A bicycle odometer (which measures distance traveled) is attach)lh 3s:s*bw luzw 3ei bb 2bjw:;0va;ied near the wheel hub and is designed for 27-inch wheels. What happens if you ui:h)*:j 3bbb; u; bilwsz3eaws w0l2vse it on a bicycle with 24-inch wheels?

Suppose a disk rotat2vr hxal7eci;fvc ct28+0 e9ues at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For wh8t i2 7hua;e92vcv xfre0c c+lich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocit ;si4zw*k: gi/gfutx (xi,b-z y $\omega$ Explain.

Can a small force ever exert a greater torque than a l vdb7p : .jpe nki)-gxtvamzu*+1:m:harger 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 /v+h0wn -y,c 6ih+rm. hjp.en4qhb uswith your hands behind your head than when your a/h .pjb-.hhiwn nhm veq0sc ,4yu++ r6rms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and wuld0.m i4c;y three at the pedal cranks. In which gear is it harder to pedal, a small rear sprock4w lmcd.; iu0yet 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 ;ihk4n)u;r-ilix i/wl1tw)ad ,s,cfon being able to run fast have slender lower legs with flesh and uf)4rsth)n iliw,/ ,ic; l1i-;dwk xamuscle concentrated 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 walkers (Fig. 8–35) carry a l-onmeth4d .9fadtlc(qq,z m 9e/e9 c 3ong, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the nno+g vzm-va yob .4;o5et torque on a system is zero, is the net forcogm;+ .o av-yb4zn5vo e zero?

Two inclines have the same heighw0 7 mgs7mn3/kfx+ lert but make different angles with the horizontal. The samews7 g3mefm/7 k+xl rn0 steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) do7 a4/rl/xbbv bwn an incline. One spher//bv x7alb4b re has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the samh ux y+,lz*r3ye mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speedrl3xzh+ y,uy * 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 aipbofn+zg23 6z .q.bp re conserved. Yet most moving or rotating objects eventually zqg6pi2obb3 nf +.pz. slow down and stop. Explain.

If there were a great migration of people toward the Eavb1.xjm cf,-d rth’s equator, how would this affect the 1m -bx.cdjvf,length of the day?

Can the diver of Fig. 8–29 do a somersault wielr7ahv;, t)sqy ;mq 8thout having any initial rotation when)e;lmt7,8sqqvr; h y a she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center9cl,8yz : dqcms.3ys z ofdy m.:z,ysz98c l 3csq mass 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 rotat 18-huy9kgl-)yfk o +rcv.kyying stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expl1-k uf k+-l.yyrg9yh)y 8cvkoain.

Two spheres look identical anq+mrbp w,nk9: d have the same mass. However, one is hollow and the other is solid. Describe an experiment to, r9qkn:pbm +w determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wej3 cx r3v:kjlu5gar,) st. In what direction is the lineauc3a5g,v3rrjx jl:k )r 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 the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wj-)bx*g2 do9hn otxf7hat happen*o79tdxox fhg n2- jb)s if you walk toward the center?

A shortstop may leap into the air to 3amwjn -ozg97 catch a ball and throw it quickly. As he throws thew -7ozna39jgm ball, the upper 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 conservation of angular momentum, discuss why a helusv4zlb 80fyw-l-7f u mos1 x46ks0r vicopter must have more than one rotor (or propeller). Discuss one or more ways the secon8 z1bum7owrv - -xfks4ll fyu40s6vs 0d propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 k,ya65-f6 gs pgy1i hf$^{\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 -w/rz k o,:zdii*rj )wcoincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eart/r -i d,:o r*zijkz)wwh.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea9bno::ac1kb w7zi; jx rth. The beam diverges at an :ak 7cx:zb9onibw;1jangle $\theta$ (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 rpmu1jm 9g6*:n n0 mwjzkt. When the motor is turned off during operation, the blades slow to res: znk0nj*jg6umm9w 1 tt in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball oz kanb- *lb bw /t6g2ouyh:6-an a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameterwb*kz- t ho/u26abgn 6ba:y -l?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manyd:k-vxnsm2r a.j8l z*tv:p) w revolutions do the wheea .vlw: tnx dsj8 )p-*z:2rvmkls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate il 9q+0r fe7uapts angulerl+q90 p7fu aar velocity 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; a5 0at uvpp0q- xuxz).ih3tx makes one complete revolution in 4.0 s (Fig. 8–38). (a) W. z;avux)q0ti-0xt x5u 3 phaphat is the linear speed 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 orbit around the Sun t o.66nupdj;; ok yi1o   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poirc +st5op4,jfnt
(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 partq(*ulzd+c 8x ,gx qw,nicle 7.0 cm from the axis of rotation is to experience an acxdczq nqw,(,+u *l gx8celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm ttt;*:v:r.,xedgi uk3y na61,q yiv b go 280 ;:tq nit6ba,v u,y*y 3 grdxg.vkei1: 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 radius a3h)b et9z )sz$R_1$ 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 xb :+l;k/ylv vy tv9b/Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during / lbkyb:+xl/vyt9v;va 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 fr(nbqr.a m +fyz0yi6 4hom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time6i4.f0qbznyrhm(+ ya ?    $rev$

  An automobile engine slows down from ro9q7h jzo,1b7ufgq)4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets q-gv7, ic*i)ri3 d lohin a whirling “human centrifuge,” which takes 1.0 m3-gi ii7ol*v r),hcdq in 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 diac.b , fhyvp r 11n:dsxd.(4wibmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled 4xb:pdn(.hfbd.y,wvs i1r1c in this time?    m

  A cooling fan is turned off when 3osbm;u 6+ uahr4(f isit is running at 850rev/min It turns 1500 revolutios; ufbu3 sm+ 4i6ahor(ns before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fth w mpz 5:q+j; (ajpts(r*,dorom 95km/h to 45km/h The tires have a diameter of 0.8:j pztm*5d ,r o;shp(jt(wqa+0 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 l uw+whc.zy+u(x* +t9fe1 qhbthe car reduces its speed uniformly from 95km/hycwfqzt9e(wx +.+ u l+b1hu*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 qqvo3by( 2c/,a(/ zturo-xsg all her weight on each pedal when climbing a hill. The pedals ro -g3uzar(bq/ ovyx( 2qt/c,ostate 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 forssitdqdu + 4d kg0t:/56f2c bdczlv; ;ce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exer i4t0cdg:dztslcf s2+/vduk bq d5 ;6;ted
(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 torquk,p/x7 k3idbr. d4hnaafr; (ke about the axle of the wheel shown in Fig. 8–39. Assume that a friction torqu4k7kad k3dnax / ;r(.fh ,pibre 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 masslesm e7xf 6tta08bs rod which pivots as shown in Fig. 8–40. Initially the axm07 tt6f 8berod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requira -m 2q9idwqx b7 (5xce-dbwv*e tightening to a torque of 38qdwemvx-qw 7* a5c9i-bd (x2b $m \cdot N$ 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 spo1q/rs:b. tb.ko4d-4n or glxhere of radius 0.648 m when the axis of rotation./dbg-x.4o r r4on1qlbtk so: is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The,p+d4poy*7d qbxql2na7 pl 9n rim and tire have a combined m+xd7 o d p, 2npnqp*ya7qllb94ass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatekk2vo+ ksvs0 z0 :t o*fpm9*ond in a horizontal circle ofvfkkzk pomv0 :9* +o ss*02not radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotatinamc .y,i5zpoy pmnk b.0 -8au*g at constant angular speed (Fig. 8–42). The friction force between her hands apyy0 ua-8ikmz b,moa.5n p*. cnd 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 oap96c1kg4 zw vbjects shown in Fig. 8–43 v z196cawgpk4 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 consists of two oxyges-eh+v8brm07bq,e)gwwb r0 fn 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 thz x:ys * *o.2,khnp;upcdad:ze correct rate, engineers fire four tangc;puyodn:*zh pxd.sa, : 2*z kential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is tjw:,bhtqcp0q:g-c9 j 4u 5hea uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcula:-cjtj4hc qtqpgb,hw e0: 5 9ute
(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 i +jwb zpfe/kd -2yax/4t from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven 5ttygw.hi6 (c 7rrl3b.auz)jy8+j nf merry-go-round and is able to .)y5r jy8zj rgniw+7fctl.aht3(u6b accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brougr eoa )9u4fnwe gr7x+9ht uniformly to rest by a fricte7raoxru)g+ 9 wf e4n9ional 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. 8x-k+zrl.us/2 dr3zfs3t9/r x, vkbxb–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg balle.8ddgmfm zyq,n67 n + is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the tricezq7 ydf+ne .dmm68g ,nps 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 thinzcwbzr4-l:sb9mn49e t-c5v c rod, as shown in Fig. 8–4e9sc5rzzcvb4n4:c l --t9 mwb6.
(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 mas e 0g,cjns(b : cje2hs7fdi.f8ses, $m_1$ 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(ad 3 aaj;v4c yuv,jzpv91sc, four full turns (revolutions) and releases it s;3, 1updz4cv y cj(jvaa9,vaat 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 haeoxu73ubvps a(; o :7lxrd/d-s 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 of 28in p e;6ue3who+r1 aq,0 $m \cdot N$ 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 slip;h s )ai00qgbdping down a lane a )g;0is qdha0bt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the E tz ihg6cdc,29arth with respect to the Sun as the sum of t2hz 6 cgc,idt9wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mas )q/cn*bgtg9k4 j b lrrd2egn-x1*zb0s of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotati* )xqd je2gc4zrr b9glbnn0 k-1 *tbg/on rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm j 0hg0wb5xdu 8pd8ct7and mass 1.80 kg starts from rest and rolls without slippi0g70 bj x cdpuw8h58dtng 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 a8kllw3w sk5 jxd 3*q6and its lower end does not slip. What will be tk85aqd s3k*jw6lwl3x he 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 /jin ckcq 65b/dk4/wcmomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an acd/k5q b4ck nc6/ ijw/ngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifos ;jzvofa/ .t2)a ufwfx w/8s2rm cylindrical grinding wheel of rad t/)aw.sf/ sozfju8xw 2af;v2ius 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 t7,v noyllr))+mb or7 at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizont)o,b +ro7ty 7)vllrmn al 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 reduce her momenth i (gybiyu:g(5ivi x-,is; 5s of inertia by a factor of about 3.5 when changing from the straight position to the tucky hugsigvi(:xii yb-5,; is(5 position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initi4;ssxh8g, (b twb(eljal rate of 1.0 rev every 2.0 s to a final rate ofx( (wgbb h4s8tjes;l, 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 yb1h,zf s65sjvertical 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 chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. ,b6s zfy5h1sjWhat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skaje-yb5 *fdl3 ssx (d;bfk4t1hter 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 momentn: a whe.q5m1jx; 7iagum 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 dropped owu6z 6npwm5b + cb,:ok*wzj s(kba5 v2nto an identical disk rotawcuvbmwwb :z b k2k6n(z,+a5*5o p6j sting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2+dnk z4p x4l,;h0pn)aimzue2.4 $rev/s$ 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 merry9h /.uckoepo.-go-round platform of rauko/ .p9ohe .cdius 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-o j) 3q) u:gwgj w3xdw83e.8wn pkyd9kround is rotating freely with an angular velocity of 0.jdude)9nw )ojwk3 q x38ggpwy83:k w. 8 $rad/s$ 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 ihni5f3 *2l tlczc 7qj;+x y,knnto 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, what would the xzcl + 5 qjhnil ky2tn7;3,*cfSun’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 inet1 i i4hy6ed3 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 q3ku4n w2wm b:$ 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 acarht r /km0x.m- ;/vot rest, that can rotate freely without f m/ o;/kxm-ha0c .rtvrriction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge0uy/bmd 0x,iv of a 6.5-m diameter merry-go-round turntable that is mounted on fricv0by,i u 0mdx/tionless 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 ground with the end ou -h7im/ l t+vy3urhmm79fiw, f the rope lying on the top edge of hltfm3 /9h7m+i,wmu 7u-ryv i 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 Eart(cv0 he1/imgg nv)g*eh. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its o01h(vce) *vg/m gnieg rbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from wb6xb.dq ,j mv5xd5a/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 uni( 2idvs9; irvoform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.02dvv sor;i 9(i-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 cylindr9vyjhzzei3kl 58m)1 s ical 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 line135 9yjzm klzih8es) var 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 ths7j 44jv s . lewo6cd3zqy+lmylk-.* we 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 times as o.3spqz 0:fey 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 inyq s0.p fze:3o the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ener1 fiv+lt uotqb 8e2p98gy stored in a heavy rotating flywheeepfi1v8b tqt289+ uol l. 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 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 illustrakdp8iai;s: )jbh 5b 72b/fqmd tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rollina4q, i)wq9+ y avbmdy;g on 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 fr(cwf c,-o g7 v +uy4d-wkvub5hiction) about a hinge attached to a wall-,u cvwduwbc4y fv 57+-o (hgk, 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. I , 5ywgo5;c7eedom7nwt is standing vertically on the floor, and we want to exert a horizontal f,5w o7;dnowgyee m5 7corce 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 spb3p 5bbg)6s-jpc ms;l eed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle as) sb gjc-p bbl5;3pm6re 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 0.7 o 6r g, fa -ckxopqyv(chj- *:guj40;vr;sux50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his h-ug: pah6x0rq;y7f u c kx*ov,(jvo4 g;- jscread, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her48m8cfba my1 r 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 held 8fm8a14yr m bctightly against her body.   

  You are designing a clutch assembly whi . mcefu8y) hgy p*)byo7,o-ibch consists of two cylindrical plates, of mas i7bufem) 8,*- o o)gy.bpychys $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rod80, d20 a tmhuz3awhough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop w3tu8wo zha0dd m02 h,aithout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dqh0 (e) e5h9lxlyt8ov o not assume $r\ll R$ h =    (R-r)

  The tires of a car mak 3 eyugkqx3.y6;p+;k n4jcey ue 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was tcu364uqyn3 .yypx e;+j kk ge;he 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