A bicycle odometer (which measures distance travele)-b* ktho3gtvd) is attached near the wheel hub and is *hogb)k- 3v ttdesigned 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 pfj8 b6u2upy k,oint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniforj uk8b2 ,6fupymly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a sinif,0k bliaox zt2i)95gle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a -*3 gwd63gv pt7zt g ywfjy8k-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 behindvm9 ,;uw 4 a3gfcksz1e your head than when your arms are stretched out in front of yo14e zvum g;cs, 3fwak9u? A diagram may help you to answer this.

A 21-speed bicycle has:nr8chj.hlwdi6 9;o.6)b4n vxufe x t seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sproc.:ei6;jh6u9.n8 ln v df owxtxb )cr4hket or a large front sprocket? Why?

Mammals that depend on being able to run fast hapah;0 6g iw4a.*njb7fcb(ai gd+q yg5w1x c)rve slender lower legs with fleshr4dci1(0na i y+;a f. pj7g5g6*)xbbwwcga qh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fixt1r, bz .cgt-g. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net -cv8 6fkwirn dd(c1lt 80ds3 rtorque on a system is zero, is the net for c 6vf dkidd38r8n 0c(l-tsw1rce zero?

Two inclines have the same height but make different an0n 4vres 4wna7gles with the horizontal. The same steel ball is rolled down each incline. On wa en n7sr0w44vhich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultac;rs(e,w2. er(7a5zblunmu1 b+t fmg neously start rolling (from rest) down an incline. One sphere has twice the a( u t.7lunfw,+e;e1c(m 5bm rsrz2 bgradius 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 bottom?

A sphere and a cylinder have3pw e2s :9 1(:ibolpauag+b arafrp:; the same radius and the same mass. They start from rest at the top of an incline. Which rea:( feslori+ubg2wprp31a bpa: 9; :aaches 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 conserved. Yetu ,c.n:pjfd3s most moving or rotating objects eventually slow down and stop. Explain. 3,:fscpu.jnd

If there were a great migration of people tou 8ld ;cl6*xfi1z3 dmpga +pshfc:8f-ward the Earth’s equator, how would this affect the length of the day us:impafd6 c3f8z;8+pdg llchx1- *f ?

Can the diver of Fig. 8–29 do a somersault without having any initiw 350pl6kp( v p uevz s/sraj6lz6d-)gal rotation when she leaves-0r z6 w /d6l)pz(a6pvslsv 5kjpue3g the board?

The moment of inertia of a rotating solid disk about anb0( ffjt 9m6co axis through its center of mass cbff0jm(9o t6is $\frac{1}{2}WR^2$ (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 eaf it8 )g:;vnmg*wusd:ch outstretched hand. If s:*):twg vm;g nidu8f you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howe.8+tst-x6 mn juik+xdhnd /2rver, one is hollow and the other is solid. Describe an experiment to determine which is k t8rd ni d+snjx-6.mxh/2+utwhich.

The angular velocity of a wheel rotating on a horizontal axle po:m*evopx jmpyc y 4)+*ints 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 tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasingp)p*4jvco y mm* ye:+x or decreasing?

Suppose you are standing on the edge of a large freely rotati93lbwo)x-afcc x r k(7ng turntable. What h9f k-3 bxawor)(lccx7 appens if you walk toward the center?

A shortstop may leap into the air to catchg7yb spo id843 ts4i:v 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. 8–3y: 4 s473p8obisgtdiv6). Explain.

On the basis of the law of conservation of angular momentum, discuss why a heli+ kcwc v54-wzwkl(je .ght(ujm4ej(mjni50;copter must have more than one rotor (or propeller). Discuss one or more ways the second 5jz(4j;m -ulnw +cehe(k 5. c (w4mvkjtw i0jgpropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (wl1zvm 2iqq ,+n85cr caiyh).v:e ue)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 co sean5n);xi/ utc;4 ikincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians)4t/asc;i)5kne i x ;un of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges .qx,ahteu33+ /c-m ywd- rld jat an an-, dx t3rdmu.qay/j e-h3lc+ wgle $\theta$ (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 turl;++ s9 uq1mgk dl14,oet acnsned off during operation, the blades slow to rest in 3.0 s. What is the angular acc + 41sltnkqsa9l1u;eg +cod,meleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flo 4hqa4ldju6 sq59u2qz or 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameterulu495zq ha6sqj2 d q4?    m

  A bicycle with tires 6oz8wyl+h+q h a/;:nfdh yvrls2 4 w:79bk 9bhs8 cm in diameter travels 8.0 km. How many revolutions do the wheels 99vqaw 2h+:8:hnf h ;rbylz7kso dl/4 sbw+yh make?    $rev$

  (a) A grinding wheel 0.35 m in diameterhdciotl/ s5 oda4hn 9o/-++nn rotates at 2500 rpm. Calculate its angular velocity in n /loh-h oid td9ocna/++s54n$rad/s$ $\omega$ =    $rad/sec$
(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 completvzselr8 z+:q0y+3 cmxe revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.:m l8vyxzsc0 3z+qre+ 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 Earto keyy:coe*j+ vcyw s(f)l *2s18rkvs7l op: 0h (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi(gnnk3k.x nqx((k 4h nnt
(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) mux lp 5e(n4bq(rmyzomi1 h+.f2 st a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration oynm.fh5 z(1rmo4b+2 l(eqxpif 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about i +woa1 0qn-td(rn+ghtts center from 130 rpm to 280 rpm in 4.0 s. De g+ rn (ho0ttnawd+q-1termine
(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 radiupyh eo oz,f7 sa58wo:)s $R_1$ 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 Moo g3zlkq7skm b 7mi7(/kn, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distlbmks7 mkg3 /k7q zi(7ribute 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 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+a st lz//,ngqpnhz.2t8 fct3 uniformly from rest to 15,000 rpm in 220 s. Through how many rev2fl3t nz, ant/hg.spc /q8zt+olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in ltl h5 a8fyo3h/c-;fs2.5 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 whirlingrjwwt3w, i* p85-id4nc6j6c r-c kb th “human centrifuge,” which takes 1.0 min to turn through 20jn5pdkw, bch i j 6c-r3 *ricw6t48t-w 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 rpfwp r n4y::k+rs2q3n9 jwwfw+m to 360 rpm in 6.5 s. How far will a point on the edge of+jnk2w+93:: 4wy wnrp rqwffs the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Iir9d. t6axc vlbm6ut,lc 1c. 9,b3pmzt turns 1500 revolution9,1v.zpti ac cbmllt.umx 6 dc96r3, bs before 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 irx cxx*4o*e4x)o.mz(a9 exv av s :g)xts speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 or gsa*aov vx9x xmeze)xc)4x .4x(*: 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 c/;iub*2n+jixt 2yv k zar reduces its speed uniformly from 95km/h to 4nk* ib2/z tv2 j;uyi+x5km/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

  A 55-kg person riding a bike puts all her weight on each pedaki7wt0qpc4:s l when climbing a hill. The pedals rotate k wi70q4ctps: 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. What is t9d)f rrq ,:(bxkiw-pk he magnitqw)kk:-(xrrf ipd,9b ude 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 e(nso0fv8)wyn1jjns- rbm,fn 7 m56 xof the wheel shown in Fig. 8–39. Assume that a friction torque of 00,n wfr -16n (bjjo5s7neyvm s)xfm8n.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*6ex*a snet;c massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releasede; t*6xea*n cs. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38ayurna )41 .th1wj( l;c-szpvtw ;uj a4hr ls-.czv1apn()y1 $m \cdot N$ 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 awn q2 dzdp8ss/hy09 8e 10.8-kg sphere of radius 0.648 m when the axis of rotati 0n8d/q2ysp98d hzwe son is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamet0f fou. s5ojt*ufc1 z9er. The rim and tire have a cof5z*fof91j s0.t ocuumbined 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 isu ou :hb,ks0y0 rotated in a horizontal circle of radius 1.2 m. Cy:h b0, uo0skualculate
(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 rotati xv(dh1 r9zo/q)lkr/ qng at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 Nq/qrzh(l/r)vd o91 x k 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 th vl 5g7eh6(3kf4f ysc-len0xqe array of point objects shown in Fig. 8–430ch v3ffyg6s en754l-klex( q 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 consists of two oxraxt1-q;6e6 wzmnb j 4ygen 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 cylindrica:vbp69b 3 w(u+niwikgl 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 force of each rocket if the satellite is to reach 32 rpm in 5.0 m 9 3(u +bwvgn6biwp:ikin? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm l fo.3voi/-lav iwu(2 and a mass of 0.580 kg/o ialv (f2vui.l3- ow. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratb 8d7 1 evhl978n2g shr+8ng;fvduuaoing 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 me9c 7mf-yq 2.k*b hd/zwqx jrz8rry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglectin9dh w. zyzqk-*xr/c jm8 b72qfg frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at l 7uq dyf.jv;/, dwrx .v/r4oe10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque ow/l4vr.d 7 /,r du.yx ojqe;fvf 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 accelera3;.f v(fg 2w6(ywyxagj*jrqm55 t yrvtes a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solx3a7sb 1g e8zk)fzj45mp sdd8ely by the action of the forearm, which rotates about the elbow joint under the action ofm7k zj54sez13 a8fbd )dps8gx 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 ine2 dk 6jmcvxwuyat- 6d3mpa+ 804+t ra Fig. 8–460c8-3tu2 +d6 tpv ay6 xrmja+emw4adk.
(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 smo.d en.nc* 4 6njlr8of 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 f(y72l dp8k0.o ajmzeo our full turns (revolutions) and releases it at a spemo(.02lajk y p d8eoz7ed 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 of inertia of -luh f- k7an *ek/vs,n$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a toe / k* *igvgywyj *q kk.lplg1h.mnm)w)1m4t+rque 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 andhfe1(zowg u 04x3-qxi( )f mmx radius 9.0 cm rolls without slipping down a lane wfx04)(izqumxxg-o 1f(hem3 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect t)8 mqj gk45rbjo the Sun as the sum of two5k jj)bq gr48m terms,
(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 kggu8sc3h ochf:3eb;v 6, :zcxo and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it ixco,h6 :zbe3fsc3ochu :8 v;gs a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k4vbyfug 31f39kfbi7re;uo n 3g starts from rest and rolls without s3b;1i93buo enfyk4f7u r vgf3 lipping 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 verticaiw(xf1)qdn5 mx +i0o ally 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 ground? [Hint: Use conservatxw a)5o0q +iidm1fn x(ion of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thz h 7i9gej h :o4apy7)vf9n)twin string nzapejh t 7v97i49ohfy:w)) 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 grinda48rkq- xsf at+/bx o4ex;b0qing wheel bxq +8x0 t;soakr/b xqea-44fof 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 that is r ivw4oa;uk (*totating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rota;ko(a4ui*vt w tion 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 Fim: gdpp05 56dxkt7oa gg. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 6om kpag7d50: tpgx 5drotations 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+1gnm6 a- :g9 hn5alqxu z1spw her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finalxng mnqs6u5 +zag1hlp: w9a- 1 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 veraytxb9: ls( /gtical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approxglx :s/(ty9b aimately 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 ofezy(mx b,: )e43x yqex 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 mome xn; l2:vec-chntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment ;-0cu i6nspldft3uh h7w)dm) of inertia I is dropped onto an identical disk rotating at angulad)s0u)dhufl - i3tn w p7m;h6cr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns+os9a(eilejmm8/ f(s w3 kz5 y 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;w tkvyscz.w7x tc/95 merry-go-round platform of radius5 wwv7kz.ytsccx9;t/ 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angu j co3b3jl+f2vlar velocity of 0.8 l+233jofcjb v $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, l8u)o xsrzs 2d+s ,lwt6osing 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 woul, 2 rxsostud6) 8lsz+wd the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve wiodn6.) l z hi9r1i;ruckr*1z +c iid;qnds 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 5pc6 za6h cnuxa0 o;;z$ 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 plan3ko5hcmm .h 0q vsg:(tform, initially at rest, that can rotate freely without friction. The moment of inertia of the peh.m :qngsmhoc(0 kv35rson 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 b5.-i94fbyuh;p )nttk fh2wrof a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170py- kihrb9t h2;t w)u5f.f b4n0 $kg \cdot m^2$ 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 ropn/q18n :qm.a2 ui7w7meb ojrre 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 persou m81qjb . r7e :2rq7mionnaw/n 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 side a ;slgy.z7 rkq3lways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about i;qkgrsy.l z3 7ts 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 zb4w , j9)tuzdat 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 shayh49fza a(l 1r xcbdw74qn(q d,i +j8rpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a b(yricd1zalr+ 4xh4 fw8,qaq jd97( n6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass ge: kyp600qoy(d n1 bo0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindric:60 yokoby (dqgep0n 1al hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how i9e n)1yhdd* -xu 4xcyos 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 with mass 8.0 times as greathg4 hqcxh+ qo((p5rf+qudw -2t fuk2,, were rotating q (5qhut+(- pd wk, 4o xughq22+chrffat 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 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 iwjp np . 3x1*n6vkau/vs for it to use energy stored in a heavy rotatinwpn6/3n1uxavvp.*j k g flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates any; hq;3;sdna c,tek 6g $H_2O$ 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 spee ij4yjguw1 :hou1j4c8 d 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 6w kkv -d0fyf*v9c3 hapivot 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 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 Fi kd- yk 6facv9whfv0*3g. 8–21g.]

A wheel of mass M has radius R. It is stand ,vlft;*.oay gw2az/y) yob0a ing vertically on the floor, and we want to exert a h.bt 0yav,ogaa2ywfyl/* o; )zorizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is 8g2.b m6x .1oq.pdhe6 *jcy zui1ciztmaking a turn with a radius The forces acting on the cyclist and cycle are the normal ..pzxei6t qjyc h1b2z godui.c1 68m*force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begi iqz2 2y0kq6fbns whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requir2b0qzk62qfiy ed to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a soli(sfhclv 0g df-/ gff.:d cylinder and her arms as thin rods, making reasonable estimates for :c/(vf dlh.f f-g0fgsthe dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consistuxt* co./dtqy rc7m,*los cj-i:u, 8l s of two cylindrical plates, of ma7:o 8yr loj*c *ucx dqtt-u,c/silm, .ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track ofs- 1y) -obko+jzv:lgx Fig. 8–58. What is the minimum value of the vertical height h that the +o-:lyzx 1-)k ovgbsj 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 8xc9o* w,fqb .a 93 q6xbv2q(a vgxlqwassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed u.mc 6lcar.v3 xbkk6u(niformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular ab kruc.v am6.3 lc6x(kcceleration 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