A bicycle odometer (6u*yi0ykg rgioa (o j kve90)9which measures distance traveled) is attached near the wheel hub and is designeu*j )kr0a99 i0 ( y6iogvygoekd for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constwk)vp3)ft ,um8k xbxpp7)g-tant angular 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 cases woult) x)p-f tgmpuxk87bk3pv,) wd the magnitude of either component of linear acceleration change?

Could a nonrigid bod :jakd+yric5ble b/4 0y be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater r;yr:fwjs9/t 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-uzvalbo-v -j5+ 6cmh7w p with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thvlvamw+- 6 -7jc5zh obis.

A 21-speed bicycle has seven sprockets at the rear wheel and84ir:j k1xs sv 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 jk4rvxi1s8:s to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to 4w1q3c,gsgj iwnm rp9*d +ii8 run fast 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 oi* i mgdpn w1,83+s gwjqir49cf mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry xab+g3+hn cr5 a long, narrow beam?

If the net force on a system is zero, is the net torqubeh. 8dn;np20g;c aque also zero? If the net top guc nn;08 a.2e;dqbhrque on a system is zero, is the net force zero?

Two inclines have the same height but make differentksp)34m ughr aono) +5 angles with the horizontal. The same steel ball is rolled down each incline. On which inclpa)5sgmnk3hur o4)+o ine will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from res17yohif3t+txk j p1 h:t) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inchyh1x t:ftij3kop1 +7line first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same rffmtha0j1 n5 vn0/gv0 7af07jql.oa)v mcwz , adius and the same mass. They start from rest at the top of an incline. Whimjj7zanfvhofv 0ga tl)fv/0q0m 7wcn50 a ., 1ch 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 conservedc*po f(acf1rapoa;9vzu3-u +. Yet most moving or rotating objects eventually slow down and stopc;pu-*f oo931 acar(upvaf+ z. Explain.

If there were a great migration of people toward the Eapv j ;, lt3-l:prmm:wurth’s equator, how would l 3p m,ru::- vtp;mwjlthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersk,h(ks8naun y+j9uli zwso* wn79 ;y d.nh-+vault without having any initial rotation when she leaves -k8ynniz7 k.uw(+ vn,dnwyhoj9su*la; hs9+ the board?

The moment of inertia of a rotating soli*ceb ncl na c1f)u)e+3d0o 0tid disk about an axis through its center of massn3 u1)e+ aoccn bt*0)f0 dceil 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 eeg zxq:z:y/ y/hgi(p*ach outstretched hand. If you z(/ie q:*gy: ghx/pzy suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howaz;hi/h ya 2u ,u,:gblever, one is hollow and the other is solid. Describe an egu :2 h,yhb;/lz auia,xperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal a,zu r4b4gfwy -xle points west. In what direction is the linear velocity of a point on the top of the wheel? If the anwbgf ,zu4-y4r gular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing 3qnvtd(wd pr.5waf ,7on the edge of a large freely rotating turntable. What happens if you walk towarn fw(va3 dw.r7tqd,p5d the center?

A shortstop may leap insv5 ukw o-1 wbzgtev,3s ,jo:m+d3wv-to 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). Expl1u+z -vev :5o jwwosd3tsvm3-g,w kb,ain.

On the basis of the law of conservaoarnyze 7.e9tz9x (c;tion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pr.rx;zeoe9 zc t a79ny(opeller can operate to keep the helicopter stable.

  Express the following angles in radians: wwwh9/ lhi cg*fl:u1 9(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/qk:,l;25el 4r *wbz tymbk wj amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and thezm k,b4kejlww:;l q br5*2yt/ Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from y)h-i 6z73aakd vqvh9 Earth. The beam diverges at an anglehq-hz 7aday9 3 v)iv6k $\theta$ (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 rf +s,fofp;h.h-(cax u pm. When the motor is turned off during operation, the blades slow ocf ;+s,-fp.a h(xufh 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 another child. If the ba,vep9 (lfk26pvh4 umg/ x a*lcll makes 15./l,9exhkv*v 2mcufp6a g 4( pl0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do . e ntj-vp + )o4l4nbhn5i*fhme(3zlhthe wheels 3nnt m4e4 h (fpl5o z*+lhv.h-bejn)i make?    $rev$

  (a) A grinding wheel 0.gi, yh2 1th2gm35 m in diameter rotates at 2500 rpm. Calculate its angular veloc 1 2,yghmhtgi2ity 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 revoludge +8b k9bph,tion in 4.0 s (Fig. 8–38). (a) Whahkg8pb9 e ,b+dt is 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 of the Eawplo,n o3*,vjrth (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 mxrlm0w,lp9b/vtj:e ml4ywm-cd4 84 point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a parti;x/1a8oqn jkuzvc u7)jq+d 3 bcle 7.0 cm from the axis of rotation is to experience an +u /1 jd3o;cxan k8uvqzbjq7 )acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter whex y z 7raqly8d7(sl5tt.jv; p4el accelerates uniformly about its center from 130 rpm to 280q;8 s yxd74(7vtl 5zlryj. tap rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius clq+e7 (xl nfu3)l1qlln3fu1$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard 9u(ri.qtuu43zr8b5hqz x* v g 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 rvgb* zz t3hqq8 uur x4.5(ui9to 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 uniformly from rest tob 2w2v:r n-kez 15,000 rpm in 220 s. Through how many revolb:v ne2z2k-w rutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpi6mkz./4r 5aitw;y hf m 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 ofx6yec4, c4dt ;h4e szn flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before r c e,44y;4n czhxedt6seaching 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 dia ys7rm9ghbe.: meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wi7s r:m.b9eg hyll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is ym4gk zfdl y;sqj+6 ;hq: u+/srunning at 850rev/min It turns 1500 revolutions before it jqq /zh4+klsgyu+ d;;6m:sfycomes 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 reduakzq /htlv /er++rkvk8-y9al x6k6 +wces its speed uniformly from 95km/h to 45km/h The tir/ z6ke+ a6k/9+rw vq-ktr8kly+v alx hes have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its /wn ax*s)cv-a.+u tuur.6j un speed uniformly from 9v-uawuu unt.n j.a+rc) 6*s/x5km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all ddwi1h, +w1:bbbubbl es (02;xaxm8oher weight on each pedal when climbing a hill. The pedalo 0bmd;wu8esxw+(: x2dbh,i1bbb a1 ls rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 kc)xx2o2fi-i sot5f- N on the end of a door 74 cm wide. What is the magnitude of the2f2-- cxtx ooi fis)k5 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.we+ b jjnf5mf;0, fk2l 8–39. Assume that a friction torqw2nk,bff0+;j fm5j e lue of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a cdtw(vz(kh o:7brl :+ massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position ank( :(dzwbch7:+ol rt vd then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a k5 ilut xye20g j7;:lrtorque of 38057j2 ;tluk: xgierly $m \cdot N$ 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 radius 0.648 m whtg/5fdt 1e7i.im-ud yen the axis of rotation is through i dmty.et/id5uif 1 7g-ts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam0 epa*r. cjgwgg,.p*w eter. The rim and tire haw * pj g..pr,*0gceagwve a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram balb zyrb5 ;u15ajl on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 b5y 5aujzbr1; m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potf,x,q8h fo z*xter’s wheel rotating at constant angular speed (Fig. 8–42). The fr q,fzh*xx,of8 iction 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 of the array of pczrj;d9o 8 qk9;vtui3(( ijk gaczi.3 oint objects shown in Fig. 8–43 aboutkizuo.cqac r3 vdk8 t;(z93jiijg;(9
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total massjz+6 4261rqej/*p9euxh*g ie j-hnk ffy iq- i 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 ch)qez .ik: :guorrect rate, engineeqz .hu)ki:e :grs 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 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylind w7r7w3tf6j bsa:6zxwer with a radius of 8.50 cm and a mass of 0.580 kg. C73f:j66 r ww txwaz7sbalculate
(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, accelerating6q 7oj r2x t805pso 5isd*c2es3osh (wmj8jac 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 tange 6y +o4jpgv3jtps2-f) /nwf4 dx lmm.antially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass oonmgtyd6jp sl 4m)4j.3 fw2/+fxa-vpf 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 10,30tz+sktf oh57 30 rpm is shut off and is eventually brought uniformly to r 3zkft+5o hs7test by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at w4j1;( 83kd3w tvis2udud ctz7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by theng -0tv4* oapg action of the forearm, which rotates about the elbow jgotg 0ap4*nv-oint under 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 Fi 6mt(3 n*njc1j4g outfg. 8–4tfmt*4 c3u 61jn(n jgo6.
(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 masses, 8g+iz *mmnzd;eyrm x9t 5x7)pem o bdq;8ir6/ $m_1$ 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 fu .sirq(s8i:kk ll turns (revolus8ski(k .:qr itions) 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 moment of inertif* (8- c8def, t7 //yc7wmkqeb ybh; )jihtcmea 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 torqu)u4w q3tiflq/s,*y:z+)nj q+y h+pukz pf xd)e of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls withoef 5in qpk1+ 5m:yr36id s5gkkut slipping down a lane at y+gpe56r3d 5 15kmnikq si:kf3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sp iu)g eh ,v)l/r7vro8um of twohel)pg)r 7ir/v8 v,u o 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 kg and a radius of 7.50 mj,9ft 0x0b-oo a pm*go. How much net work is required to accelerate it fromooat9 g0*mpxbj fo -,0 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)rw+ wcv/d) ex.0 cm and mass 1.80 kg starts from rest and rolls without slipping dox)) w/wrevc d+wn 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 balance1hewk;e3. bz/8csejj gl 8 7vwd 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 ground? [Hint: Use eg; 1s7j8cwlbeh3z/8.vwejk conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating;sz qhcb,76ed-:/cys9 xt z er on the end of a thin string 7ycxsbh d:rqsz ec/;z 6-9 et,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 cylindrical95qcz, sqqkvu lyh 2j-:1clorkg6 yj,a)8f9 m grinding wheel of radius 18 cm when rot1) f2ojk,l5- ch9qgyazs8j qq6uyr,v:mlk9c ating 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, handln3xi e1oh50av t;w ,vs at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his av 0;x we5io,vah3n l1trms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shodb, 3m l.kvfss09ug *lwn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the ldkv3 ,0s bm9fslu.*gtuck 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 from an iz pk4 2dyfcpuh (2zc-* ;.lntu6;jyernitial rate of 1.0 rev every 2.0 s to a final rat l;czpe4t2-yy(z.hcuu2r *nd j kp6f;e 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 ax5y.wu rj+x 0r n,l is.aw-r)hris 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, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clayau0yr.,l)r. is rj-r5 hx wnw+ sticks to it?    $rev/s$

  (a) What is the angular momen/*7 .wuj.vk 4 vspjj;etuahd:tum 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 Ea w5/l;d:jafb irth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is drok. 48ud-o yolgmr1 gy:pped onto an identicalgo: 8kyd4r1 gyu. l-om disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsh ,dvm24cw g(h 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 +5c6h7;3o9m jfi kqw+axbvo2zhgq k : stands at the center of a rotating merry-go-round platform of radius 3.0 m and mqbc2kagq 5 owjo+h i7h fm+6:9zkx;3voment 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 isnqy +4nuls-h 7/ kdf.vd,cwg3 rotating freely with an angular velocity of 0./vc,h.ns-kgd wfy+ 7 n4qdul38 $rad/s$ 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 f.e5 w2fuw(g*p3ai q cdwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost i(g f25 w au*fq3cpew.mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exces g ckm;zlz2bk5:ci 1a-s 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 k7t1qauw2 7lf$ 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 -z-ssf7ua qqfs-n5l. without friction. The moment of inertia of the person plus the platfoau5 zs- s 7-fnsf-lqq.rm 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-ronw9m9iz)n;juw rq; /t5 zri ,ound turntable thoiw;zr 5,w/uzj9nt)mi9q ;nr at 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 grou (a1ono4yrrsyv50wlnhd6 w9 hw(z . 6hnd with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behw .w0 9hnv1w( h6yorlh ns y65o4adzr(ind 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 sucgdw2m4qv+a1q 5p(n3n)r kl ol h that the same side always faces the Earth. Determine the ratio ofqw3gl(+1n o2qkldarvnpm 54) the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest b0p9qf, h9b z7mfvpx* 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( irxc3xovf3v,aj z: 7 shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torv 7jrix,:3 f3v ozacx(que delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylw j 1xw1mst 2+(yd)mfcindrical 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 thdmx(w fcsm1 +j wt)y12e 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 speed of the rearecrc n36.)rz 0yee ej: 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, 8)w1zb t: jdlgow54 lnbut 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 the sz:8 g 51 wl4)jtwnldbotar is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy ;1sa6 c/smivumg -ls *stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel ofmv/c ; g*a-sulss1m6i 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 lp( qp ai0bbl,ej*r/xz-f1 u4an $H_2O$ 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 on1ffqp75opt * smtd0. t 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 le 5 zf6szpziid6.v+q eh1+y )opngth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wa d+z.qzs 6z i5ov6fiep)p+yh1ll, 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 Fig. 8–21g.]

A wheel of mass M has radius R.nixzs(amm8 *vbks ,(vm t yp,o 7)l2m; gqmo.) It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stg msmv2(; 8.,)t)bm zmpq lns7k,o m(iyox*a vep against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wsvm4 dqekzww5sx622 ed 6u ;uat )b*e7ith a radius The 2 dsq6 xu6zwwm2tesv;)b*7 uk5ee d a4forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling oj h 34;ocdb2 b dv/ztai;sxzzhn9 :-,df length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What 3z:jdhv xt;i-49n ad,h;cbdzo/ b2 zsis 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(t sc+kl 9v8igmoh9 ;h her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the an ol;kt 8vs9cmi(g +h9hgular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pclefvj*o .p+ 0(*k*/ xbjrysnlates, of mass*s. vjxklr eon*/(fpb*y jc0+ $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 akjgd1x4 y-cj; y72kkb nd radius r rolls along the looped rough track of Fig. 8–58. What is the minimum valu -b2yk1cj k7gdj x4y;ke 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 assvm. o6aq 7l km0k/om:kume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducekyov53i0/alelv q s)*s its speed uniformly from 90km/va5* /)ov lels 0qk3iyh to 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