A bicycle odometer (which measures distance traveled) is attacheik8lg5 q )uk*rkpg 2:yd near the wheel hub al2rpyukqkk: gi8g* 5) nd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on t,+50uipmhohgem d +nd 3 m/(aahe 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 which cases would the magnitude of either component ofdnh ap 3 +gom/u0(h5+eami,md linear acceleration change?

Could a nonrigid body be descrfh d8zrn tiihj54sf ub.d9 1; /4.spzfibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever kwp5c 4n h5mh;exert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your he(+i svlm-jc(oad than when your arms are stretched out in fr jsicmv -o(+l(ont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven spro+d9hxg)3gs6xebq n b,ckets 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 rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a l dbbg,)9gq +h6ns xe3xarge front sprocket? Why?

Mammals that depend on being able to run +xhkd ,44lm wxfast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advanmx xk,lw44hd+ tageous.

Why do tightrope walkers (Fig. 8–35) cal.,qu0yynd+xi(a(5 rh x cw8 frry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the netzw2owwtc2 /o; torque on a system is zero, is the net force zez/t2 c2wo;wo wro?

Two inclines have the same hed5gg6e1eh7b3p q(u mvight but make different angles with the horizontal. The same steel ball is rolledqv1gpuhm(e6 7egb35 d down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultane)srycs*4v j. oously 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?4c*.s)jo yvsr Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the 5x4d n cm6c9wm8ac)or same radius 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) 9 6cmm corwn48dc5xa at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momen5(e- lup+tudau27n i atum are conserved. Yet most moving or rotating objects eventualpia 7-a+uu2eln(dt 5uly slow down and stop. Explain.

If there were a great migration of people. f-qu c/ua+bp toward the Earth’s equator, how would this affect thef +/a.-uc pubq length of the day?

Can the diver of Fig. 8–29 do a somerszzy wsm5 t:om0:kv.zsi a/73vault without having any initial rotation when she leaves the board? .szty 3v 5m7zmio :v/szw0:ak

The moment of inertia oq l4*8 jxgmjwu n7:5ljf a rotating solid disk about an axis through its center of mass 4w7j*m 8jgq un: 5xljlis $\frac{1}{2}WR^2$ (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 onxyz4gau:9 ls-km zy(6) lgfl.,pss 6 m a rotating stool holding a 2-kg mass in each outstretched hand. If you ss(my l 9-:passyx) 4.fzg, zglml6uk6uddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hoi2l xislj216 nwever, one is hollow and the other is solid. Describe an experiment to determine which is2 6n2j si1lixl which.

The angular velocity of a wheel rotatin*f z4bttf* ql5g on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangentif t5*l4b*tzq fal 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 z rz(,3fy-k +vl lb4hgfreely rotating turntable. What happens if you walk hz+(zgbk3yvl,r-4f ltoward the center?

A shortstop may leap into op l/iqbq4m3*w/u 2k6s rx9ap the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.mr3p i k*2pws4o9 q/b6/lquax

On the basis of the lawk zg;womyy )+xo*10hr of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helico* +r;1oy0zgmh kywox )pter stable.

  Express the following angles in radians: (a) 30wgflb0fv 3:s . $^{\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 amazinjmyn cg090irv0)un+qg coincidence. Calculate, using the information inside the Front Cover, the angular diamet0yumvn0 +i 9cjqn)rg0 ers (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the w dmckc .0ihgy -vn:9i/sw+w4xch*g / Moon, 380,000 km from Earth. The beam diverges at an ainwk*id: s -4x g0.h/+y9wcv wcmchg/ngle $\theta$ (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. W3u.k 7ru ;b)/0p js5whpnycluhen the motor is turned off during operation, the blades slonrulp)wj30.y/u 7ukhcs5b p;w to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothem8bpe)wzi tqo69jst, a9/7 x fluc( s2r child. If the ball makes p9/m7t8i ql xtze obc,w sf(6)sj9 au215.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revaj h:fqaf 8d+4olutions do the wheels make?8 hfa: qaf4d+j    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2tu0* 1le )8*zgkjt mc8 -tu)avkwlox1500 rpm. Calculate its angular veloc clut8*t11)tzw8x*jgmo 0keva -k ) ulity 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 (Fs1mstrz 6e4v0)n7kr xig. 8–38). (a) What is the linear speed of a child se sntmk1rxv4rs)e 67z0ated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arounii2hc6zfkb ps5 .508pqtpl g;d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of u 20wis9xpo 3ha 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 11;;ovfmo b;q g + vjx(jp2y-evg8tn fa particle 7.0 cm from the axis of rotation is tobmt+o 1- ;ygfqvj;n1o2f( vex;g 8pvj experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a ..demn/vwa7j bout its center from 130 rpm to 280 rpm in 4.0 s. Determinewna.e7mdj v/.
(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 /oy. rvk4j139.rkcs* grfsn tpe d2)ds $R_1$ 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 abor x-9tvnfaqf* o/i /nhi94ip3ard 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 n3v9 prq-fxi4/nihf/9a oit* 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 fromss5zzc r) ck8ade9u1. rest to 15,000 rpm in 220 s. Through how many re z81redsks )a.5c9uzc volutions did it turn in this time?    $rev$

  An automobile engine slows down ,y.n+ypgw 1nwfrom 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 whq5: gt hw6nf5uirling “huma 5gf wq5htnu6:n centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm) vms +yzhze+2 in 6.5 s.2zz)hms+v y +e How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned ouc /d+jt y ysjzj,,8lciz :,d(r,o8ioff when it is running at 850rev/min It turns 1500 revolutions before z,u,olj i, c8it8:,j djydc r+y(so/zit 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 ik7+ign8v.sv/yarc:k f jo 2;pts speed uniformly from 95km/h to 45km/h The tires hv7jn f2r+kak :os. ; p8icy/vgave 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 uniformlymcgg f/:f3c3l+a/i wkjf jd 8: from 95km/h to 45km/h The tires have a diameter of 0.df fwf+/lj 3cij:8k: ga/m 3cg80 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 pedaw 5zzyn5 -a3oql when climbing a hill. The pedals rotate in a circle of radiynz w355zaq-o us 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.f jzi(ipn((zmw)-8kwo g9 zp of a door 74 cm wide. What is the magnitude of the torque if t- kfwm9 j( n)zipwpz8((z.g oihe 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 du78be7*ezpy a3uv4ewh os * (Fig. 8–39. Assume that a friction torquw 8oseba4ez7u7 veh*y *p(3ude 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 +p1xp thr -kbdqv)z: 5 wq+gmc.8fa h3a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction f5+tgxvka)h-8r mz :w1.+pqdbhq p3cof the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torq+iv 10nhh,jva:.am 10 goqisf ue of 38 hq vi0f 1a . j:asghnmo0vi1,+$m \cdot N$ 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 sphere 17 );7tqo3)qrfanizt4 ql)qczwbp g7of radius 0.648 m when the axn))o7q )g7btpiqwa4t 3rq1 lf cq7;zz is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm i ,b2j ecs9*w.x l8ononzj;xt4n diameter. The rim and tire have a combined mass of 1.25 koo 9;n,l jbzs*nw 4c2 j8xtex.g. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizonnkz)8 pbxfbkup.-:sov /3q e5tal circle of radius 1.2 m. qb3ubvfn/e8- xo):z5kp ks p.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 pottecj+aeyc5+4o p r’s wheel rotating at constant angular speed (Fig. 8–42). The friction forojc+a4c5+y p ece between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of poq95 g uqsls*c 3wn18fsint objects shown in Fig. 8–43 about wscqu qgn9f*sls 3581
(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 consistsl5+ efao+yu,t 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 cylindrical satellite spinning at the correct racu+j6e 8fqrv(te, 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 ie+cf6qv(j r8 us 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 cylinder with a radius of .f+jb1 fa jih;8.50 cm and a mass of 0.580 kg. ;hjif .a1fb+ jCalculate
(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, accet d42i e+0zb0vajktp-lerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round spbk)mz qn394zm9s e,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 hm9bk)ss,n4 e zpq 9m3zas a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is sz x 7 b8pv5)dmss:8d4z4xd6pui v;lidhut off and is eventually brought uniformly tz74s d: bzi v4xx ;m 8pd8d5lud)v6piso 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 acceler5 0 m o(wdj d:yn(rye2gn4+srwttc )k6ates 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 throwp )sh-p *f,y1:af thfa9upa. sn solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated unifor y9 )1hapusp.afh-f, f s*ap:tmly 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 l18maycs qhyyq5 zt u )8-x5ur1ong thin rod, as shown in Fig. 8–46 h 5-5mx8zrysau)qq yy8c1tu 1.
(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 pto6 pkbpvut2-qz* jw92*w9qmasses, $m_1$ 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 accelzy ng o 1,(.sal(zuh9ferates the hammer from rest within four full turns (revolutizzg.f9(yunao1s (l, hons) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mob w:t12hau gn6ment 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 d1 8aif s+aw(jwzpxff 1v.rg*z +rt)i7evelops 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 rollc78px q9q ohsx k+r tb40. 90wyjttqk2s without slipping down a lan.tqqh 4s98 b7r20 yx9o xk+0ckjwt qtpe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to trsp5 wqy13-pqa+js,v he Sun as the sum of two w-p 3sav+5 1jrqy,spqterms,
(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 7.50m 18nj 7 eqcp(f e)hmxh/jid:) m. How much net work i) mjp:7jq d1mx()/i neh ceh8fs required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm 6o)o0*kjgz bzand mass 1.80 kg starts from rest and rolls without slipp6jzz0 obog*k) ing down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced verticd htn6nu-f9wa; 6 7 2uyd3dszially on its tip. It starts to fall and its lower end does not slip. What9d zht;-fw u n236ydisa7 6dnu 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-k,kn5auy/gg*ngb.t b 1ai5 .yb ur6m0 rg ball rotating on the end of a thin strinygab t6au mi*5 br5yk r1g,bu0..ngn/ g in 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 grindz4sge7oe2t, n g sxzpmr096mk 8;l-taing wheel of radius 1sk8t9;me-zgp, r4 s z6a2 lx0nt7oge m8 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, on a platform that )hakm od(8l85 /lkad wis rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotationk k( o)8 wl5al8mha/dd decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her oi ;ta refyl0f 9- b,czs0 /dgk9vc)0amoment of inertia by a factor of about 3.5 when changing from the straight position to the tuck pos,0odk9c9baff0;icryz s vlet)a g/ 0- ition. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initih7c 8mlm z-ja q-d(nymp)p7t/al rate of 1.0 rev every 2.0 s to a final rate ofp a-)c-ldmhmnp(q/j 8zy7m t7 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 atu5oror , fh)k8h4if :v 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, ontov5iu:k oh r f)roh8f,4 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 figure2n1r+ n1v cif*; tk-m kemix6i skater 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 momentumqp .gbndc;x.- 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 momm/yz0va 1vhz1 2b2px/hj6 k; y;lry8cgqhv5 j ent of inertia I is dropped onto an identical disk rotating at angulazv;1/pklva zxhm62j2h8 y y0qgb yv1 jr/ ;hc5r speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns75junem;uh ;0x 1 fsff at 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 stands at the center of a rotating merry-ge/mrziye. 87+g; 7qaoj c /vdio-round platform of radius 3.0 m and moment of ijq c iaozy 77e+ ./gdi;8r/evmnertia 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-m;c4b *iu-7b.jcosu dround is rotating freely with an angular velocity of 0.8 *d-j bi4s.7bu cc;mou$rad/s$ 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 collapse;lbhjll7gupm5tgy z . t5s2a5xf 3/m/ s 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 rotati b f5 2ammt /;lhxt/.y5jl5 7gps3lguzon 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 prkl-f kpf;;9pwc * j-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 (dpoflf c;cm d)lzsd e)8sst5 u7 6f*4$ 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 withou()srl4poaj ( gt friction. The moment ofp4j(( gs)or la inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg persoqs yd)k0hdl+j 1*)aqd n stands at the edge of a 6.5-m diameter merry-go-round turntable that is moundasyhq+q)*kljdd1 )0ted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying rx5lk m /m,t9ogv1pw8 xt0 .lrmuz7w0on 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 tzow5v/klw mgm 1t07 ur0mp98 l,xrx.rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the sa 51-sg /qymjesme side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular s1gqe/-5yjms momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratlp3vd m 2eo(fip x,x.4e 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 unifofpj7jb1 1r(a zrm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angulap1j af r1j7z(br acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s-kc i/-gh;m ovunr93iolid 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-longoc ki-/i;m9 hugn3 r-v string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, ho *zhvl)ksr8p) w 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, exo9:q)z aon*c-u*: ndyf-b*b v6qb pbut with mass 8.0 times as great, were rotating at a speed of 1.0 revolutioq f)y *ebpc obn:- duv96na :-**xqzbon every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a loerk/dgrj xw8c:l.h c4qs yh5*)de7.rw-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diametex*e5g.cr . lrk e:yq hd)shwr84/7j cdr 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 5 vw,ybai ea2,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 roo fuc9a6t4lh 9r8jo:plling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frz xq (xdsondd ,k,9na;)/vbh9 iction) 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 centeb,qadh xd knn9)sd z/o,xv ;(9r 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, anv f ovag+7yid69kem7 7+ y4uxxd we want to exert a horizontal force F at its axle so that it will climb a step against whv7u +xy7e a+fkdxogv6m7y i94ich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with se*vlvzf8 54 fz5lme;h mj,/oruwcst) 3,wf*t peed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the norwtwu*5f*erlof ,zc )l 48e vt5,jsmh/f; zv3mmal 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 u7 (yv9/ e)ufkaaz* mddzd5)6a w3xr yinto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, z(dfamed9 y xuv rz ) a/76ky)3d5w*uaaccelerating 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 tho nza cj. d7s1y4f*ozi8;f p0min rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and 8mi0fyf7o dc1j *p.anzo4zs;with arms held tightly against her body.   

  You are designing a clutch a3pw 42qqyeoaizv+g:fk ,g qi.q 7 gu,p7o76br ssembly which consists of two cylindrical plates, of ga,fg,2qoekp g7w+r3 qyzo p7iv6u7 .ib:4qq 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 and radius r rollsvjew +w0mfmn,c8 p 9u 1td;ky0 along the looped rough track of Fig. 8–58. What is the minimkcmm+nw8f 0 evdwt9,j1up; 0yum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assupj6kk h7, 0;xvkrr*z(dw rm6ime $r\ll R$ h =    (R-r)

  The tires of a car make 85/za) a;9hoy,kz cn/z r revolutions as the car reduces its speed uniformly from 90km/h hazo znzka ry9);/,/cto 60km/h The tires have a diameter of 0.90 m. (a) What 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