A bicycle odometer (whic:, rmh/r7kp uaewu t6;h measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What h6tpruh:7,m/ erwk; ua appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on tka./j* ahdaf 8he rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For whi8a kd .f/h*ajach cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value o6:jlq-qo -qik f the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explainxg3:m4d 5mh;:xlf 2cssqe .vr.

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 sit7 yuufa pqg8gd*w 6,,f-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to afa*6pgfg u7d, q yuw8,nswer this.

A 21-speed bicycle has seven sprockets at the rear wheeyb,5jazo k jrx)/ 4z8jl 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 l)zk,z 4o ax5jr/bj y8jarge front sprocket? Why?

Mammals that depend on being able to run fast haveso3to ):5bkl*z 1nxb sn: ycj/ slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain whocz1:nsnj bsy)/ox3k : 5 lt*by this distribution of mass is advantageous.

Why do tightrope walklm6v:v ghw+6ywyfcm5 a 7r09 wers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? p)lmhfy1wip l3r rh00i /vr-+ If the net torque on a syst- m rwhly 3r0rli v0php+if)1/em is zero, is the net force zero?

Two inclines have the same height but make diffa6m6g u lc6y .hkrgv 43ekb*.jzr9h 6ferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottk6b jg.r*u9 hyre.z6v6 l 64k3fahcgm om be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)s)1 ue t0;w37lmbk odn down an incline. One sphere has twice the radius and3m; 0wt)1n7e uo sblkd twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius an-5 5:e 16jcg4.oro+vm2 refbzt ujw hjd 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 5o4wbe 1:uc e2j5jmog .h f6jtv+z-rr the greater rotational KE?

We claim that momentum and angular momentum are conse:p m pyubhd,j7;m,sq.c/bth 8lf ;e3 arved. Yet most moving or rotating objects eventually ssa.pmt8 m q,fhb;7cl hjbued, 3p;y/:low down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wor/csh 3x: bg0huld this affect the length 0b rs/cghxh3:of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation *:;g o7ij cgwhn8i36dk0 r c /7kpvqhfwhe p6:f7 ;rg7kcqc 8 hwhgi*no0v/ kid3jn she leaves the board?

The moment of inertia of a rotating sof 8.b-8ow6eb.rzor g;*eche olid disk about an axis through its center ofo8 ;chwe.oefr8 *b-beg r.oz6 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass e-tvnvxkm.4 t9) rg4 gin each outstretched hand. If you suddenly drop the masst4gen.t9x )g4 k-vmrves, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holl4g0rfs 3) xnnpow and the other is solid. Describe an experiment to des)rfg3 pnx4 n0termine which is which.

The angular velocity of a wheel rotating o+ ej6i6yojfz.rc( o ,usn)(obn a horizontal axle points west. In what direction is the linear velocity of a point on the top obr6+ejif z onc,juos(.oy) ( 6f 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 3qbcy rcteki2.48 7mhlps8:d large freely rotating turntable. What ha:i8 yt l2c8.rpk7m hcsb43eqdppens if you walk toward the center?

A shortstop may leap into the air to catch axg 83e1ip0l vt ball and throw it quickly. As he throws the ball, the upper part of his body e8i3p1vxgt0 l 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 coi js u5q*f8em)nservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller).i5u8mejqfs) * Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles infqz/ (,rv3,fghs s egm wv6:.s radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using67:sgh i jnesja9q4( m the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eae j :ijs6q4 m7nah(s9grth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earthanqnw ez)9(k ). The beam diverges at an (ekaw )nq)n9z angle $\theta$ (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. When the motor is tx kwo7p((p. ,pea0-lc,ha iz 29znnkturned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? l0 (cie,w koaknn,xz- 2pt.9paph7 z(    $rad/s^2$

  A child rolls a ballh 4pxlc(vo:xa*4sv 3cpu*jbl 5lzd6: on a level floor 3.5 m to another child. If the ball makes 15.0 rsalou*4cpp*l(65l :bv4h djvzx:xc 3evolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter w0p . ao/zd b*/hw/qwztravels 8.0 km. How many revolutions do the who0z/w/a* zqbpw/ . hwdeels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcu8 1sydr89qwapa;b ,j ilate its angular velocitp;, yj89d8iaa1 qwbrsy 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 (Fi 7xj90xk w*njf.1wxdz g. 8–38). (a) What is the linear speed 07 wwnx.jxxjd *k19z fof a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in itss7q,j o(:jlt g orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speo0 *4ify8fra *i9j uuved of 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 centrifuge rotate if a particle 7.0 cm from the axismze e i2 jm d2n f:+-5wp,y.,qkepqbk8 of rotation is to experience an acceleration e ed 825yknz bmeq2 ,.-i,pjkwpqfm+:of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rgugd q/-87vn-o/b0ac :mh ahqpm to 280 vn g/:7uh bg-0 q/a h-mc8oqadrpm 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 radiu4l9rxy,tz cx/ylo* 4xs $R_1$ 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 Ap4chxn8 dli4 ;ynm61mdl1v tw: ollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time intervwxmt:;v4d8i46d1lny mn ch 1l al. 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 16t2n2lju3p- qp3 jva s5,000 rpm in 220 s. Through how many revoluu3nj japs lv2-26q t3ptions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1223oe,gwf d8g/f9jw lx7n l j3c00 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 ,jhm/e0v*fvq-dw gu+idxc -9 flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before dg /w,q+ ij09-vcud-*hf vmx ereaching 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 fra4qgr+tmgy*8kopn6 fg9 b -*nom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this timp* tfrgoan n8g+b*m69q ykg4-e?    m

  A cooling fan is turned off when it e6sni.htn.ha z u/6 r+zlax 3zh7ox*,is running at 850rev/min It turns 1500 revolutions zanz+,.6.t7r 3 e*h ah/ozxh 6 lsiunxbefore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speedqsvj+9l/bvrsv8m- k ) uniformly from 95k)/svskrjm+b-q 9vl8 vm/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 the car reduces its speed unihtzlu5b2/y5hmr qm r317y xn 0formly from 95km/h to 45km/h The tires have a zhy72lx h/ru513mqmnb0r5y tdiameter 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 eachu+hilrmm/4b 0; (kvud pedal when climbing a hill. ;m uidmh/+4bv lk(r0uThe 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 wide z:*j jfe:kx nsw70th:pqz77t. What is the magnitude *xn70tjw7fs:je zk7 :h qt:zpof 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 uuow 7i apd4og9z19p:in Fig. 8–39. Assume that a f99g:p uw1oou4zi7dapriction 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 whdhd;c.d kx1a3 ix 5;obich pivots as shown in Fig. 8–40. Initiallx;kcbad ho;1d3ix5d. y 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 tightenicvfh1f051 kss 82bnf 6yj5 npung to a torque of 38 v1 c6f 15n n02s5jukhfy8fbps $m \cdot N$ 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 inertqh)j(n,;hpsk ia of a 10.8-kg sphere of radius 0.648 m when the axis of rophhs(q;, kj)n tation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whesd,uc4 * 2,naefw)nilel 66.7 cm in diameter. The rim and tire have a con2 ,s*wena)idf4 lcu,mbined 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 rotat7ord7o ubf2go/ f*to 7n79g:thazt* ked in a horizontal circle of radius 1.2 m.oo*2h f*7d a7bof tz:u7g tnrtko/9g7 Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rot0 q6t g ) f(ohte4lhugbt+g1h6ating at constant angular speed (Fig. 8–42). The friction force between her hands and th60 4h1)hl( ge gfguot6tbq+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 rgt e 3i+2/esnrc2h/y of the array of point objects shown in Fig. 8–43 y t 2 rn/ierec/2g3hs+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 cons tr/7dwgp-a(ug3we+n ists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindric 1/g1hftnyxlw ;a1h/x2tyb *xal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required stef11 ht/2; t xgxx*yw/1la hbynady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder ) qt556xeaq qvibx+d5with a radius of 8.50 cm and a mass of 0.580 +qdbiae5 xv56xt)q q5 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 swi 4ocgvj+, b4mhngs a bat, accelerating 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-driv.ele)6 1 s8haygf)rhxen 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 mass1x erh8le6.hf) agsy) of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut ofw6qb0 sof;+3bmbx: ry pa( gh7f and is eventually brought unib xh:07q;arb 6bfp (+m3swgyoformly 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 baltmm+fqjzaug -0p3 d-i58,) kf6 qzluml 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 action of the ,*tg+euwzbd j(t+- iqforearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly+,dz-jqtw b+u et *(ig from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig;eph )7ny jc w-qk)mt-0+ x*uz-ycfsp. 8–46.wp+c -py*jc))x f7usyeq t k z;0m-nh-
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two mx)i( zzhbc)ke 6 (( rzpf7vo5hasses, $m_1$ 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 hi*8wpnz. kt8 fammer from rest within four full turns (revolutions) and releases it at a szi *n88.pwf ktpeed 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/vh,).pmjv 9aobvonk( 1b0 t x 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 devegiej) w q wxb57ev3m6*lops 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 d5j 7t ywj 3qfkz;:r./g ,ihsveown a lane at 3.3,7 3qy/:sk;vtj5zj hw fegir. $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Su30kcnfseapdt.( 5;jd4 ji;h yn as the sum of two t;4pk(5isdh ; 0t ade3fcjn .jyerms,
(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. How *xal iul179zwmj8m )s jr1xo. much net work is required to accelerate it from rest to a rota o91.wls zx mlija78u1)*jxmrtion rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and n6i 7.4ra vo-n-l*aht-h psg8j f:clvmass 1.80 kg starts from rest and rolls without slipping down a 30.0 nj--h4cra8o. ns tf*hpl- ga7vl6 :vi$^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its kw2s; fsy)*p 3z5ruzx7m9ei(oqzfd 2 tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole md9(uo)e 5y;s pr ifkqw3 fxs2zz*7z2just 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 :6ou; q6 zquhsof a thin string in a circle of radius 1.10 m at an angular spee6ou uzhq;:q6sd of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whl ax6it2ndo e1h9e *:deel of radiushx*9o 6ite l21d ne:ad 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, o ma pl-z3,vb/*fniw,d n a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rb , nfi/* vml3zd-awp,otation 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) cang:5l1e y ;gi1zbxq)r z 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, wq5yig xl 1z1 z;bre):ghat is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spiiv+6y46a,k o 7mgulqs n rotation rate from an initial rate of 1.0 rev every 2.0 s to a fi 46avkl s7gu6io,myq+nal 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 at a freq zv:vz,/mc waa1 j*4wudwmy) n-al v7)uency of 1.5revd a cza jww um41/yznvv)-m,v)awl7* :/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 wf, uuy c fi3)l6/g0 0kgxvr(d$rev/s$ 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 momes //ckkk/b:9qpbj0f wntum 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 dlzc468+-kmewdfg-znxe wv b svx .8-2isk of moment of inertia I is dropped onto an identic f2zzs xdvgxe-6 w8w8 cl.bk n+4-mve-al disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atd whiwpo83wc2b6pym,l5pr9 s8 u *2p n 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg st5379ag,casrsg*2 xqexs* t hn/oq,yx ands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920g7 xh yc,9g* 3aetx 2osq/*qrss 5na,x $kg \cdot m^2$ 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 ) 9rv/*e oul7j(gd nrpfreely with an angular velocity of 0uplr (gn v7 ro/)ej9*d.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually coll+ckshhyihbvo9 ,824sapses 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 b8v 2k9ch, hish4o+ys?(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 involveyk 3mkkh n76/x winds 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 oc v1i)d8xcv :$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withotff,(ec9 *ng .r.ibjj b0ptgcwi e)*. ut friction. The moment of inertia of the person pl*w9( 0)gitbeci r*. fpe. gtj.cjbn,f us the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merr)ma0 rgk4ei)63u xx ivy-go-round turntable that is mounted on frictionless beari3 aiem) g4 x6r)0kxuivngs and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the ropb+ 0xpl2* vzpl9yon(pe 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 rope unwinds from the spool? How far does the spool’s center of mass m0pnl yv pbpl9 zo(2+x*ove?

  The Moon orbits the Earth such that thj 5apd0i00 geme 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 particle orbiting thj p00id5emag0 e Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate eo3q m m2,fqc0of 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 of7vqbagm69 s4 q radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelerationav4 gqbs6m7q9. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg nczlnczzx:ct 7+ swe5une5*z d0+,8v and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservationnl*nee,zzcc 5: 7zxcsd ++vz8 ut0w 5n 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, howc yu s1p8:of vf i55h(qjcu;v, is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wi5ycoalkd8t,m 8m nby8h1+hxa0;tyo l 1)c+jw th mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitation;+yxlt, h 5oknla) 0+8c 8d18jbhymwatocmy1al 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 stored 4z(wc 6v bnbgqy btmy-x159l9in a 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 wc9y(-tmn4b16qb5v9 zbxylw githout 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 illustratp4:ff9h9gx 8lqgd hjq+tss/18yhq2b es 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 horizontal surface at speed v=3 vnf04k qao+ 7bbsiig:64e/yl.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 pivocxh6:i3- z*l dxfh7f nt freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acce ifh7:nlh -f3*xcz6d xleration 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 s(4 g2koy0fhw d- h2tbltanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it wil(yl- th0hbkf2 w24ogdl climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making((/fh1kx:xjrqz o2du a turn with a radius The forces ac2(/qj(h1 x:dfok zr uxting 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 rm5 ynlw+ o+g1yn. uhoa1jg12a ock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating j g+2ng1oyahw1+5uyl n. am1o it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thid9p+ij jho9ti.1rz f9rn /xt3n rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.. ttp1n hi/rxr 9ijd3fjo+ z99   

  You are designing a clutch assembly which consists of two c; hnoeh+j3kngw*(r; r ylindrical plates, of maj;+n ghoke 3(rr; n*hwss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roughqsb zuc(;(c 6z.rex;p 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 theper(;b 6cqc x.u(zs; z loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bzd6a1yy 2x(yyrp 3j+ qut do not assume $r\ll R$ h =    (R-r)

  The tires of a car maqxd02in.9i ipr*cxyg 0 xw),w u2 t0swke 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was0* , xsq u0i2ynx02rid).piwcgxtw w9 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