A bicycle odometer (which measures dist5,2w 6r ky(4u9:wosfo cggzicance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it cyuiwfg(r :wgk o5, sz46 co92on a bicycle with 24-inch wheels?

Suppose a disk rotates at*ngkdx+q*d, pmb *y 9m w*3hmb constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity i*dhx9b*w gm3,*k*m+dp mbqynncreases 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 single value of the angub8d9 cc ujj 6x.b74erzlar velocity $\omega$ Explain.

Can a small force ever exert a greaterw l.lt0 *y04j karx2kofvo(7u 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-up wsmwmbl.74 efj+6sr8 oith your hands behind your head than when your arms are swm8mlsb4 e js .6+7ofrtretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whu6 5ufk r; 9(rlweicu0eel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a lar0ueuk(ilwuf 95; 6crrge 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 besm+f :obt 2,bhing able to 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 th+bh m2,o:t sbfis distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bea0yj5 jvo9uyyien) (oqm1; 3nlm?

If the net force on a system is zero,s)m56tm uw3)onb py/ r is the net torque also zero? If the net torqs /yu5)mw3p nmto)6rbue on a system is zero, is the net force zero?

Two inclines have the same heigh(maomrvp x59maa0c 17t but make different angles with the horizontal. The same steel ball i015ma x97rvmpma aoc(s rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from u9bw.-;*ft2vgnsimoy1 8s k krest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the b*wi k8.su;t n-2 mg kvfybo19sottom?

A sphere and a cylinder have the same rastsp0-ojhj ,i 9. 1(xbn -jqn+zn2zhqdius and 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 the greo -jn ,ztqshbsnhj01 .q-jzxn+ 2 (9piater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most m39dsoh/0 tq*uh-tq ) bv 7imlaoving or rotating objectlbqd/o0) hh9s- tt*uai3 7vqms eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how would8yiz5vylpim gnwv 21ba6 /f+3 this affect the length of the dap8l wv ymv1/afin3iz+by6g25y?

Can the diver of Fig. 8–29 do a somersault kmdn59; 7bvvb(a, yudwithout having any initial rotation when she lek 5;duba d7 (ymvb9,vnaves the board?

The moment of inertia of a rotating solid disk about an axis through its center 0b)mk r4o p-veo1ojh) of massk b-p j0)4orooh )emv1 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 mawfy9mjl*ugn -x*k .e( ar69avss in each outstretched hand. If you suddenly drop t w*(ruxa 6n9-em y*k9l fjga.vhe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the saya- 8m-*ad5xiz0sge wme mass. However, one is hollow and the other is solid. Describe an experiment tomd z 8 yeiasx5wg--0*a determine which is which.

The angular velocity of a wheel rotating o8 (+ycwarrv,)c 4ymr rn a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular amwa( y8cr)4v cr r+ry,cceleration 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 turntable. 7ae.je 4 b hwd2wuuarx8p683yWhat happens if you walk towardjwweyu76 8p xa4 .buea28drh3 the center?

A shortstop may leap into the air to catch a ball and throw it quickl )9 ,wkb50g: fy:djop t0d5za0 eyq7hrvovs6ky. As he throws the ball, the v oeoy:hrj , v6d0y5zksk qgfw5dpb90:0t) 7a 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). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helic14iik +7zfio,yy m7ob/f9ru mopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate tof4 z9,mmo iuikri7 +yy17bf o/ keep the helicopter stable.

  Express the following angl7g- v7;cu.e vkzr.7korz 8ksues 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 coincidence. Calcu9lnt-c 3xg j:2jn4ok2ibegv;late, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earthe3c 9v:ix-go4tg;2nj lkjn 2b.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divelb xjz c8s)/e) 3dr:6v5djdbr rges at an slzrb8dc5v6e)jr bx d:)3jd/angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When( clx2ydp9(n7m4ee1 kn bpm;n, 8xa qp the motor is turned off during operation, the b e pd x n1,aepm8 xkn7(q;y2n4p(lcm9blades 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 floor 3.5 m to anothg4 2v5/q6a7k 9cbi6ahzoxt pzm14utx er child. If the ball makes 15.0 revolutions, 7acq5t6oxhuz 12 m9xk4 4v iatb/z pg6what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How man7.k 3d0v jvvjsy revolutions do the wheels makv0vvdj j7 3.kse?    $rev$

  (a) A grinding wheel 0.35 m in yk5v.y xe/+0/aqf8z peelw/ ediameter rotates at 2500 rpm. Calculate its angular velocity 5zay.8f0 p e+ex/wvkl/ eyqe/ 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-r (b2l 1cp23d6l xyqecagq8n 9oound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seatl2ybnqpgld( o 23c9aec8 q 6x1ed 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 Earth (a) in it;ibh2- ivvz3vs orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sbbew6 h ;vtoo)lf0nxn,v54 :dpeed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a partici3yh -;)xz1gc(wku zo(a(b vlle 7.0 cm from the axis of rotation is to-b1;ugx)y( h(iz(lk3cwa zo v experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelvoq q9ivdgqj *,1r/)l erates uniformly about its center from 130 rpm to 280 rpm in 4.0 s.dli,/vgq1q)*r qvjo9 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 zphnny; 4*3yl $R_1$ 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 spacecraft put wq uv75 o(jjacu*u6e*riz, 21bc9nt zthemselves into a slow rotation to distrvn(*wco* ij r,5a c72q91jbtz6zu ueuibute 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 rest to 15,000 rpm in 22 yk jr1,(gjzmu;w4 /jn 0qf-gv (hg(es0 s. Through how many revolutions did it turn in thisg0/ nwm(jv ;zgk,jqs-yefj((4ruhg1 time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in ,5vftyfkw7;g,uj5f k , digg(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 flying highspeed j.jzwubc 8b+m( ets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reabm(j8 c+ub zw.ching 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 diame4fs *0we. v6lcoukkm c7h9tf-ter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of t7c*wlhm-t k. e k49cv6f fou0she wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 faz70q:jatig5ej 2 /qrevolutions before it comes to /qgt j0zeqa5 :fai27j 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 revokw/1ug /q-c9uf fhu:amnab,4 lutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ofhun-q 1gu/fbaw kfa c/:u, m94 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 li cieil5;sc/2*c cy0*h ko*i speed uniformly from 95km/h to 45km/h The tires ; ioch*ll5 ie 0c2*c*k/ycsiihave 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 g3jivf81ks)7ki, cq l each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.ci8g,31if)sj v7kl qk
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is tnt3f*seel 0(d he magnitude of the torque if the force is exeren dtf*30(lseted
(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 Fivafx3f1 k6w7 n5 q/ bveu7jm:rg. 8–39. Assume that a friction toa5 3/n mfb6jwf77 kv:rvex1 qurque 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 endsahklm-a5n5 hhxw 374qdi wg03 of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held l3h7mh05gd 3q 4w5aa- xnikwhin the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteninvrtcxr hc:ofmf;.nmc(a 1 e/2z+2 2w rg to a torque of 38 m2;2fa1tcxv zher: fwm/2r(+r. co cn$m \cdot N$ 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 ra )v,95+4nemopjj d4z .nyvl3+2 0torshr c ciedius 0.648 m when the axis of rotat.s4 +re,lvnzp90tcr e5nvy2 ocd3jj m+ io4h)ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diametey;km+34xe2k umk6 m1xcqy 9ymr. The rim and tire have a combined mass of 1.25 kg. Tmxcmymy3k 1 + e;qxm9k4y6 uk2he mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the e5k yp/ aj;h) vv;:+e+(xbnb+tl mx qidnd of a thin, light rod is rotated in a horizontal cirqdk/ jbh; a+y)(x+i:;x +m5n etvvlpb cle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheelzu) ,u5czx5bqi2 hc)x5gu ;xf rotating at constant angular speed (Fig. 8–42) cxguzh )ux5xq ;5 uf2iz,5)bc. The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment og - 7u0ad1al*-4 jinq hnbb3vnf inertia of the array of point objects shown in Fig. 8–43 abon - b-n0un i*ld g4aq731ajvhbut
(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 to42oa,.tmjgiqmb v2/ 9lb3 ys btal 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 correct rate, eilj(lpf+fr4n:w .gy * ngineers 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 rockfg rl:y4pf .wj(iln*+et 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 amc5ojb o ; j+ct:;lseqv+3jq 0+ bz*jw mass of 0.580 kg. oz ejcj+;q+wst *l c: 0mjbq;oj5+vb3Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating ,erbh,ji/x/kj,/ky pit 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 l7i9o;- bdc.ngyhe.p ( 1qsthhand-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 of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, nege.qg dypl.chith(n9-obs 7; 1lecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,3lc,e k(hn + a4m:1iryb00 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 4km arye1:bn, + ilh(c1.2 $m \cdot N$ 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 ap t66 r8nilh(zt 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 q;eclgowlzh+)lq e4(4/nv ( eaction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accele+n(4h v c(q/qlelw );oe4zeg lrated 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 considel//grt5*kd: pb0if ) zy3iyle8p3dxm red a long thin rod, as shown in Fig. 8–46 il/yr0/b*)gm: 3ef3 ddxyiptl5 k pz8.
(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 cons;dnz -di;uund45 s(*kx:yrlg ists 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 accelem i)6 tx,c6;cvul (q0txyk)jyu)l, tmrates the hammer from rest within four full turns (revolutions) at,q6; )ku ,txc))m0 luyyvmj6lx c(tind 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 inertia ot.qj/adz3qx )f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developfrt9 z fn (1zen3h:c2ps 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 rolls without slipping down a pidhfik1zf 4 tpiq)1n(m/(4l lane at 3.3(lifk t( )phimp /qd1f i44zn1 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earq s3 sdq.lh:36koxz)rth with respect to the Sun as the sum of twoxs.o3rsq:q h k3 6z)dl 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 m. How nkp otn vg90u/1gln1+much net work is required to accelerate it from rest to a rotation0tno9 /gn11nkv gu lp+ 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 fromf27501 *olemv bwh jki rest and rolls without slipping dowl bj he0o1i w7fm*kv52n 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 balanxe81f 9qw c)aoced vertically on its tip. It starts to fall and its lower end doesqcfwo) e19 a8x 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 4si.n -kgde c kqau2070.210-kg ball rotating on the end of a thin string in a circle of radiua- 4g0kc7q.nskeud2i s 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angulad 1sm0p,mcz nb4(iqm9dl8. sa r momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wnbqd,4mi(1 pdacs8s m9z.m 0l hen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a9hzy;+:kxv o c rate of 1.3rev/s If he raises hisk9+; ozh :ycxv arms 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 shown in Fig. 8–29) can reduce her zmm qj 0u6p+8cmoment of inertia by a factor of about 3.5 when changing from q 06cum mpj+z8the 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate o:h31k 1ls8 bbecy;h pvf 1.0 rev every 2.0 s to a fi1s3bpy:1 hke;bv8h lc nal 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 ;jrxi 4lee cvqf)kdv .,f31 t8center 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 aftei.r t4q v)31jffce8x;lk,dev r the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning : 4fomm (w5/5n9jjla m+xflqy 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 theuwdfmnnx 1t: tnfivg 31)2v6 * 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 vz1tp537w s cv gwao2)onto an identict7gvap3oszv 5w )1 wc2al disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at (kc 0s1 b-zmm 2p :is8wqk9wua2.4 $rev/s$ 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 staoheb 9l/:qudy jf. . agkcq+(8nds at the center of a rotating merry-go-round platform of .uco 8d9 hjgb +ey ak/f(q:.lqradius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular v)-vxz2m8 cpobm qr d7e 8k/8ogelocity of 0-r g/m 7xmb8kpcdo)qo882v ze.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 u0vv (p 7biziltm00xr (//u s h//t.dsh/brtu collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentmttphux(l0/v7h/izbb s.(tu /00u srd //irvum, 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 ex s p.cq,nok35xcess 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 6j-roc kr 9gc7$ 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 brzl u1 r/ir u+a0w/j4rest, that can rotate freely without friction. The moau jr1bu04+rwz ri /l/ment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stanoi7l w9b2hbzbs)b093p+od la ds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and h2s b+bl3 bl) aobp9owz7 hi9d0as a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyingq21j rn lk0*(g5myxbq on the top edge of the spool. A person grabs the end of x*kqg lmy1q0 2rj(n5b 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 su,a uh 3e s6pg6mz1bg*ech that the same side always faces the Earth. Determine the ratio of t p *gazbe16h, s6m3guehe 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 at a rate of 1 m11y, evq ity9 nom/y.g/$s^2$ 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 unifo6e, a1avqwp ,srm cylinder of radius 0.20 m acquires a rotational rate of from rest o,1s6evp a qa,wver a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eachpn*gd )4ezq-wm gy(m,2 h q-jv of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub- p4g,nhvezwqg *qjmm)-(dy 2 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 is the angular speed g9 p-yw(-lp vn*unv zg5n,s;kof the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, wejt zbd6m,*2i g c.523k wtvlussr*(bsre rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitation 3(jsi . 2k r,w6vs2btdz ms*l5c*utbgal collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobil x*bix1 77dua)n9thf ue is for it to use energy stored in a x tn*u)i7a x7 bfd91huheavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates, 5jl3 jlh*;/lycivbz, pye/d an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizon,h b)d:5l(ddy1v gtx btal 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 anf 9omm1bh)h;g6ohq rd(5,ysk *cq:g z d 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 lh6)mhgkmgq;q: 9crd 5sf y,oo*1bhz( inear 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 vnexzp-,(s)a jx w 2u,oertically on the floor, and we want to exert a horizontal force F at its axle so that it will wj,2 nxpxa)-oz, seu(climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn wu v5jt6tl5u 3rj;*nnkf/ysmz.nmd 05ith a radius 5 j nnz5sf3mjuv tykt6 r/ nu0d;lm*5. The 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-kgj6h cyl.nbdn/h,/ +fd rock into a sling 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 is the torque required to achieve this feat, and where+n hb jfd//lnh cy6d., does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a /8 qzql) msd7ksolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wi8 s)mlz/7dqkqth arms held tightly against her body.   

  You are designing a clutch assembly which cow a61usk6 0oyzclr, kt:udj2g90i. lo nsists of two cylindrical plates, of mjikldo.g c: rw0 uz6a2s y0ot,19u 6klass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig.ozft; 8:l/hqu 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;zqf 8: t/lhou loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d)f2fa) t 5p +kez8mobzo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the rug)h 3 qb/j0qcar reduces its speed uniformly from 90km/h to 60km/h The tiu3q0)rjhb/qg res 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