A bicycle odometer (which measures distance traveled) is attached neeslv5t4tzj (r 8wr4vxkm sct 4d792z/ar the wheel hub and is xjls24wetr(/r8v4d4 mz zvk 9 7sc5ttdesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rimydavrs 25x1 nz 0(2khm have radial and/or tangentdzvs02 ay h xm521krn(ial acceleration? If the disk’s angular velocity increases uniformly, does the point 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 angdfiq42 q:tr; h;g f-rwular velocity $\omega$ Explain.

Can a small force ever exert a greater tojfjb-- c oz4x)rque 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 head th c1zf9h(5fj4axg/ x5os6r gjw an when your rg /j(6x5j1f oxgwz a54s 9chfarms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle ha7c)gage,lt bs c-0qu8s seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large g8t-7e ccq lasu)g0,b 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 withjrc nk*u3q8x b: *c6dz flesh and muscle concentrated high, close tocx n8* *rqbjkd:cz6u3 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 long, narrow beam?vfg m.uc0 6lp ;w6icn(

If the net force on a system is zero, is tk0 osvr n:l.xy/ b)a+ohe net torque also zero? If the net torque on a system is 0 s:no.l)/var ob yx+kzero, is the net force zero?

Two inclines have the same height but make different j9ii0 u ep(a9sangles with the horizontal. The same steel ball is rolled down each incline. On which incline will the spee9u(ii esa90 pjd of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouslw;z ) 5mz tzc10y 9yjzc7rtjy0y start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has yyzc7901crw zttjz zj)5; ym 0the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start from 2 (oiy7a3ri 1yuppl;arest at the top of an incline. Which reaches the bottom first? Which ia;pyiu(yl12 a3 op7r 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 angulr:.cnelc )j j*ar momentum are conserved. Yet most moving or rotating objects eventually slow down andc cjljre). n:* stop. Explain.

If there were a great migration of people toward the E.el7, dxzv khh7q0k4-nx : nroj9b7y rarth’s equator, how would this afnjkb-zr 70v.49dlho,q x:yx7kne7h r fect the length of the day?

Can the diver of Fig. 8–29 do a m*;wuzfygpwy6r 0p6 -somersault without having any initial rotation when she leaves the boar zug wy6rp6p*;0wm-y fd?

The moment of inertia of a rotgdvz 5fsnq 7, jp8m4o4ating solid disk about an axis through its center of mass is 58q7v zd4s 4pmogjfn,$\frac{1}{2}WR^2$ (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 hol bg-i3y3 xz; nzhr)9yyding a 2-kg mass in each outstretched hand. If b9 -hiyn;y 3gr3yxzz )you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holyc. wxr(zbbtyd 4z ;49low and the other is solid. Describe an experiment t.tz(z yw9;44y bcrbd xo determine which is which.

The angular velocity of /zz+xa2w4b5m dvs5 efa wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a xs z da5/fw2mez4b5+vpoint 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. What tvb nc:u)c,h+happens if you walk toward th c:+uvt,nc )bhe center?

A shortstop may leap into h h6;ujv2hvn :0r1w kothe air to catch a ball and throw it quickly. As he throws the bj1krowh ;06 v nvhu:h2all, 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 scoeu142ra9z/2 nofb conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second prop 2or fcso9eua2zn1/ b4eller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)ztg )5t d.ll 0k6gdfw:l8;lp o 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 coincidenc zq9a+tzx4 /bcu *p;grmq 55uve. Calculate, using the information inside the Front Cover, the angular;9 5qmx az 45zu+pc*gqrbu vt/ diameters (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 Eartp.ky d/n/4qnpm z;y,uh. The beam diverges at an 4qz,/y pypu /;d .knnmangle $\theta$ (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 rpm. W6r*uyh jme zy 46kx72q2n x(ehhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slxyeum 2z 76 krxeyq4*2nh(jh6ow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the balee5ooa oc 3,n9l makes 15.0 revolutions, what is it9 5ao3e,oncoe s diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 k8a9*,cifk,ub 2v0pp2ag2la hn g-e b km. How many revolutions do t0l2 -u9ka vfh*icb8p2,a b,g2 nagk pehe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calto7 ts)o5vhd9culate its angulav sdho7 5o)t9tr 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-roundp+zqr.dx2g - r4*i5we ct 2mcr makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chilrmx r*di5- 4tw2e+g .pcz rc2qd 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 itwsrgbxtny zq h-1z(8* hj)e;7 s orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ak3nkgzf:.ob l )eb) s: point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 4cpq84apyzg(dg 6 j (fcm from the axis of rotation is to experience an ac d(pfqgya8 z4 g4pc6j(celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fr28 4fq. krsi-p ioj3 i,tmde3com 130 rpm to 280 r8e qpsc.i, jo i 2-mif43krdt3pm 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 radiuso65tcw7x al3l 8x0 g); tcilvf $R_1$ 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 aboa7)ukwzcr+)2 y5 rel6elqoia/rd the 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 minutele/+ ar2 e c5yrl i6ukzqw))o7 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 uniform neqy :2m/x44ysyms qk /.hmvn91zkp1 gw*-z mly from rest to 15,000 rpm in 220 s. Through how m-k2qwmn4 nes/4mz p 1y/sv 1 hmz.xy*qgk:9ymany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpmgr vhvrvcir/ 5jbz8h):6 +y8 e 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 r.qj snb3;.g b7 izts4npk8y0 highspeed jets in a whirling “human centrifuge,” which takes 1.0 min tonpr g74;zbi. n s skjb83q.y0t 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 diameter accelerates uniformly from 240 rpm to 360 rpm in 6.7vrqz : em2;rb/1w st3zd6im m5 s. How far will a point on the edge of the wheel have traveled in thiv qzs dzmmb2 wme:t31/;rr6 i7s time?    m

  A cooling fan is turned off wf2mo4o5oybv/ aq2y2 s hen it is running at 850rev/min It turns 1500 revolutions before it comes toy2oo q 4f/mav22syob5 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 itse (bt:7wc 8bziok7 -wd speed uniformly from z- 7ecbtd8w (io7wbk :95km/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

  The tires of a car make 65 revolutions as tw..jhpc3z g)/b7 xr aihe car reduces its speed uniformly from 95km/h to 45km/h The tires have a diametbg3hx).wpza ic/7jr . er of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when ;q2 9pz/2kw9zuzvmuw climbing a hill. The pedals rotatvzpkzqm9u;/ 2w9u2w ze 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 b u5 xu rr5-t4/styismk e *i*9wpv)/f55 N on the end of a door 74 cm wide. What is the magnitude of the t/emi*rbu5 vp r95 kui4t *f ss-/ytwx)orque 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 shownjn7u 4zobf*i,g lp 4q1 in Fig. 8–39. Assume thati4 1bu4nfjpgqzl ,*7o a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod lat e;y38g )dmfx q*c,which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posi)ax8y ,3eq mclfgdt *;tion 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 torquerk. ;f bn7)imv of 38 b)7; fk mrni.v$m \cdot N$ 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 spherz3c+u i4nils +d c)2bte of radius 0.648 m when the axis of rotation isblnci+)u4s 2 +z 3cidt through its center.    $kg \cdot m^2$

  Calculate the moment of inertia5z21p ew7 dels of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub ca7d25 1epeswzln be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the eyn,q s 3bmp 5)xz- 7rggse3+hznd of a thin, light rod is rotated in a horizontal +sbznmgr-3ezqh)s, y75 3pxg circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl ol5q:2z :dxzqmo p/- ejn a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands anjm zdo:-/e 2x:pq5zlqd 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 8+ryr86 4 avedxv/ zuoFig. 8–43 a+x86o rue 8/vzy advr4bout
(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 massvde l)c/:2wszztu.v9 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 a( d,svea4 5vjwt the correct rate, engineers fire four tangential ro(4dave w jv5,sckets 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 a unifori2hx e8/q+y9m v9 vctgy-u ml:m cylinder with a radius of 8.50 cm and a mass of 0.580 - l gv+q29emyu/ 8vxic9yhmt:kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accedqg(:udud u 7v.e:5d tlerating 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 xjh6g ulab tk6j 4vn)0x;iy /8small hand-driven 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, neglecting frictional torque.648jvxlbja k n)h g/0uxi6; yt $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 *2fdwlfq+n1n: gwwv*7au 1jtrpm is shut off and is eventually brought uniformly to rest by a frictional 2wvjg1fa *+l :t dw fq*w1nu7ntorque 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 acc3d 8qfcdolk.g *iy0(ucu q4j)elerates 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 ball is thrown solely by the acu 8vpv3d 29 ktyr4l/-wixkz l;tion of the forearm, which rotates about the elbow joint under the action of thel4pzvk;rwkt-yi3 92v 8u /ldx triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a lon/lnnl )5a ljk+e n-a;fg thin rod, as shown in Fig. 8–46-)jlfknnl+/ae l 5 an;.
(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 coj0at -j1 hae19upkff8 nsists 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 fc2r 9vu)fod: srom rest within four full turns (revolutions) and releases it at a speed of 29osfcdv 2:)ru 8 $m/s$ 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 opm70-mz2s qc 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 deh,w *j.we3vbvmz ):wyvelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down aw nwzy1uo1oj9u/ ; js8 lanewounuws;1o1/jj y9z 8 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energyle2 4;fv)1c.ch(uhim mj emat;:5bqd of the Earth with respect to the Sun as the sum of two termbcmt q)m; 5.fde(h;c:liuejhv2a1 4 ms,
(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 opdwjsw m+0fts;3x7:*ia cieobi u- .,a)qp6and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00i0;) .:3pjmatuwcf e-woq+* 6o id7,spasibx s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wihir v clg2vl- 2t nuhs90tv+,.thout slipping down a 3vvulv20l gnhsr2 9,ch.t+t i- 0.0 $^{\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 8gk:w jyrik*adh;gwxo(mb 6f0. jv( 6vertically on its tip. It starts to fall and its lower end does not slip. What wil ixkym.6 j(kv :w0;wj ghg*abr6 f(o8dl be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a t cr(; 0lhdg1wn4wer g/ z;u)lxhin string iwxgg(r/ 04h ncu;)1; d zlrelwn 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 6yrpcebs4 zw. 7y2:b 4xdjj(c:n 3r+-ncizvh cylindrical grinding wheel of radius 18 cm when rota+sib6 bc r:zjyw.3pv4(42 x-nezc ndhc: y7rjting at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a sg2z*xy 09m f5 rjlaj.platform that is rotating at a rate of 1.3rev/s If he rairgj*.2yljx 059m faszses 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 gk v) fapd2abl /(ocl 545x1 og7 pjh34ce;njbshown in Fig. 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 rotat1;/vcl 3lkxng)5pa5fg(7o4 4d beo c2jjap hbions 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 inem) bk715tdxz crease her spin rotation rate from an initial rate of 1.0 rev evetebm7 1)x5d kzry 2.0 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 vertical axis through its center ;8 9soplct,crat 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 dis8 9,ptcrls co;k 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 atph:y(k5kg2 re-8 h mws 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 angularad( mrz nt7l )mnty6vl*:m.l6 momentum 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 9l37x c4irm0gq*x i m(cd cf.sdropped onto an identical disk roi*x(i 3 cm9.xsqc d7cgrmfl 04tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a*0 vvn.4+ hhazdplhqlc / q kee.lk+.kpg4k8/t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 6 s :4n;r-o*fdn)cy yp xjykw,kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 92)fy4,n6jdcon* px:;r -swyyk0 $kg \cdot m^2$ 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-rosc;1 xtlpi05gund is rotating freely with an angular velocity of 0.8g01;l5pti xs c $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its mqeg hs-r:3+r6dx ;jgaass in the process, and winding up with a radius 1.0% of its existing radius. Assuming hrgjsgx;+e3r: d- qa6the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 8ngtxg5 5egv e32p-:jhg*hrx120 $km/h$ 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 8fbgkgi5 hs ,vm8r4)7; ebads$ 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 glza:d4u0e uunf h8x2:x: bu8at rest, that can rotate freely without friction. The moment of inertia of the perso u:lhfz0xuugn884 u::dba2exn 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 stanks2f*)qnn+ct ;2d vm wds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless beas +w)ctn2kn *m fv2qd;rings 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 grej j)l :w/uxs8ound with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of ro ljuwxj/se 8:)pe unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thatn yb;:o;dt68eoqev 7cg n9g2 e the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a p7vggqey;o 96ot;dn82c:ebe narticle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/;o+aobs6/1. db b kchp;zxt -nx m(wv.$s^2$ 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 rar+xnz,j j ;jsh43q+qmdius 0.20 m acquires a rotational rate of from rest over a 6n4+, q3xszrjq ;+ hmjj.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made 0*cxf*v 0:mf qi mq5fkcz(9qnof two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it *v0c(mffc* q: zf90i qmk5qxn is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed o 4nty- v+b*loef 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 tz;w9 pjt5w4vn 8kp 5drimes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gra 5zt;wdwj58v 4 p9nkprvitational 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 energy storezms*v5p: kadlh o0,;vd in akzh odl5v :,s0 ;a*vmp heavy rotating flywheel. 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 illustrates an+ f()ohy /c,v)n2epezc5p hly $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 9o t5r,hvmq (xva*0uu$m/s$ 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 l/nc 6w6,onttrp-g)rv r0jp*.i p5 qnnength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine o ,trj 6*p- n5t)/np6rrncg.iq n vwp0(a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we 216mt2b smvus;spn9 v want to exert a horizontal forcnsmssv ; m91tb 22v6pue F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/z j,km9k 5arz:sb//b ds on a flat road is making a turn with a9 db:/szj/k ,zbkam5r radius The forces 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 length 1.5 m 9zo k 454pm9f01 je,klo achnwand 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, andj45oc,ako plw z9 e4m0fkn1h9 where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods,)xz 77jj*qgjpz 3b o(v making reasonable estima*xqj7oj( vp)j7 z 3zgbtes 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 assembly which consists of two cylindr(hhl207/xr dtumdg 5jx f(su3 ical plates, of2t7hfd5/du0 (ms lhux(g3jrx mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 andy(m rj(:tcw 8d radius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without:j w(y(m8 ctrd leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a 9c-hl f201ko drkhw3assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car red.bc:vj ,3z c1iw,advgq+e nx 3uces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wdb nv3giwx ,:z .v3ce,cj+aq1hat 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