zero? If the net torque on a system is zero4 o q*ajt.-jbm, is the net force zero?

Two inclines have the same height but make different angles wj2nhw el9 1zz:ith the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball a1w l:ejzzh2n9 t the bottom be greater? Explain.

Two solid spheres si1d9/m7cui .0wshrcm qmultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed therei0 rmw 79h/s1c dqmcu.? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same r ekm999idm2gw4a 0,llxe(pbn cr +pk; adius 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? Which has the greater tota0kme2 9( k4bllxirep ;, wp da+9mnc9gl kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and a+6l, o9e,tkf3uga; 3yilg n djngular momentum are conserved. Yet most moving or rotating objects eventually slow down and;6 dg+u , 9ytn efl33jakligo, stop. Explain.

If there were a great migration of people toward the Earth’s -onh kci )qhx y(e;q d.f88(cjequator, how would this affect the length of the day?(d iq8(h -h)q jo.nxk8ey;cfc

Can the diver of Fig. 8p/wmtm 2dpt9z:q)ptzg 95j z7d zv6a1–29 do a somersault without having any initial rotation when she leaves the bz92ptp5gd ) q1v7t z/jp6:wzmatz dm9oard?

The moment of inertia of a rotating solid disk abouttu -4cw*9 c/+70q ygegmj* npwpeja( k an axis through its center of mass icempjy4 *qk9p* +ugw/jcta-0 en g7(w s $\frac{1}{2}WR^2$ (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 9f),:sd8pfab+ esgx xpip7.fw 0 )pncgs)mm.holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decpsfc pfx)w+ mn).fdb 78x9g) :g0.p ps asi,merease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holk6ch,hnw n0(yj7/hgs low and the other is solid. Describ6/ whc7 ,ksygjhn(0n he an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizonm7u/h wz x3nerj99 ujvhz61.7lg -chatal 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 acceleration of this point at the top of the wheel. Is ther9xhch e/gjhmw7j.6zazu7913uln -v angular speed increasing or decreasing?

Suppose you are standing on the edge of a lar1ffs ma8+* qlde wr1h9n.gd2 cge freely rotating turntable. What happens if you.ddl8+21w cnqhfsf er 1*9mga walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quics;vbi+ sh 7pu0kly. As he throws the ball, the upper part of his body rotates. If you look qsup+ b7shvi0; uickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dinx(ovp s*l9( fscuss why a helicopter must have more than o*no(l sxv (9fpne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i2,nqnnghe 3/ws* kmatm1 -2oq n 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, using /iu.-l1tzo y1opbr y3r:hr 7p the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eao3yrryor/tih-up 1l.7: p zb1 rth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divnpwar0 ia-,;-p3bdl g 31 gljeerges at an anglealieg a 0w;nd-,j3rpb31-p lg $\theta$ (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 3aw, p ( xhz3ekf;ia*jof 6500 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 blades slowei zxjhap(w*3; f ,k3a down?    $rad/s^2$

  A child rolls a ball o d*1r;xojlj/p *g2:huq w(qrkn a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its dir:drhj2glx; 1 op*(uw*jk qq/ameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. au(z qon:c;)xHow many revolutions do the wheel);xqo z (nuc:as make?    $rev$

  (a) A grinding wheel 0.35 m in diametej *3c -mx,8s/o 9reibnviw2ewr rotates at 2500 rpm. Calculate its angular velocity in w/eeivbcix,rs 8omw*-39j2 n $rad/s$ $\omega$ =    $rad/sec$
(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 revolutiontu2q6eq8b -w 0(nksxk in 4.0 s (Fig. 8–38). (a) What is the linear sxq bq-kte2wu6 ns(80kpeed 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 its og sj( 4g+-2ufxtfcd* ib8yj9grbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point d6 jwj6 hf)9,pl7gegk
(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 partiq z.-j5b 8 thiyp1+bibcle 7.0 cm from the axis of rota5+8thb1zbq-.i pi ybjtion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceluf4 01qjyvt -0igjv2xd ;xi*jerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. j ;2g4v0xj-qxt f1i0vuj d*iy 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 radius8mv 912kze7pvvueb,k 3mn4 nu $R_1$ 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 themselves )upvy jdr8mek 0/pa t6rnt0n5 /.d6ib 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 can be thought of as a cylinder with a diameter of 8.5 m. Detrm0re ji/ . d0n5t8t) 6p/apnduky6vbermine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. s3(m gk:; mzagThrough how many revolutions didz(m;: gasg3 mk it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Cal ,)js6uo epk)wculate
(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 stressep21d y+ )krq1tgfe -wzy.rxx2s of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn t1.)yf2ek q -twrp+yxg 1dr2xz hrough 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 acce.c aw5 jgayn9c;2m, idlerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled ing;wcjnda.mc 59iya 2, this time?    m

  A cooling fan is turned off when it is running at2 5ej,;.n 3r ekwujar6+rk sbk 850rev/min It turns 1500 revolutions before it comes to a stop. , 5s;reu+ rwj .kn6brk2 ea3jk
(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;fgcj 47 s.oee car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 c7egfj;s e.4om.
(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 ma1i 2fd,3m qmvnke 65 revolutions as the car reduces its speed uniformly from 95kmidnm vm31qf2,/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

  A 55-kg person riding a bike puts all her weight on eacao g82x2 c0dkyh pedal when climbing a hill. The pkaxc 28 gyd20oedals 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 cm4.zvo0 )p qxrucv( g(u wide. What is the magnitude of the torque if (0qupgxovzr)cu. (4 v 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 v cdba,6wz:d; l;(aoyabout the axle of the wheel shown in Fig. 8–39. Assume thl;c;bvaaw:ydoz ( d 6,at a friction torque 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 the ends of a massless rod whicotg3zg3p*,-xa pwc. xry).ywwi7 q9bh pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relea p)wqy, atx3w gywg7x z.*irp9c-b.o3sed. 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 tw g1d jibe92n 0if9va3orque of 38 0d 9 avni193eji2g bfw$m \cdot N$ 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 oy f4sx)wq t68i4vo3ehf a 10.8-kg sphere of radius 0.648 m when the axis of rotatio)ofhysi684w tq4 x3ev n is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a ai.34qth n+whbicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub a +t i4wh.hnq3can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated inkqmgfc 0 io(38a(d ie)kyc j*pb:i 0,u a horizontal circl8pboc(q,ucid0iy3ag jkek(: 0f im )*e of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a . yaibxc 4eg+;potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force b.i +a cg4xeby;etween 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 point o 5lrlqd2/f5 deaz r44vbjects shown in Fig. 8–43 about4efdvra/4r5l 2qd5 l z
(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 o:w 2ihb8a-igo/ soy 0tf 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 satellitab 4 :8ug.w5elhep1 uke spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radigul8u5p: k4eb. w a1ehus 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 massdnejd 5qw:;*4jt9 hqs of 0.585; tdwse jn9d4q:hjq*0 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 rest z49hwjd )g(qnh y8d(nto 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and il171mska;*5ym8p fjr, seiz m s able to accelerate it from rest to7mrzs5*1emlk ap;yi1,jmsf 8 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,q62,1a p rs z0emm bgplat7u)/300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 a6zms02 )lmeptr p q/,7 1gbua$m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-k1(qffiv*jo/dd3 x k 6 se/g, m,zxl:)dmjzn,cg 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 ia rtq mi80awt ,w9dvddo/16 n3jb)jf+s thrown solely by the action of the forearm, which rota rq 8 m0na6fw di/9a twdd13)jbvjo+,ttes about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long th-2q8y9* re m z:mfxwy1 tcbst9in rod, as shown in Fig. 8–46.bfm* m 8 92:xrqctyytw9s-e1z
(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 con*a6tib- b heg;sists 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 res(y1 mio+ 0akb7gfu,xrt within four full turns (revolutions) and releas+bmkr u1(,fg7 yi 0oaxes 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 haoqe1ar 4w+a.caj)1 6;z rkjqps a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o9z o5w: wbw4m5 ez,mnxf 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 p rb)zq)e86 (u3x-dsi3o nn+eazqvg. without slipping down a lane at 3.3e)abgz ro (n8 xqpuz3 qn3)d6e-i v.s+ $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thep a) y/x8r s70 j0odbilros-,: jary7w Earth with respect to the Sun as the sum of two termsry-7j y/b s0:7aw sjro palr)0i xo,d8,
(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 7.5k3+l./gqxwq 55i/xyz:jl 5 nb poiz,q0 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revol 5zx.oi3qp5qxj ilnq:zg/yk,b 5w/+lution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1 0oadxdryhgb z2.-3 -z.80 kg starts from rest and rolls without slipping down a 30.0.yd -gbaz oz3- 2rxhd0 $^{\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. Itrz , gp4s05k2pw rsf7r starts to fall and its lower end does not slip.4rrg0ps k ,52p7wszfr What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thinj4 5;: 9hvdc7l04nfw sapibkb string in a circle of radius 1.10 m at an angular speed of 10 ij f4vc74a bnwk;:9ldhpb0s5.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wg9;jx aec qm d,qjei0.41 +hkaheel of rad9haeiexj4; k0a.1c+m , q jgqdius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is r-57-bjgt pchs)p :z:hzl,jqbotating at a rate of 1.3rev/s If he raises his arms to a horizontal postc,-:pz bz)5b j:gs p7hjhql-ition, 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 reduce her momentolz: i9 k ei0(wgd*mz2 of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes g9eomw* ikl0:dzzi(2 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 /wt315tc .7 i l1 ki1nolmmnirrev every 2.0 s to am1i 1tinori. wmc 7/lt5knl31 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 f m+ajsf 3q13fc :srf(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 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 stisaf:ff 1r(j+ c33 fqsmcks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning av 16ng4 u9d wr jxjeb.a0taiap9c );2vt 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum 0/4+j2vwfr v sikwni;pqfz *2 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 inerulcbb g +*ry(-tia I is dropped onto an identical disk rotating u-+y blrcb*(g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsln +kvt 3z 0,jy3odx6b 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 a*gl:n( rxqg 5at the center of a rotating merry-go-round platform of radius 3.0 m and moment oag5n( xqg:* lrf 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 mer*l*wtd*t m j8zcgw5z ldp* d60ry-go-round is rotating freely with an angular velocity of 0.*j zwg8 ldm5c0*dzl**td tp6w8 $rad/s$ 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 event:wg-n ,eei 0yv6cyy x0*j vko7ually collapses into a white dwarf, losing about half its mass in the process, and winding upy*,wj ync:0 vgx0iek 6- voe7y 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 oftuevva6a*zwuk) p *5- 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 mx2c ,t opl6 p.+,ibom$ 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 rotateckb rh/91,q ay freely without friction. The moment of inecah rqy,kb9/1 rtia 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 20ujif5.0ca jodx,cfc 4( lpttwo(hlu5zz87 a 6.5-m diameter merry-go-round turntable that is mounted on frictc 4f,p dz(t0j5j7u(zwocxoa u 85c2 ftlh0.i lionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the groutr+*b;h+k;)7thha9 a f xd tpgnd with the end of the rope lying on the top e gx;dhhh7* f ap9k ttra)t++b;dge 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 noon4y;bb4wb4q1 + er that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its owo44r+yqnbbwb ;4eo n 1n 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 rcrkeb 76hc2pk+gmjua 0*h.1west 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 sha wq / mg8mjr*9.xtf5.- kanvayxs3jg9pe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelersam.jan 8- fxgx/9q .y*gwrjt3mk5 v9ation. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made ofrj.ogd)9 l 1sk two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches thedg rjk .1l9s)o 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 utf,x2a/a. ea the angular 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 tiu2 crm1 4zgb*bmes 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, los4 bzgmr1*2cbuing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it xy-f/-p xc4qg6n vwiu(gc4l+ to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.-x4g yuf lw+nxpqc(c i6-gv/450 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 afq8hzoegrba -v 6zq(j1 ,le(4n $H_2O$ 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 horizonta(u;,iu udehw l)1 sfc622fchkl 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 lnaax, -9e6p5ab)yn xL can pivot freely (i.e., we ignore friction) 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. As 6y) xeaxp9aa5nl-,nbsume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor,jga tsx /n-g,, and we want to exert a horizontal fgtn,,xg as -/jorce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s p0srcxh gxqrzqfk 7/* ).cf+6( mah7son a flat road is making a turn with a radius The forces acting on the cyclist an+fs*.kxqc h0 rf q s6hg7c(a/x7)zm rpd 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 lenpx)hpn eo;0- u 9(sg vbt-ps5cdoi9/ jgth 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. What is the torque required to achieve this feat, and where does t 09i9jto)- ps /b(edg -5ophn vxupcs;he torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin vpn363og5cl5e( kuram oh;u 6 4om/mlrods, making reasonable estimates for the dimensions.m6ah o6evo5r5 ;cpm3l3/ ok4guln(m u 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 cylin1i ch4 2kwg70w8 )d i,ypwkd.mvhn w;tdrical plates, of m27,0 pw8 vk.wycdh4hw wkmid)i n;1gt ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 aq6kk9kymb9zgi5mp+f cq826s long the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is t 2bkkqyc sqmi 9kp8z6f 6+m59go 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 iv ,-x35xfcvaassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uni o 7f/j/jee7usyjuf r5w;(/faformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? f/5 jao;//uuwr7 fs yej(jfe7$\approx$    $rad/s^2$(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