A bicycle odometer (which measures distance mleeut42:h+au - yar98tcaq;traveled) is attached near the wheel hub and is designed for 27-inch wheels. Whamr8ael :ayut+ q9-ach4e 2;ut t happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular z nk+.9en0qxwt)s t m:velocity. Does a point on the 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 cases9tz+xe 0kw :q .nm)tns would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describg((b e9itxryz1.kdf6 luhi )+f s ),jjed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explk lf/uq /e(, l1oakeg y)s+0tyain.

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-:*cdda4yji0(ml j w*saap3( aup with your hands behind your head than whld34:*awsap0ac( * jyjia m(d en your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at t x-wmnk j lyx7(c2c)1yhe rear wheel and three at the pedal cranks. In which gear is it harder to pedayc2 lnc(7 -yjmxwk)1xl, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowernw:1)wx3yo yf 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 ion3x yw )f1:yws advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narro(1wuztr x 6d7iw beam?

If the net force on a system is zero, is the net torque also zero? If the-* rhl(3zz aht net torque on a system (lz-rh*tzh a3 is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontgg0he / j.tt/kbm 7y1jal. The same steel ball is rolled down each incline. On which inclinkhyg.1m/t 7j tj0bg/e e will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. On 9x5khcyy3)h ezc8t;fe2v qg .e 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 the greater total kinetic energy at the )5. q892;yhh gevykc3etcx zf bottom?

A sphere and a cylindxlsukd jhq +7 aqr,9- 44fnou,er have the same radius and the same mass. They start from rest at the top dxjuq 9s4or4+ul7n, -k q,hf aof an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conse* .oapk) 3kbt9yc++ bexh7tnv rved. Yet most moving or rotating objects eventually slow down and stopoc+b x7pvn)k.hytk+ 9bt 3*e a. Explain.

If there were a great migration of people toward the Earth’s ecwxg +1q ( -dbxa7muilp:g.a2quator, how would this 1ipl7-qxbm:a+( dggcx uw 2a. affect the length of the day?

Can the diver of Fig. 8–29 m6q 1bw1im bv:do a somersault without having any initial rotation when she leam 1: 1iqb6wmvbves the board?

The moment of inertia of a rotating solid disk about an axis throug;, jnaszbz/jw tuehg7j :87j+q7 vme9h its center qujsha, z;8eej/g :jt7+bnm9v7 w7jzof 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 in each outstrec 7uy 7a(zf72rmsulw*l/mg y 7tched hand. If you suddenly drop the masses, will your angular velocity increase, decreas 7mra* fumu( ygyz/c2s 7l7l7we, or stay the same? Explain.

Two spheres look identical and haa+f q(g-k* uxkh*ukj7y9hk ) zve the same mass. However, one is hollow and the other is solid. Describe qukuh9h k-) (ykk7fzx*+ j*gaan experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axep ,vvi e4j xh.u-olp:;/ -wqzle points west. In what divw, x vepu-zo eh:j.i/4l;q p-rection is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stan pkqkh*cor;hu4vcd ;z+t++c,.k r+q tding on the edge of a large freely rotating turntable. What happens if youk*c+hr+qdrk ck,;zp+cv. uqt4hto ;+ walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklycys .qcl ; 5m5-6jwmnd. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hipslw q j 6-5nsm5.dccy;m and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wh :2 j(m j fxwrahe;jw90kb*-yuy a helicopter must have more than one rotor (or propellekj(29m-:; *uaewyf0 xhrbj jwr). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following8 a-)y(vdp wc*kd) lpa2pxm z/ angles in 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 coinciv3xq 4l2 /xmnj7tkv4x dence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as/xx v q2 kt43jlnv74mx seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M+ :x ;dkqftli,oon, 380,000 km from Earth. The beam diverges at a l;i k,:+xftqdn 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 turnes6g) vqz-mfmi g d8rl2.ygq/4 d off during operation, the blades slow to rest in 3.zyfs- .r lmi2 gvgm 86/qg)4dq0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball 8 eu(w*y 2hyt r*bkb*fmakes 15.0 revolutions, what is its diameter? * wfybbr*y28u*e th(k    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. x0p/n:c li41 zcjn*be j1g)-ehnmqv1 How many revolutions do the wheels make4/n x1b1e c*1jn:qjmlz ngc eip0-)vh?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcu gk x-/b+c(wzo 8b h)fby :ywpr0o1d,ulate its angul+/o-hu o y 1rbwf zx,yw)bd0 8b:pk(gcar velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round p/s1j2 vxi*kjcsgk jd r/4z.2makes one complete revolution in 4.0 s (Fig. 8–38). (a) What ix4s/kjd /grsv21j icj * pzk.2s the linear speed of 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 o6ui*m e.e.v8qck u ,wqf the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe naide,5d )hmxz4g u17ed 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 iu )bsodg -ux i.sm :gwq4i.h/8f a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10g/:.84oqg ds)ubhisw x-u m.i0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 1337d-/ukha0iis y n4 re0 rpm to 280 rpm in 4.0 s. Determiy 7e0ra-id3n k4 iuh/sne
(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 radiustr4(c( p yfluf9y,5ya1kb l*scw ;q6l $R_1$ 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, astron0wgv+a b/w gl5 :n/nqk 17hodzauts aboard 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 minute during a 12-min time interval. The spacecra/:kq5on7n1/g h+ gdz bawvlw0ft 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 1gr4l6 n y6o(s 9oyv2vb5,000 rpm in 220 s. Through how many revolutions did it turn in this to4syvgvrb9 2oly6 6 (nime?    $rev$

  An automobile engine slows down from;qp . onud70ws 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 fdh n7i m), :8zbqrfd *w6hd/eklying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cbq)w, hfmd8:ne7 kiz h/6*ddromplete 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 ri3m,gnr1 v mjdgi+( b;pm in 6.5 s. How far will a point on the edge of img g(,+dvi1bnj3 m;r the wheel have traveled in this time?    m

  A cooling fan is tur- ws h1a bgkl(2wgq1g9ned off when it is running at 850rev/min It turns 1500 revolutions before sb wg1 aqwkg2-h91g(lit 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 redu 5mv/th :ug/xvces its speed uniformly from 95km/h to 45km/h The ht //5 gv:mvuxtires 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 frwk0+fk7 p.0y-nb /ui ur;er the car reduces its speed uniformly from 95km/h to 45km/h The tires hakr 7br+ui0kuy r wnp .e/0-;ffve a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts al n ydp96a xwr.0c93coxl her weight on each pedal when climbing a hill. The pedals rota0wxyrnxcp99c6.a o d3te 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 ct . v9ro3lawqt(a7 ,xN on the end of a door 74 cm wide. What is the magnitut xo,twcaa7l(r q.v93de of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Askfs h.zfdr;xd8,kxk+ ,kg( n :sume that a friction td+z kfdf8 x .kkrxn ,;,k(hs:gorque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached topyoa;zhp4ojwai,wx 1 7/* nw8 the ends of a 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 dir jwapp84wxn;/ 7*o ai,oyzhw1ection of the net torque on this system.

  The bolts on the cylinderlz4c/izsn tg bz 3+1d0 head of an engine require tightening to a torque of 380st4dbl 1n 3c/z gzz+i $m \cdot N$ 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 of radiu*j/z j qeq4wz)s 0.648 m when the axis of rotation is throug w)/zj*4eqjqzh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim o2 -dsvodm,:rjhz)czw 1b f3.and tire.z 21rhd,:ms ) 3 zocbovw-fjd have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a tanr s7wj8f /m4,h2 slihin, light rod is rotated in a horizontal circsaf hn7/4j w,rs28mli le 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 muoti8348 kb( mn 3lr,(ftkx ka bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–4 bm3u8lmkrt (,t 4kn83(oxif k2). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia o-p0kcpt 9p:smbfv a 7b 9 9bnn2wgj/1of the array of point objects shown in Fig. 8–43 b:njkw1g0av fbb m9otcp997/n p-2spabout
(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 oxym0i 6(9xu7e btxs 5xfkgen 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 cylinf2wn mii8epa .qx7(h 6drical 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 steady f67a mn.wx eqi8fh2(iporce 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 8.50 cm ann qf2w6i ,, p 8hxpq7mlll3b,y.10sd kbpfit: d a mass of 0.580 kg. Calculatebwy8,mp. pn x,:kb7236qid, ipqsftf hl10ll
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest tbyce:bn89 8nf:nv 2e no 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is ablv2 6 w)dq05v vdtot4mve 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 kgt0dm 6vdv4oq)5w tvv2, 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 rotatin8; ab6hzxa lmy (dts v;l:vxp3;7 ust4g at 10,300 rpm is shut off and is eventually brought uniformly to rest :(;tlvzy3 mp;vhsaxx;8 u6l sbd ta7 4by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball ah rw -a,02db)cn(xb aht 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 enmg/vw (fa:-ball is thrown solely by the action of the forearm, which rotates about the elbow joint under-wmf (v:/e gna the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fid;sw1plu-ww1j 2*d ;o27 t,igsk/ljho, cnqp g. 8–46lljjgwo2 1c1s -pw n,* w/pdthk;o7i2qs;du,.
(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 of81+ s) byinpid two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (re.x1vr4j:s5f fcypaarvt ( 1e 6volutions) and relp av5. t1ejxvsf(:yc 4r61far eases 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 mo)ea 0jtl :xdo53wn bi;ment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a t5 p2z7tt,q h4kxm ud-jorque 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 radmb2l h02nk loc5:;plw;dx7ceius 9.0 cm rolls without slipping down a lane at 3.hc;2lxcnwk;mob7d0 pl :el5 23 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t(s3 :3qs6 rpk ;camj+t) ebjcphe sum of two ter t+c(; p)bkejs p6cm3qs:jar 3ms,
(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 msb:mbrt::0 gzy1 d6fx. How much net work is required to accelerate it from rest to a rotation rate 0xb :y tgzfm1dr 6:b:sof 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and v,mfm*y. -snzuxt6 8i mass 1.80 kg starts from rest and rolls without slixf.iyz6ut 8n - m,ms*vpping 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 bja-7z:u. 9owi (q02jtwmcm eg trez( 5alanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the g o7 z(i:w.grmw-09aqe( z tutjcm25ejround? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentu )81lzxvfrs u8m of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angularusv flr1) z8x8 speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniafl8*fj6 8u9dduj*lw e mr c.3form cylindrical grinding wheel da *6fllf8c.u8e3u *jmwj9 drof radius 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, on a platform mw9 kdbatc5*7that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal pod *75am9 tcwbksition, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fignx;l3w(;hy xl. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to tx3y; hxn( llw;he tuck position. 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 frobp4 a1w8rf6 cmm an initial rate of 1.0 rev er4a6p 8w1bfm cvery 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 atyye: nypcm .y .zpag(hb02k // a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheezg.paym(yey.c/n k:p2yb / 0 hl after the clay sticks to it?    $rev/s$

  (a) What is the angular momenpi6jb+nt.(d -9qmk ix );r-.arad b/ g3vvgdvtum of a figure 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 momentum of the Earg ew 17+dd:su5upro*v th
(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 cylindrslv t;)rqm )q/ical disk of moment of inertia I is dropped onto an identical disk rotating at angula )tr s/);lvqmqr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at+8w a(gorlmtoh87z p2 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 me epzv8t. horsx3t2.q) rry-go-round platform o xt38po .h).vzs2teqr f radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating f.: pspk(* o3nuoji)qbreely with an angular velocity of 0.8q*jk):so p bi u(.3pon $rad/s$ 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 collat pemr 3,i;hf* .k55b5a xivsnwxx)1apses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integeaf.kx 5 ;sw brvt1)ha55x *pi,3enm xir)$\approx$    $rad/s$ (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 involvne nk zf4,3 wci5ig//xe 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 ( r;nbhb (y. riipbhy :9qj4/bbjx- q5$ 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, tha6g.xql7 acd -xt can rotate freely without friction. The moment of ial xq-x c6d7.gnertia 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 person stands at the edge of a 6.5-m diameter merry-go-round tujk)6n w.ju .kqrntwu).jj.q6 nkk able that is mounted 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 wi7:d aesanyubbg3mr7v,k7)6ld 20s utth 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. Tda v27e sayr 306gb7 ,bt d)m7u:ksunlhe spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sidq1ujs+4-e6lzxyak l 72pfa (de always faces the Earth. Determine the ratio of the Ms1+e2ju-4x k a adzyp6lq7 fl(oon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest wfc; e (v8.vtzgb4:+wcc .sqw at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radir g; o++smmq6t6 dqi:sus 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the s6o+d 6mm+ sq;tigr:qtorque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each e- 9ai*af3r ffof 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 f9- 3aa*effriof energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed0hc4 tq5rq;;wke )xdv 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 of2u que 1y-a2lb9ei m-c our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the22i9euuqy1ma b-cl e - 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 foz(-7 zwobdvh(r it to use energy stored in a heavy rotatingdob(h( z7v -wz 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 illustra *s3zh3e9 rwqlbvnigai1 :n /:tes 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 surfmxe,ci: yaf/5 ace 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 pivgacyvk- 27 kgvw.pw-y6b6 :jw ot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and the6g 6ycgv kyww:-akj-7.vb2 w pn released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing b p,w2mfycs:x *3jygo. q q+1fvertically on the floor, and we want to exert a horizontal force F at its axle so that * 1scfw2x,3y:gj om.qy b+pfqit will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spee).fs 4ij(tde0u3: q7u kbeuh ed v=4.2m/s on a flat road is making a turn with a radius The forces e(3e0h:u)7si qtk4 jduf.b ueacting 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 intof-dmga4op9bie r q1q21, dtb, a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from res-p dotfa1bq1,94 br,2 dgmieqt 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 an6j9yh,dvl )b ld her arms as thin rods, making reasonable estimates for the dimenj9ly)bh 6 d,lvsions. 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 clutclw3np6-v,q xo h assembly which consists of two cylindrical plates, of mln-qox,p63 wv ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roug1szyeihc/ 3k 1h track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop 1ks/1e iy3hcz 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, bubvlz8eqe8 4b ;t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the zhh *g g94l//sam2v,z1b zaip w07r etcar reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angulh/i9p274h gm 1talzzzw*/,esr0av gb ar 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