A bicycle odometer (which measures distance travepb+dpwhp9ql :g1ez5 7 led) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use 9+ e z5lqp1w:bhp7pg dit on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on t) x4ce:d; aqffhe rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial aq :4d)xfacfe; nd/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sixdl3 b+ghx u.3b-7zadngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explainunyv g jy;o:z+ 1had.,.

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; v::klj0f5 ,;lxjm4 zxjc mki a0unt4dafb8: your head than when your arms are strettjjfi f lla:kd;jv0 n0z4k:a:5x bxm ,8c;u4mched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets atu k3g; u(m-aui the rear wheel and three at the pedal cranks. In which gear 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 s-gik a3(uumu ;procket? Why?

Mammals that depend on being able to run fash ksf,9lnq5.b1xl.eh t have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the bl.nkbhs l q,.519xefh asis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lowf+m+g) pjn;rn9c 8hplbu(u3ng, narrow beam?

If the net force on a s pnm5a0nv66w)/yqn p3j kl0liystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net f)i nn6/5j va6 0p0wyq3llmnpkorce zero?

Two inclines have the same height but make different angles with the hor ( 5lbbwl4q4me-fj:hxizontal. The same steel ball is rolled down each incline. On which incline will the speem4bhqx4l( fe: wbj l5-d of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inc qs/d c5*.,::q pr6yzrezfkjhline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom y.:q:jrsq6r dcz,5f/hpzke* 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 radi 0qkrp8b v4dv8us 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 total kinetic energy at the bottom? Which has the greater rotationalqvd4k8 bv8p0r KE?

We claim that momentum and angular momentum are conserved. Yet most moving ,4se;4ujscf;3 k0jp s0rajla/g3oz eor rotating obj es;jzlcjj 3spa;k30 r04eo4uf,/sga ects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, holchlq8fsi3 : t)- d+nhw would this affect tfsd )8nh3l-qhl:+ cithe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w-a:bnm zv y6b+hen shyb-am 6 bv+nz:e leaves the board?

The moment of inertia kc;;bz+ bjz :f6zexj a.u,,xuof a rotating solid disk about an axis through its center of mas zbfz c,+jkej6xau,z; u;:.bxs is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretddb8,g6p b+i/ z)q loiched hand. If you suddenly drop the masses, wbdi oi6p)d lq8g,b/z+ ill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have t5ybr c x,op+ -f4tlpz;t;lp)k he same mass. However, one is hollow and the other io xc,+-yl4f5 ;rptp;b zl pk)ts solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Iny-wxz6 :nfq-no wcv7+ 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 whe: xcyfzo-+6n wv -nwq7el. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rmsdf*kcpusr1l;o pnyf+ e)8m;h*+ / potating turntable. What happens if you walk toward the cf*no+y1dm hclk/*spmfseup;rp ) + 8;enter?

A shortstop may leap into the air to catch a ball and throw itx m0mo;uwg3s ,-lf)pw quickly. As he throws the ball, the upper part of hiufox3-0l mg) ws,w;mps 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 angul dc9m i)rtr:a k 5kr5*heq6;uyar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can 6 ;icuqe yk9a:)r 5dtm*kh r5roperate to keep the helicopter stable.

  Express the following angles in radians: (qme8 c 7*u;pufa) 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 coincidencqwk:m 4m 3*pmre. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the 3qm :kwr4p*m mMoon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamr.ao 0)6.we7ytxynml diverges at an angleo0ym6y aw.tl7)n. xe r $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blend/k jad qe2h:3der rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades sl:3djhea / dq2kow 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:*9hsllg us( 4 ash6ki floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diametel 9 u6sskalh(isg:*h4 r?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionsd9cixb:d 5)sg do the wheels d59ic:)gbdx s make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rocg.z,:,te /gn2jamwv aq/w(ttates at 2500 rpm. Calculate its angul n/gtz/c, .:aaevw m(wq,gtj2ar 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 makes one complete revolution in b/)i,ub rrg0r v9ney14.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fren),1b9vuirrr y0g/b om 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 c e m4bo*,uwc6w0jk7 sits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of6sv e-4 h7jx:wghbxc2lp8o -z a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a cent+s 0)lzmxc.agja +;sdrifuge rotate if a particle 7.0 cm from the axis of rotation is to experie. xdjsa sgzmcla +;+)0nce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a,bpo: /x2ctb ilu 4m16a6 ffsvbout its center from 130 rpm to 280 rpm in 4.0 s. Determi vu1f/b6,p: sx4fblto2a m ic6ne
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu-t :wgzh+:h )y/ 6ep b7kn64alobt cvts $R_1$ 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 spacecraff(7 ku.+rxfift put themselves into a slow rotation to distribute theur (ifxfk7 .+f 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. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 hc-gx6py+ rj 0( lqr68pnj; dcrpm in 220 s. Through how many revoy8xq (j0 g j+nc6hpdr-;p 6lcrlutions did it turn in this time?    $rev$

  An automobile engine slows down fr+nic2e gh ;tnv +*vj3 vkgs.*wom 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 in a whic8jg9di6)ii qrling “human centrifuge,” which takes 1.0 min to turn th i d8qgicj)i96rough 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 diam1cu29 ) hesqnp3ke h0yeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. 3ph q1 esuyech09n2) kHow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rl4obv7; t bij.2vwqpo 1vqc (3ev/min It turns 1500 revolutions before io(.ij w v;b vcqlp12v3q4tob7t 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 cjji 9k x2ru(/mar reduces its speed uniformly from 95km/h to 45km/h The tires h2( jm/jkx9 riuave 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 revoluv c kfo9-*j6dltions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a 9ovjd-f kc 6*ldiameter 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 a;j7r m,d0m:(rr*x*91ga megff yl*hd o ejvg0ll her weight on each pedal when climbing a hill. The pedals rotal gx y;,er:rj9f(g**orhfj7v* dmme1 dm 00g ate 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 toiej.dmf50o : f;o) nvhe end of a door 74 cm wide. What is the magnitudemof o;f.o 0n jd5)iev: of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fiull4g (udax,0b t 0sy-g. 8–39. Assume that a gbs-a0 yulu0(lxd4t , 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 endl2t rdshdb- *vmr):ymwyq+/.s of a massless rod which pivots as shown in Fig. 8–40. Initialq .d /+yr)rwhl-tbsv:mm2dy* ly 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 tighten n *3:7iw/xbdc q*jgmd fi:tto33 u2jqing to a torque of 38 iu3*qjt:o/gm2xnj7 d 3q*3cdb: itfw$m \cdot N$ 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 spher hsfhqdmrp/6. nm:eywm8) 5 (ae of radius 0.648 m when the axis of rotation is through its centerds (.n85w hh)mqfpme: y 6/mar.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The r z7dsec1kqvga93xw 9hh5br9y u0 v8v.r /1lhzim and tire have sqhg1/5 du09vkrz8b lv1 a h9wr v.e9h37zc yxa combined 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 rotatyei24n60w a byed in a horizontal circle of radi0yenb4a 26iwy us 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 v/ o4b. 8rff/r gcy guify;7)ca potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her bvf.8rcgr)4fgc o; fy /7iy/ uhands 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 then; x5pny(za zc6j57eu array of point objects shown in Fig. 8–43 aboc75u;6pzez5jay x(n nut
(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 mastqfsv )wy u4,qm/q o,3s is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corrk-ar;3duty1 e ect rate, engineers fire four tangential rockets as sho; 3tr y1u-aekdwn in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinders xzaj 9t(+iop 4wl aj*,oc5,l9c1fyj with a radius of 8.50 cm and a mass of 0.580 kg. Calculac91zjof li5j4wlsyjac,+o ,* a(x9p tte
(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 frd gj:et;f6 -n(+rrrfcb+kh 32ms *y omom 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-dri16;w c+4vcqxhnjk ed 2ven merry-go-round and is able 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, neglk 1xc;q cje2dhn 4w+6vecting 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 brluk+ 3y kjoy5r.0ovu 0ought 0yry.+35 o vjlku0kuouniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7anh k 77 vqfeh9uw79t dk33/g.hxpu c3 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of u 4 xtlsymq)-t5mm8g2the forearm, which rotates about the elbow joint under the action of u8mx)m 452q stgtmy-lthe 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 blad1nvyina0 29bsaa t*.l e can be considered a long thin rod, as shown in Fig. 8–46. ni 0bsvn29ay*1 a .atl
(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 consa e4eha0 02ldpk3/*rzj6aes g ists 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 within four full turns (repked y-a:ayf7 3ueivn0 k xl);vmy)37volutions) and releases itpyi ldm y03v -f3)e uxn ka)a:e7;vk7y 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 moment3(hsbnn.co0mc o4 ;, e5maid(w f,abq 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 toroold6 +a1fcy*w/ a)8ir +y xqwque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a v .j8oxsy2.d ilane at2jxydo i.v.8 s 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofhx*h o4c6ku ;kh 2lyh3 the Earth with respect to the Sun as the sum of two tehc3 *luk46h2h;y hk xorms,
(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 radit j,urfm dmz. 8 lp/rugz(*k+9us of 7.50 m. How much net work is required to accelerate it from rest to a r8jg mm( 9kudf rr.t*z pul+/z,otation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls0 /-()fmu .+8oj ekc cyyzoezp without slipping down a 30.0oeyo. jke /ypc0fz( 8+uczm)- $^{\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 vertn( 8scepl 8 lr8ik 0j*bvim7+vically on its tip. It starts to fall and its lower end doi7ve8 0 *j8cnks(mlprb+l v8ies not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tg sl0n rt;r-05+jyne3mx7s hyhe end of a thin string it s-eyrr0 7j+x0m nlsn5g3y;hn a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grle(zaqh:1k0:av.ji h inding wheel of radius 18 cm when rotating at hq1(e::akvih.aj l z01500 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 thn/bk lunoqw1i*pyc5j 2jnsl+8; -s1f at is rotating at a rate of 1.3rev/s If fq8 1*s 1w-jocnun lijlk n +2p/syb;5he 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 Fj8 c9:h aka tb7f(cn5 s9.nfdlig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her tccb.7jfnkda8(l sha 9 n:9f5angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from2tgi-n(p *ynfw k1k-z pvt3s- an initial rate of 1.0 rev every 2.0 s to a final s g-twkf2k--npyi*v( pz3t n 1rate 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 rot, xn,rsau 5xi6*rmh s. c0jum.ating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is tirnxrs.0a, ,5ujhmxcs. *u6mhe frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure ga ys*fm6fzy69wl*/ jskater 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 momentum of qv0 ;b+bynkwg)0.ya0ttxko1 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 onto an ip7i4pp/7.gpg ir3j + om ajhl.dentical dig3.pmih gl .aprp7jo7p /4+ij sk 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 2., 5kxn45yf / otukaa7e4 $rev/s$ 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 standmd/ hny./t:pihhv .6 fs at the center of a rotating merry-go-round platfortd ./yhi:f v.h6m/hpn m of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0hi g4kux v4k7hb, 8 ztkf:g.s5.8t:fgzkux8b45., vih g4kkhs 7 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, l5jpjumvm xl.l)y x /6;osing 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 nem l5 x.jmp;vjx)6luy/ w 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 windzbi: yw ia7ud)p:3 eh* 4zik1us in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 7ki*l+z,r lyc $ 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 z 3zxzaz)d0rb -g(2pnrest, that can rotate freely without friction. The moment of inertia of the person plus trzn x d2-)z(p3abzg0z he 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 personbzilfl,w6;24 b+xid v stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings ,l4b2;lwi+fvzi x6 bdand 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 zl(gu a5ngzv ,9em:;mon the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without sl (zvgn;ealg5mmzu, 9:ipping. 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 sidem.-bh cg21n zr always faces the Earth. Determine the ratio of the Moon’s spihg-2.n 1zb crmn angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate nlnlsa, (vgj)nex * 4-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 rad/ooev f9)8u d(zjx zc3ius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the)e3zoxjv/ (fu oc9zd8 torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylinh b4.gh4qhpo k7lt(4f7hy d o4drical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hpl(4fh7toh b4qk .oydhg 44 h7ub 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 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angula8 ,)e8ql(jg psu*zmpmyoj6pffi4h/ /r 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, were rotaae1y+q5r :t:i4 hlwhqting at a speed of 1.0 revolution every 12 days. If it were to undergo gravit4a:5tw e: yhriq1+ hqlational 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 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 vn c)(lq/o 5mwenergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diametewc/5mov) q (nlr 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 illustratesj,yo 4jdi)8 yn 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 rolling on a hori)1e ici 3zsxva2h0 6pbzontal 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 frictionh2m5e ym: t*fpovj h o(er:c:8) 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: Se:rh:vc:h 8f5p ejmeyt o*mo 2(e Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the cxo2xuk3kd ; .floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 83c dkox;x2ku .–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flatzwjr f va;g4xa3nu..9b2;lj 4a sluu8 road is making a turn with a radius The foaj44rfu; lb;aun2ux8.9jswag v3zl. 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 0.50-kg rock into a sling of l0a4t daa /nm fv,ry16eength 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, an41n/aa r,e0df6y mvta d where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid9ry5 te,xq .vl cylinder and her arms as thin rods, making rea9, qe tvy.rxl5sonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch asskm6)jb*yr fcei- 9*gs6 y5zhi embly which consists of two cylindrical plates, of maz*5i g)-rh96b cymsky jfe 6*iss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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:y qhf )2 krhdg81od*f 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 highestf r*h dfoh2 :gkyd8q)1 point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notug1:o: 1iugvu assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the carem4tn v5t-h 0dhl+n zd9me81 r reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? 5n-h1meezt9m 8 n ld4tv+d0hr $\approx$    $rad/s^2$(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