A bicycle odometer (whi-mslg7(m ;pa0ch y7f 1z hfem/ch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bfs01;7fceh l z7amymgm-h(/ picycle with 24-inch wheels?

Suppose a disk rotates at constq)+vvs ) x0rzawj- a cdf0yfip.s 5+6bant angular velocity. Does a point on the rim have radx)v if5 dpfa0w+6+ )qyzbjc0vr.a- ssial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a si s 3ypm-fl wz**zp:fw bchc3;6ngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forc kw, *xtqbx7a/e? 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*7h.lemon/wfnh t 3)p with your hands behind your head than when your arms are stph.fwm)t7nl/ o 3en *hretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets e a:rbou:29 jd6k h:dvh.jru dgd2l98at the rear wheel and three at the pedal cranks. Iru2h:roe 2 v j:bu djl8g69had9d k.d:n which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have sls.r/wu ck*ac1(h*jciy mj -/zender lower legs with flesh and muscle concentrated high, clo /skjzc.cjyh**wia -m(1cu r/ se to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lo82/) fy4nlitr(6pzs .ycp2 x z+j daxang, narrow beam?

If the net force on a system is zero, is the net torque also ze q*dfgx+rq9 r-ro? If the net torque on a system is zero, is+ gx qr-d9r*fq the net force zero?

Two inclines have the sw q4gjr-p;nu yxhjq2+;sz,.i ame height but make different angles with the horizontal. The same steel ball is rolled down u;4+gn-wi; yh qs2j xqpzr.,jeach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously 5c8eo jkb5 o)khf/wm4start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which rea/bo)mkwhk4 58cfo e5jches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. nnqr13fb h*r8blvq (h0z1 lg7They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic b1f07qbrqn hv1 n8 3lzr hl*g(energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movp:ubj) em bcs*3s:vg; ing or rotating objects eventually slow d)m 3cgje:; *suvp bs:bown and stop. Explain.

If there were a great migration of pe*w6aei1 qc 0hfople toward the Earth’s equator, how would this af6icwq*e hfa0 1fect the length of the day?

Can the diver of Fig. 8i v*7xb qr:mw2–29 do a somersault without having any initial rotation when she leaves br wviq72x*m: the board?

The moment of inertia of a rotating solid disk about an la f+4aj-: 0tc tvewl m.,y7fuaxis through its center of mass-e afl7w:c +,ta.luvjtmy 4f 0 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--vs9mjv 9-u deno 5.hqkg mass in each outstretched hand. If you suddenly drop the masses, will your angular velociev-9-djmnvs ou9qh5 . ty increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow an+h,)g 3 qea7)lhlad ewd the other is solid. Des+aqa wehd3)gl7le), h cribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizoco dm5zt4ui.6 3bj4 bsrbm+5bntal axle points west. In what direction is the linear velocitym4zitbo3 b6 .+b5sum djb c45r 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 standing on the edge of a large freely rotating turntable1i kisus) :hf t-4/4zm xyj5cs. What happens if you waljzs-x1iu t)y f/im44 scks5 h:k toward the center?

A shortstop may leap into the air to catch a ball and thjul+)v fbzmm7i ,p:8 qkwfv8(row it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips anvjmulvb+(8 7m:qipfw kz8 f,)d legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mt7(8*byu lqo1o8lmf z m.fy b1omentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the bo(m1uq.y8 lo* 8mffztyl 17 bsecond propeller can operate to keep the helicopter stable.

  Express the following angles vi6eprjo,gdx+b 3u/ )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 coincidp- ox s dy0a8j4p3f9wdence. Calculate, using the information insidj4psxa9oy3d wd 0- fp8e the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at nq /.m ne)vmx5ejo- vu( b+zhpcy6;h: the Moon, 380,000 km from Earth. The beam diverges at an a: /v-cz(.; jhemu65ovn nqhebm px )y+ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderlzitx*nniz-+fy 8* ,r t nk0n0;vi;dc rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blarzlxn+00f*,ictyi ;z i8tn;k-vd * nndes slow 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 flo--tkdp 0,b zmuor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its,k0z-b t dmu-p diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manyqo/n u4 0lkcw:*rcae( revolutions do the wheels make?:na/4lrc*0 kw (eqo cu    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anguu7).4 lr0 y sme+phg3g btlij6lar v.l thg3mu4 +yj0re7)ip6slg belocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. hi):n: 2uo tvo) w .lmqu2+yej8–38). (a) What is the linear speed of a child seated 1.2 m from the ce:qvu+t2: e)y)jm. nohu 2 wlointer?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around t ep.gl umv670wgqh; f+he Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point oq:,g r,dypa1
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm fromg2uf5z qq+j9kdsw h3 8 the axis of rotation is to experien8f k3qgjz hd2s9+5qwuce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlyw5c0w wu 3zy1szd-o2 0y7bmfh about its center from 130 rpm to 280 rpmd1 ouw75y0fym-swh23wc z z 0b 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 3 dh;aak5e.wq$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecra g 3 os+ thd+l6p7t18ptgkf; vo+9vm ipt7s;lmft put themselves into a slow rotation to distribu tt77sio;+p dgmp;l6flvgv9p1+m38hto+stk te 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 uniformly from o6 q,yo)sd:aa ta9uq6rest to 15,000 rpm in 220 s. Through how many revqoasd),9: 6 au qa6tyoolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpt6e b -r89wuhcm 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;w)uz t+5odm duap1d 6 flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed. ;d+mz)d 1o5u pw6atdu
(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;en*fho( 6sr -wmazn7 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel(anh wzrefnm ;*o7-6s have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It hhc*r-usemuf5*+o;iwpk *dcb,9g h7 turns 1500 revolutions -ckm9f gpw*;,5hhibhrde + *us7uo*c 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 revolutio8azg ixspj) 77ns as the car reduces its speed uniformly from 95km/h to 45km/h The tires haveg7ps)xaj 78zi 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 revoluepj -)7q t8to(bq 1a) fp;rqjitions as the car reduces its speed uniformly from 95km/h1 qafi78e q-tppb))( ojtrq ;j 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 her weight on each ped imwig1m ,:ec7sw w9rmy/c6((efhb +mal when climbing a hill. The peda6 ic ,mm/h :(m7c ye+g1sw9b(rwewfmils rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 l j +89alzl);hsvum9ccm wide. What is the magnitude of the toa8;jvll9lch) u+sm9z rque 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 ax 4ut k*38hrsgwle of the wheel shown in Fig. 8–39. Assume that a friction to kwr3*ht84g usrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a *xz(d:,ikd6i aqe)u dg,a j jgj88z9jttached to 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 releaseizjegga)z dd ja8 ,:x6 ui9(,jk8*q djd. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder heag2 t 5pfp)c,nld of an engine require tightening to a torque of 38 l2gc n5fp t)p,$m \cdot N$ 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 ra7x(st0p nfes5dius 0.648 m when the axis of rotation is throughx 7ept5sns( 0f its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thezmxs*,9be1.n. fs8n9 lnc7 ccx wbk3l rim and tire have a combined mass of 1.25 kg. The mas*n,mnx8f b.k9nc.se cllx cz 379 w1bss of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,u d7wfsc4;,:: nfw kfc light rod is rotated in a horizontal circle of radius 1.2 m. Calcula,wsccn;: 7dff fwku :4te
(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’spd) j -6 (jeaebikwh21 wheel rotating at constant angular speed (Fig. 8–42). The frictio(-j 2wj1k eibhdea)6pn 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 arra ,w7l-xscgw.h *5fapm y of point objects shown in Fig. 8–43 about -pcw75glw.x*af,s mh
(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 tj, 3+c7s3nesw,b0 e/w irflz hotal mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corred, ;lswwkpmi4wi 0;9ib j8 (gect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sa0dmw( ekww;, ;i4g 8ilijpsb 9tellite 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 cylinder with a radius of 8.50 cm and a mass l +au,gpl(a.b/j xh+g of 0.580 kg. Calculat /gb xlj+h.aa ,u+l(pge
(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, accelerajp48 t f1prs 297- n6-iyr uhyojc8bwfting it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is abl 5.y ogy :olgu8t0dc1ne 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 thtcn.yg0o 8y1:o ugdl5e acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut 4gf5i3:g 6c5nwmr oej8ox j:1chkri5off and is eventually brought unif gwg j ocrcj53xr6 he5:48 :nmf5io1ikormly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a h)i)du mf9m x23.6-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 solely by the action of the forearmo r/cs ku*m4+b, which rotates about the elbow joint under the action of the triceps muscle,ou/br cm+* s4k 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 h a; )jj:8kxdqthin rod, as shown in Fig. 8–46.ajh)x;8 qj:k d
(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 consiscd1uj )lq*gu6jisnn*q* 5u3h ts of 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 tc.pb q l1s q(m0;moc8zurns (revolutions) and releases it at a specqbmm l( 018spcqz o;.ed 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 ofi.yje0) g;p xl $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque o7+ rx6d)al pdvf 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 ai xeah2rg g5;, k0cwv9nd radius 9.0 cm rolls without slipping down a lane wx,kh 5gar9 0c2vgie;at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy2ehsn: +dzdl 2f0(/ td*nrr ev of the Earth with respect to the Sun as the sum of two terme0dl/n rr*dt2h (s2fe+ znd:vs,
(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 a falzdq .63l36e a -mp 8r05hw5krdjdfnd a radius of 7.50 m. How much net work .ep3 az36dhqlrk6 ld0jf8w-ra55d fm is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls w -irzo7bn; u1bithout slipping do znorb i1u;b7-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 balanced vertically n69q5lv agj3e on its tip. It starts to fall and its lower end does not slip. What will be the slnj5 vaegq369 peed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotat:tpqlb* a7 y-5d;eyr o/bl )cring on the end of a thin string in a circle of radius 1.10 m at an angc7elo ;r*y:/5lty d-b)pbaq rular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylimqik1wwh )8o 3y.lvec1lxz9v . xvf5)ndrical grinding wheel of radius 18 cm when rotating at 1500 rpm? wv e5vcmyovfl)q.x 8x3)l11kwh.i z9   $kg \cdot m^2$
(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 rotating at 1d.zlpnat2l qm1 2d5t a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, themt2 1zal.d 5lntp 2qd1 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 Fig. 8–29) can reduce her moment oe rad wj w*e-46as026sdz7wwrf inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular s4e d7s6zaw0ad6w rwj2 -*se wrpeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ratby8ey dn7+uck ,,tfr r8(ilx3e from an initial rate of 1.0 reyl d( in fyr7te,8k 3xc+,r8ubv every 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 ro.q,s bckafrw):ix(5*(lypvxtating around a vertical 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 chbsa(5 rxf( y:k*ilxc,pv .) qwunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angulasih/ih y75 en3r momentum 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 ofuo*l gm+/hd r5ebu f62 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 ohcf q(:uifco:mw3 : uj l0x9.z0hxda4f moment of inertia I is dropped onto an identical disk rotatinwfmil zchxd03u: c(jf ou qxh.:90a 4:g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atw w0fj*c4-wyo 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 t cb8gx,42d culhe center of a rotating merry-go-round platform of radius 3.0 m and momen4 uc2bcgxld,8 t 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 hxo5,aka aq:0 rotating freely with an angular velocity of 0aqok a,0: hx5a.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losi2tvwbu*2oie0y2po g bxbf j-eh,g/k ;27b3yong 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/0p t i ujhbe7yg*,ofwb2x;2 y32bo- vkgoeb 2 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 excess of 1odzzj4 l*.vgqyy947p20 $km/h$ 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 m+ ys(u, t*dj*agyk(gud-u),igzsy;$ 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 withczp.h +z )ze :yan3vdz1;-r;cfkca(m8 .xxq kout friction. The moment of inertia of the person plus the plat.ax;yhd)8zqavr( ;z3+fcp.k ex:cz c 1mk -n zform 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 dizyouodj .789pameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ofjop8 y d.zou79 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of tf/i ;mfwmqkm5+:k4p an9 pe+qhe rope lying on the top edge o qw 9/4pfqf+mk5emkai+m p;:nf the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces th0a j-qp 9cp i6jkpn.9re Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbi 6p0jqa9-p. cik 9rjnpting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates f io,f.ejmhxq v3ua7+6 rom rest 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 ofa:fpv/tj4 p(ob 8xur 8tc.(nuu4k e;j 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. Calcula( t o8/p;4pj:jeubkvfr (8 .uxacu4ntte the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two so .bni3sls -lf(lid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrib s lni.s(f-l3cal 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 i0us c+j/: uwlfs 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 tqva23txjq 7i6 imes 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 praqx v 67q2ji3tocess, 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-pollutilw 5zmiwv/d:nw *e, /bon automobile is for it to use energy stored in a heavy rotating flywheel. Szw*w5b/wn,iv e d: m/luppose 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 anbtkcb sd 5i :79wg8w4d $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 pa3 ymd(6) zl;5kbhrp ,s48en8ewhwx $m/s$ 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 mass96ejs*2w rsq33ko rzt M and length L can pivot 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 9e3 trq3s o2krw* j6zsshown. [Hint: See Fig. 8–21g.]

A wheel of mass M haiges9bie7 e fi7u97 2us radius R. 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 step against which it rests (Fig. 8–55). The step has height h, wherfs ibu79eie7 9eu7g2i e h

  A bicyclist traveling with /ec4:eq zt4bg.pf w/m speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cycl /z wfqe t4:/m4e.cbpgist 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 of length 1.5 m and begins whiz)*2svpyv)l erling the rock in a nearly horizontal circle above 2z sv) pey*vl)his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body a02xs9938a n ni8tr *dp ptq o*wpgkc4ys a solid cylinder and her arms as thin rods, making reasonable estimates for the dimen* *9 t i89x ogcanpy3w402r8pdtsq kpnsions. 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 consistsvh *r7qfm +g+g of two cylindrical plates, of m7qrf*gh++g m vass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiy 7ah wh0vp1 ztls3j-(4nrds( us r rolls along the looped rough track of Fig. 8–58.rh1s0pa4v-dn slw jy(t 7(zh3 What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buttnm/n4up .f ;5) cxfrh do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th6 jb-py:jc )0dw2yavy e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular 6)d20yy:-pvj byawcj 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