A bicycle odometer (whk5 vi5*hqg ka9ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wik h9q5i*kv5agth 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poinv++hnc;lhf*lkz -+bxt 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 would the magnitud+f*+hvxl +bklzn ;c-he of either component of linear acceleration change?

Could a nonrigid body yc; ,b .0hjw(t4-uw2hs o f+6obd nnkzbe described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explah9 p5uqr5,lcl-m,2qstb os,min.

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

Why is it more difficult to do a sit-up with your hands behind your head than wa,, zu:p) rxswhen your arms are stretched out in front z):s,u pa xr,wof you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thrg6nay gu628ufajb b8-h al2.x r 0+igvee at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or ag6 0xuy fibnh82a r2vl aug-g86+.a bj large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs afnn;. vzmnd*w 7;,d*wz; psggf26cnwith flesh and muscle concentrated high, close tonp2g*;w;nnzfcaf,vgn .ws * 7d;z 6dm 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 long, narrow mae1+ 6jo 4r.rrs f*mw0mz;jdbeam?

If the net force on a system is zb(t ; 1tqnny-zero, is the net torque also zero? If the net torque on a system is zerz;bn t nty1-(qo, is the net force zero?

Two inclines have th*lexi-sy 4zq8b0jx* mut o ;:me same height but make different angles with the horizontal. The same steel ball is rolled down zm y0j ox*4me8xbsql*-t;:i u each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline0g;09 bsqxlzx(aq 9 kh. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Whibx0k0a(;qqz9g lsh9 x ch has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius andf.sorl xqx q0:a q+ wmcof aw-+(:h-i2 the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has them2q-q(.:o+ swcq fr+ax0h a-lx :oiw f greater rotational KE?

We claim that momentum and angular momentum are cons+aozio /r t/4served. Yet most moving or rotating objects ev +/szt/o4 raioentually slow down and stop. Explain.

If there were a great migration of people toward ti0y;say//wr2 yl/q dh8yuy;yhe Earth’s equator, how would this affect the leqywyy; 8/yry syi/ 0a/;2 hudlngth of the day?

Can the diver of Fig. 8–29 db/lrf1vs4ejw m d,v*/o a somersault without having any initial rotation when s v,*/ jbvs/lmerdw4 1fhe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centqp4 vmb,, 1aecer of mass isqm, ap 1v,c4eb $\frac{1}{2}WR^2$ (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 hr2ixcs2*x7+l*3xcsga ..na cy kl*yrolding a 2-kg mass in each outstretched hand. If you suddenly drop the masses,*2 x3.i*rrs+kccagysc 2 ayx.n ll*7x will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevwl r7it 0y.ax.n2/fc oer, one is hollow and the other is solid. Desiw2on7.t.lx fca0y/r cribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In w1ouuze52)i0q tmhc 4 qhat 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 increasing or de5ouq 2 mhetzq)4 0ui1ccreasing?

Suppose you are standing on t8tnx4h + s)2e1tz mlkphe edge of a large freely rotating turntable. What happens+metlphsx2t)8n 4z1k if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qhti.hhvc42eu6 8 . fzcuickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips andh4 i.hcfvuc. z6t2eh 8 legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of 6 t4p4rg e2 /e nfpbpjafr3-u,angular momentum, discuss why a helicopter must have more than onau r,f4ben 2 pg/4j63pptfer-e rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following w(6alrpyq**ma 1y: ljangles 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 Earth9f, jy35vp 23d4a(hiqixjtped: rvu: because of an amazing coincidence. Calculate, using the information inside the Front Cover, the 3rjfx4a 9pq vj3 iuh5 pi(:dt2,e: ydvangular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, m0)emw k,v)h:yictnf6oqjv0k ri* 1.380,000 km from Earth. The beam diverges atoktvm:i*w0 , fi )yhce10v.rn)m 6jqk an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 d:vna ))jpf 9o. fdb-em8ys v4btx kju, :/zf4rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as xn bfpvzays)ujvkefdm8,9d jb f.:4: -t/o) 4the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child.o4hkez 2bv4m*esg- v 9/pibq 9n5mju xn*fw 3; If the ball makes 15.0 revolutions, what is its diameter*3m ;vp4hf-bk2e vn nsg/ mzo9jux 4be5*9wi q?    m

  A bicycle with tires 68 cm in diameter travels noi+x6 my(3j(j7gl3s;qff k*zj )g8hkn wbq2 8.0 km. How many revolutions do the wheels make7 j 3iqofk3 ;f 86lhgg y(zxkq+w)* (mbs2jnjn?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C:9 hvca3mbfgu671 mo ey7 fqe0alculate its angular ve eh 79b6yv0qf1m7f ca3m g:uoelocity 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, fjc vesu5 a7/qbr9xxf)g h(ga6,9uq one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m f, 7f / avf5hb6urjecx9,ag(s)qgq9x urom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sie v pus cl ;f9i:g:yls52cr-z0e48t cun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointx foj81e s)nn,tqdi-xgn 8u; 8
(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 from the apun;y2 us2oq c x.x2(ol:;omy xis of rotation is to experience an acceleration p x2.xo(;co suouy;m2qlyn2: of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml3xis3 2(va nltc6yp g5y about its center from 130 rpm to 280 rpm in 4.0 s. Determinapc(x t3 5s6 iygn2vl3e
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu ffw5zyc+tpn::21 gacs $R_1$ 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 A:evq : 6:3cm;om1n rnweq.tfspollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy rmen qw .cv: 1ft::3o; me6sqnevenly. 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 rest to 15,000 rpm in 220 s. Thtb1fv)u ks:hs6 2 5)o5bit eau0*ddkurough how many rsd 2akuu)6eto f ud:v * s0b5b5ki1t)hevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calc,3x+ p xzfkbt4ulate
(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 flywo4 + so ykp7*t*;0yrkgzali0 ing highspeed jets in a whirling “human centrifuge,” whichywzkkpg4 a*r7y;0s0lo*oi t+ takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uny1i (s xvokkjk d,8y2g qipj,5b 310qyiformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the ykqd s85 jiipb3gjk01oy,xv,2(q 1kyedge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/minsn -kr (f;1pcv It turns 1500 revolutc( nf1-v pkrs;ions 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 its sq j*d9+y*og 14zh:tkm8r en+ zi s3jwgpeed uniformly from 95km/h to 45km/h The tg jm*r84 zdhzoijt3 n*1e+:q9kyw+gsires 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 speed u-au vx5htvim +t ,c)uos*;mmtn;/l) uniformly from 95kvt;m ot);hau vmm5)suu l/nc+t-x*,im/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 her weight on th w7d,-ltynt1;6rqk each pedal when climbing a hill. The pedals rotate in a circle of radius qy-wrk7,td 6nt 1;lth 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 74l2zwo k*sg ar o,5(-bb+ado*c cm wide. What is the magnitude of the torque if the force is e (osb5 ,aw2bzcda o-gklo +r**xerted
(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 a,n: ke )+krl d)hyl1uhbout the axle of the wheel shown in Fig. 8–39. Assume that a frice + ,u)hkh :dy1lrkn)ltion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of massb* zzg+ 0lj.sj m, are attached 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 released. Cg*j 0b.zjl zs+alculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an enginepuus5wl 7 c1v5 require tightening to a torque of 38 uulc5wvp7s51 $m \cdot N$ 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 ineri x+qdkz,jgtgdp j w k534,.9xc.b1 xrtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotatq9k j5.g px1b4, .cdwz,+xgrx kdj it3ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of p;9m9vs,aq2 trv)rvv a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The vr;tpvqrvs 9am ,9v)2 mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a ho h-zna cw7hl2eli:kg1, r 4s)wrizontal circle of radius 1.2 m. Czhlkwc ra7 4i1,2h- le):snw galculate
(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 rotating at constant a(9y.lcq jfu,zi 2k;nsngular speed (Fig. 8–42). The friction force between her hands and thel;isz,9 q.f 2jcy n(uk 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 4e kc ,+7djgmnky6 j1earray of point objects shown in Fig. 8–43 abk,j eje7gnc 14m +kd6yout
(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 whosei m c878x*zon7bilug7 total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at mbtg/ kq)7+tyh14cz hthe correct rate, engineers fire four tanghm)h k+ t1tzby7c 4q/gential 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 cylinder with a radius of 8.50 -, h*lv nbkc ryz3v*4hcm and a mass of 0.580 kgnv ky,*3*v br4 lhcz-h. 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 swings5am o( 2qdcrb; a bat, accelerating 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 i3: lahcs:6t.2xl wkw bs 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 btl::laxs 23c 6 wk.hwwith a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at+9jkz/b+i9r 51qolwhcd (bpdx 8, ayi 10,300 rpm is shut off and is eventually brought uniformly to rest by lr5b8j+/i xdqzhp akoc yb(i +,dw919 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 acceolcl,+ 5afyhb wwmxsn2fc/ 7m b.h*-7 lerates 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 solely by the acti1j:;a, 7jpwl5zvakkib)n -t bvmb*w2 on of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–5z;m *lwjnjp ,vai)wkt2k7b - b: va1b45. 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 conk1mt b 2muhfhcab,.6 *:mc8 vksidered a long thin rod, as shown in Fig. 8–4, h:81k*hvbct famcm 2k bum.66.
(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 cqp(lonq5 ;f)konsists 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 fu7ng ex4a)tw e1ll turns (revolutions) and releases it at a speed ta7)wgnxe1e 4 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 ; vw94uzqbf )v$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine m rrrsls;:(-ubgc j) a1vl,k9 develops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rup 777mx b w he8a;ggk4,eedu4olls without slipping down a lane a,7km;edw7b4e4u uga7 8pxhget 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the 6zx; br7lkh b a*fc0d)Earth with respect to the Sun as the sum of two termsa)z *xk7bf6 drh 0lb;c,
(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 radiusp )tfq1yu (0vt of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.()ty v upqf0t100 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cmaw;lwxhl65+ 3 xmrbe1;l psk 8 and mass 1.80 kg starts from rest and rolls without s m5l6awxe;p1r8kb3+ hx l; wlslipping 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 isomm f7s3w n7( l(vk)gd balanced vertically on its tip. It starts to fall and its msf(7nlw)mk3vo( 7d glower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on t7gvql -fd4 razdv 2 7evf;r82nyn36 / jako4qthe end of a thin string in a circle of radiq4r 4n -v takd2yq;of3z7 v/nv2d7gfa j8 elr6us 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 cylindakd9km4 yv2llx0ds ufc e26(3rical grinding wheel of radu(0dflk2lyxa4s2m3c v d9 6e kius 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 5hfkjcfhzeeh9d8l62 ) qq0bh,5i ,oatcag, 4, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decredh,h2tj c6lc i ,5zfgb459qq)ef0ha8 h ,aoe kases 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. 8–2f v0-qebikwa6* oz; r7nt/ql9 9) 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 rotations in 1.5 s when in the tuck posi;n 9ekwlzb-*0f iq o vaqr6/7ttion, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin r(k 0mi9u 2)qw4hhiv haotation rate from an initial rate of 1.0 rev eve 2hum0v h akhq4(i)9iwry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a u4gx)ycf)foell0 hua)9 4nr +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-kunx o)l0y+fhu a))gl44f cr9eg 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 clay sticks to it?    $rev/s$

  (a) What is the angular mo/ool / 8qj+wytjw : )1*hm-jf,glvf vbmentum 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 momentu)rrvn,fhn .h p6 6n kaztk2y4)m 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 of inertia I is dropped onto an identxv(pg7 b;o3q bical disk rotating at apx;oq bg(v73 bngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2. yoh6 2e7i6ahk4 $rev/s$ 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 rws d: wd;n43npw1 yd0k1o;x mcyxvj62otating merry-go-round platform of radius 3.0 m and moment 4;ywxdxd3pv26mwnks: d 10 n1cy;w ojof 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-gz 3 mx4rxp7 ay48vldhirl+f 12o-round is rotating freely with an angular velocity of 0.8 4lxmp82zf3x v y1h7r4 r+dlia $rad/s$ 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 tl-7otob deg /.ir (deu1d- x*into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing r 7x .edrouledt io- -/*dgtb(1adius. Assuming the lost 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 excess of 120 5e wvtee9u p85$km/h$ 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 o0 7 al2l1ue3cxho44b :tw fmlv0,mro$ 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 r me-10/ia( hzyu qkbp5est, that can rotate freely without friction. The moment of inertia ob0ym-a/ uhipk1(e q 5zf the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edglf/c7krz n2j1e of a 6.5-m diameter merry-go-round turntable that is mounted on frictf1ln j7/z2rkc ionless 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 o* 3so,m89k rvrl,yln ground 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 behind the person without slipping. What len o 9kl3m8*lv, nrosry,gth of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that-s, ;mppn *b8(ryso pa the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (apr,pn b(*-8 msys;poabout 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 rw3clg *h y;m1pest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylin9l k )ht 9p8a.6wqjqmrder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval kt6pq)9lj. 89 ahrmqw at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli(gyk a5 puw i:.3*m2bf2 wxwbendrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1:ke wwga2wuy i53f2*(bm bx .p.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 -y.s blvq ,y1 -ca3exaof 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 dq+ra)2vd:i1 efs; y vof our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that vd i+)fy:v;ds aq21rethe lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lowdpr:)j2o6fc2ts 0kd k -pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flyw sfp6o:)jkt 22cdd0rkheel 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 5i 29*du9af o*uirr crk68oao an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling mp,) 8czz:s igeo7f 0fr2.uy aon a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i 2wsl6ay246 sod1vzps .e., we ignore friction) about a hinge attached to a wall,6z 14ydov ps22s6aw sl 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. It is standing vertically on the ;:km* 9ugird * vb-wnkbwk;,o floor, and we want to exert a horizontal force F at its axle so that it v ; dww bok;**ikgnbku 9m:-r,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 fla0pohw 3f;a- fqt road is making a turn with a radius T 03wh-pa ofq;fhe forces 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 slunhz)2/9ajdanuckh1xy+ p4 1dp,+3 sydys :e ing of 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 in /n aju1y+s+1ue yc:p42,h3a xpkd9dz shd)ys the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as avv90h far +4ye+a(g*cep) i dg solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for 0+ )fcv4 id9+yer(* ahgvp egaa spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly z qu6qi88qp 17vnm/9rrb z2k xwhich consists of two cylindrical plates, of ma r9xzqk81p6 /zqmi7bu2 8 nrvqss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radis twzc1bfple9h2,uw/hv+4 dj /ny b44us r rolls along the looped rough track of Fig. 8–58. 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 leavcy9ew,tz uw fljpb/v414d nhs+/4b 2 hing the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assus/+i0tqr t9 8 z9zfge ra8my4vme $r\ll R$ h =    (R-r)

  The tires of a car makexv9i2yyab:jro , 3cv) 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the vv)rcx29y b yaio:3j ,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