A bicycle odometer (which measures distance traveled) is attache6v+ci flun9o ;sh;(emd near the wheel h iv(+snlh9uf ;6eo;mcub and 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 an x*1. fha)ypmk8zuqs:l3pc/ v gular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point q pp zky3xf8*ul: vs1)/ahmc.have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angu /b fg2gw16lmylar velocity $\omega$ Explain.

Can a small force ever exert auq.up +ev ve*of)7o 9f4 -xfod greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your he8tfbsq,-g iptba:/;zo.nv ; wad than when your arms are stretched out in front of8n;;ft.qo t: /z vigab ,psw-b you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three,*nqt snvub1d : zr0q0k(/rmz, vxll- at the pedal cranks. In which gear ,n (zklqz0m:q /lnxv 0ut r-d1rb*vs ,is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with f7tt3,3b)bf hl2f(i fvnzhp8 dlesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotati7bfptf2,bv( zlnhfhti 33d8) onal dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrypz/9akp+7)t6l ptdwy .sc r;v a long, narrow beam?

If the net force on a system is zero, is the net torque al l+kd 4a:jbm h.r-sktv qg98 ifutn)k/7d.z0vso zero? If the net torq7kt8.fs4 lmrb q-.0:uj d9ahit)+ gdk vnzkv/ue on a system is zero, is the net force zero?

Two inclines have the same height but make differentp.er1 j2 14orebl0alp7eca w7 angles with the horizontal. The same steel ball is rolled down each ir20la.c aepj1 or1 le4epw77b ncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) 3ag9rl uv1+ k+geypc 2iq 0a)hdown 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 there? Whichli+gqvr)e ca19+2hypg3a u0k has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same z,t52.,xjaj b(hch me mass. They start from rest at the top of a(a5 hj2xbz,tm. ,ehcj n incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are c d2qnuzct9ct(0 tw )6ponserved. Yet most moving or rotating objects eventually slow down and stop. Ex9zt td0ct2p qc6nu )(wplain.

If there were a great migration of people toward the Earth’s equa gc wnwt*npx u0x,my/k 1y9rs80u.a:w at2lp) tor, how would this affect the ler p :tg*ynwu2 9l )pwwyt xax.,8ksuam0 /nc01ngth of the day?

Can the diver of Fig. 8–29 do a soxrz3e)wl(f j5si 8ja 7mersault without having any initial rotation when shw)fs 38jl (iz5a exjr7e leaves the board?

The moment of inertia of a rotating solid diskohg +oy,l) h11lnw,o u about an axis through its center of mass is +on, 1h )1owlolhyg, u$\frac{1}{2}WR^2$ (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 ron c/*0wnfco:g8d+vx g;ov;(:r4 vvlep6h wvdtating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increapglnd(8n + e0ovhw ro:;gx/4*:v v6fw ccvd ;vse, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow andb/ sl6 q94zd 4cvtjs)fxy-2ip the other is solid. Describe a4p- f y6jvi2ss) zqt9d/cx4bl n experiment to determine which is which.

The angular velocity of 0f6ooamv.q. ca wheel rotating on a horizontal 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 the angular speed qf.0ovmc .a6oincreasing or decreasing?

Suppose you are standing on the edge of ai nokae4 )g;e7yz* 7qf large freely rotating turntable. What happens if you walkai* 74qn 7kf)y;goze e toward the center?

A shortstop may leap into the air to catch a ball and throw2 . xditofj./39aasa l it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotatea t2o3 xfja.dsi9la/ . in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation ofnq o,uma:).67yruhru +;ar xs angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the heli.ua7sro)6q+, rru ;:nyxmuh a copter stable.

  Express the following angles in radians.b4 d/(ww,skuj mnst0: (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 beca5sc 652/c/mrkx q teo,prc, z*rczqy*use of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamete2c/p r mkyz,q, ocxe*c5rsc5q/6*tz rrs (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth.s -7epnhtk j yb2;7mv j+0eaiu1fh* h6 The beam diverges at an an2 u-yfva67* nhebts;hekip j0+mh71jgle $\theta$ (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 w+nd; afewm, *me+ ;ow6500 rpm. When the motor is turned off during operation, the bladnw m,w;*d eafo; me+w+es slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another mchzsf-k 3pa f-s 0*:c+0 wfdfchild. If the ball makes 15.0 revolutions, what a+fs zf-0 pks f hf*3wc-:m0cdis its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions d1+pge bec;+y 2tl+ yfao the wheels ++l2ypbayt+e fc1g;emake?    $rev$

  (a) A grinding wheel 0.35 m in diameter ny:hc7n 2dn-k6z: oie rotates at 2500 rpm. Calculate its angular velocity:d:-7 6nny o ein2khzc 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 complete revolution in 4.0 s (Fig. 8– o6n2m(kp.k0 sym ;o(lgk; tni38). (a) What is the linear speed of a child seated 1.2 on6o( lkky.0;p(k; sm igtn2mm 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 j)ljjd908b jlno* g+jsvf3*e Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of;wg27glkorjb6m0c vea ( )i7i 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) mub )h4bq9xim i/st a centrifuge rotate if a particle 7.0 cm from the axis of robim4 )q9xh i/btation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlxe m9if :dfefrw1o e4* .s1i g/3f8rypy about its center from 130 rpm to 280 rpdf:*f/gifoe1fm18r iw3 4 s. erye9pxm 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 radiusra 4) ix/+mbzd $R_1$ 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 themsel pq-k;m 1-c 7l/a,tjs0pl6fum hlh.zn ves into a slow rotation to distribute the Sun’s energy evenlpsfnch la-1.k6p 7l tlmhuj/z;0-m,qy. 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. 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 tf9x2wfz re q3r q*ox93o 15,000 rpm in 220 s. Through how many revolutions did it tqfr x293r wf*q9 xe3ozurn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 9 q8vcq;-uojiyrn5 +-zt : rgaudo,,m lg31ay2.5 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 hk(mv t ,q,kp /5x6k6mgsvbpa) ighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throug,)q(kk6b pgx ms,/kpv vt 5a6mh 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fr 6 gx.kgnw1jll n(2h1eom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tim(6n1l2jegg.nwk h1 xle?    m

  A cooling fan is turned off when it is running at 850rev/min It turnsz7,p+0ptr-t-/ushuchsl yfck ,z 4g 7 1500 revolutions beforfts7 /,0ucgp csz +p hlh-,t4- ky7uzre 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 r ew9 h+tjglg87educes its speed uniformly from 95km/lj+9g geh8 7twh to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as th jo lh.i3::afve car reduces its speed uniformly from 95km/h t jl:: a.oihf3vo 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 hb65qh8gjcx 95 hpdd3fer weight on each pedal when climbing a hilq 85j x g65ch3pfdhb9dl. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm widzu15,grw 0dsd e. What is the magnitude o1gzus50r wd,d f the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of) wbi7ke,ajp.l /tu.k the wheel shown in Fig. 8–39. Assume that 7epkbaut .iwkj)/ l,. 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 a0++edyz0td1(unnq gu ttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initun t y(d0+u0negzdq+1 ially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torquew r*7x 54loeil of 38 leiw*xlr57 4o $m \cdot N$ 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 at d(hws cp s g9dqty..e;rs:) 6zr;-al 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centz;.ar9)p ttesrqd-: lwh(6csgs. ;yd er.    $kg \cdot m^2$

  Calculate the moment of inertirl.k6q d;dx+ea of a bicycle wheel 66.7 cm in diameter. The rim and tire have a com6e+ .xldkq;d rbined mass 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 rotated in a ff*u-6iol 68wadm6zw n*ob hvqg., 2ohorizontal circle of radius 1.2 m. Calcula*.oif-d2w * 6fvnuamw qhgzl o6ob 8,6te
(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 ftg/t16. lm89w.uijz f+jq2ray-f s u potter’s wheel rotating at constant angular speed (Fig. 8–42). The frictf2 t 6w.z8fugf -i9l1y. aqr+u/jtsmjion 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 point objects shown in Fig. 8–t: c,-r7 nku rnlz :3f /pyhrwcjx,br33l h+-m43,/clb-, k3zp jn3 r: yhr7lt3rr +cm-w:hfnxu 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 oxygen atoms whose total mass iwds8ju2q2i 4. p 9kkq 1mbp *fyjpq0l2s $5.3 \times10^{ -26}$ 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 corrufj9r(q2xl52adf1i e o x 57lm*wbl k5ect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each roe2 27lib5m 59k*dl 5farfwx(lojqxu1 cket 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 cylindex j0)qxn 0+54o y/riy 8iwhi*kdabd2w r with a radius of 8.50 cm and a mass of 0.580 kg. C*a q)wiidjd 05nbo2wxy /04 rki+y8 xhalculate
(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, accel*022a/hyxe(ot xsn 5b9 ar69jru wfcverating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is able zcvh*:f)tqh 2 to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torqueq:t* hfv)z2h c. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut ofr:/ p l:eswu:dly,g3q f and is eventually brought uniformly to rest by a fq l yps::ewu3,l g/d:rrictional 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-kyfz dq: ni3x.7t7pm8zz*wr 8p4k upfz6 -al u-g 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 ball5;2sxjkhh1(p u1hc : hfk l:ki is thrown solely by the action of the forearm, which rotates about p1shkhfk;kulhxc :2 ij5h1(: 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 b6/ixf6y bc9 vyyeat2vm30,hhn +x.p zlade can be considered a long thin rod, as shown in Fig. 8–4h+byp9zyvt3an 2iv/,ef06m. cxxy 6h6.
(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 consi )ku7rng wk-dzdty) /0sts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest withiub4ko lw. )fq:/ku(qwn four full turns (revolutions) and u/kqu .k:wbofq4lw)( releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inx5x5qi1 +ntyj ertia 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 94y kvma,5anxb2tv9dtqh. (r 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 slippi -h:h5a4ngw ,vxdo i8ing down a lane at 3. 8n: 5i4i,goxvha-dhw3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of ut)(8 vlmxz r*e7qg:pthe Earth with respect to the Sun as the sum of twz 7x8mtu re:lq(v)p *go terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. Hohaz +; 4pv1aljz/8sbtw much net work is requias +th alj1/;vbzz4p8red to 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 ghf oa 7tn.7;vfrom rest and rolls without slipping down a 30.0 a gthv. 7of;7n$^{\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 balan9x7nnjnj f,s(ced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservationj(nn7 n 9f,sjx of energy.]    $m/s$

  What is the angular momentum of a 0 + pye*z gp-ciqs9(3hg.210-kg ball rotating on the end of a thin string in a cishzy +3c*iegpqpg- 9 (rcle 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 uniform cylindrical grindf j:.ili8b 9nb-eu1p) 4ci phping wheel of radius 18 cm when rotating at 1500 rpm? hb1unpjpie l.:i8-c )9bi fp4    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, onz6.+e ht86 ikzn/a zbn a platform that is rotating at a rate of 1.3rev/s If h+z zbh ze.nn/i6t8k6 ae raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) cajre,pip/ 9 )h-i ;ky *rubp7iln reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck pp e7)h by9 * kiui,pr;-pjril/osition. 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 heln rw 3n3g0;o*idkj.tr spin rotation rate from an initial rate of 1.0 rev every ;n 0t3r g* o3wnjkd.li2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a ve myxorepdo3 3pd /*f2;rtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of 3/ pe*fpr o;dd 23xyommass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at h czl;)svow(; qv7ef8 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 mhal(na e/2 bc,4bd 4cx;b 7levomentum 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 disk0pnp*f9, hzn z of moment of inertia I is dropped onto an identical dipp9nn* , zhf0zsk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at x jmp9hljl j+bagrod x/, +0/92.4 $rev/s$ 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 t lsra7c5/sx *zlu fs.4he center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 4f/s s*l s l.z7uarx5c$kg \cdot m^2$ 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 rotatib5;:pba zq f2tng freely with an angular velocity of 0.8 :bz5 ab2qft;p$rad/s$ 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 4)lfdnr: 2w j+gvpvj.ually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nejln 4 +. vdp:)w2jrvfgarest 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 excgkg oaw9gy2;e8- jxj)ess 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 ) w y6lzmz7)tz$ 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, thilo5du0s-r1.wl0;ya w gpbv v: a- oe3at can rotate freely without friction. The moment of inertia of the person plus the-5wau0og0vei abyl v;w3so -.l1:pdr 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 0hxdl/ k .kph5at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearin5hdhk /plk .0xgs 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 thea (a:e-h 5 kjr pg 7341sb0geujttxnv/ 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 pejg4 g5pes7th 3ju -axanke: 0v(1t/rbrson without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. D;td4dd wr osvx-5v gu7;*m0 tzetermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital a ;otw 7;drdvg vuz-mtd04*xs5 ngular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest a cya r()s,yxf/t 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 radi j0tgwa0qr1ahubt2: a8 3 /hpjus 0.20 m acquires a rotqtj 2:u380ph1awag0 ha/ rbjtational 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 twk gypt69 .pl*i9 som5nv/ica,4gg w c(o 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 the end of its 1. 5tcwo(p kvl/i*6an94gi cg,. psg9my0-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 ang)j+9k: s rwhvka( z:oeular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, wey,hs48 t/b wiare 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 is a uniform sphere at all times, and thats4th8 ,ya/i bw the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tp .9gh l5n -m+ug6lbhwo use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform c+9 l.h-hbgu65pmlw g nylindrical 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 ankazb-g6- fz / jrhv7 yq(2si1l $H_2O$ 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 fkt53)kjpj 4vy eu4* xa 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 free2u4i(ba f/z6cz/n z: cvpcbd8 .uov(yly (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. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.4 c abv y zzd((.2uo: /zibn/86vfcucp]

A wheel of mass M has radius R. It is standing,:svmgn-ip .y vertically on the floor, and we want to exert a horizontal force F at its axle i.vs:,n-m y pgso that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelinij.-i) s+ /hzr q tb1j.ig/hdsb s3w7xg with speed v=4.2m/s on a flat road is making a turn with a radius The fo/s7t/x1h irg- jihzsbdj).bq.s+3 w i rces 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 hcwty zovj gz9v7 ,xemz-5 z r/c:75g50.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque com5c z5 7g9c7 r zgv -:zmzowt,/jhxyv5ee from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mauh 8.gt7nu5; dk)rcbj king reasonable estimates for the dimensions. The5) ;un.dr87ub g kcjthn 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 cylindrical platsp( ;m*if9o (/sme wcles, of maome c9pm*fl( s( sw/i;ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–58. 5x8xaqwun i*)k62fnd 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 without leavingf a*xw2 nud q5xk)n86i the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume5wxrs byp v10l gbxm7 0),-dyl $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly f()o wo5o hxc+7co xqm 66ffxg9rom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whathm 7xqf566o oxf9 (wcog ocx)+ was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s