A bicycle odometer (which measures distance traveled) is att dgkjdt(ca8; :ached near the wheel hub agc(j k: ;adt8dnd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at const8zp y j.5knq7gant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For whiczq.85np7jgk yh cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single val,+w aw+4lb9 2;rowpsnplu9fs ue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larg2*k aagw4n3tdem :,toer force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head vqxzs g1z-27j8cao ) dthan when your arms are stretched out in front of you? A diagram may help you t1) q8a-7xos cgzj 2zdvo answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and88rl)zumqon6 three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sp lou8) q6r8nzmrocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs witint- guzjts; ( w6tz2b.s-rl2 h flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution ofjz-lbw;zis(r2g nt .tu2- st6 mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ca/v ddzg cav((r (3xkk6rry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net togk7z xc f llf4wu8n4jz qda:- +ar8u15rque on a system is zero, is the net force zerjk8a q5f x 47rf-8z1guncuwl4 lda+ :zo?

Two inclines have the same h;o0l0x czez5cjswx:d a b84/ feight but make different angles with the horizontal. The same steel ball is rolled down each incline. On bf8jxosc 5xw/04:0c e lz zda;which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start roluc4 7 /c/o)q/oqkfnw hling (from rest) down an incline. One sphere has twice the radius and twice twccho) qu/ /fkqn/o4 7he mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinde 6p, nhussq wt9 2wrw.xv68fi/r have the same radius 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 6q2 p ivwhn8st,xwws.6 r fu9/the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yeh8 zeyebj 6/*at most moving or rotating objects eventually slow down and stop. Expljyz8eh*a b6e/ain.

If there were a great migration of vcw ya9g9p/)kpeople toward the Earth’s equator, how would this affect the length of the )vw9/gkc9p ya day?

Can the diver of Fig. 8–29 do a somersault without having any initial 2fbzo: 6)prujrotation when she leaz f)jbupor62:ves the board?

The moment of inertia of ad ,2lxkvfz x)4pqn1 t- rotating solid disk about an axis through its center of mass qxv4xpt,1 nfd2 zkl- )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 r, qn8s.)tqihcj j+hati26l, 0ia kqx9otating stool holding a 2-kg mass in each outstretched hand. If you suddenlyt8shjna ,i.jq2 i t+ql6,a)ihqkxc 09 drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and :p(befb6 93 germ-enthave the same mass. However, one is hollow and the other is solid. Des39gp ee: -bfme6 nrbt(cribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Invshdo 7sy bk-np vhx**/-di/3e z,*bm what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration point*3i dshkvmbxen -/ b 7so- /hd**zpv,ys 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)j p/ zf+zm6mo freely rotating turntable. What happens if you walk toward the centezjo mf+)6m z/pr?

A shortstop may leap into the air to catch a ball and throw it quicf29svy.p lj+(*hy 6tz 4ykww qkly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Expl9q 2yv4h*ty+pwlkjyz f6 .s w(ain.

On the basis of the law of conservation of angularmo bs)5h:w1ex/5o czqyh; vr0 momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter sc1 eh0/zoqy s;bxw m:oh) v55rtable.

  Express the following angles in radf/c nl - )ttrsl:o+)e js4j2iuians: (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. Calculate, usin*+ lp vt/vhrzi8/t o2cg the informz/it +8lv or*pc 2v/htation inside 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 the Moon, 380,000 km from Earth. The bedvb 8/uly s.2ddu6l t3am diverges a2b/.3utl dyv6ds8udlt 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 ratekubpg0ox0;r9bdo p7 moh0b0 p9c,) py of 6500 rpm. When the motor is turned off during operation,0pok hbcg p0y p70 dxo09rpob;b 9,m)u the blades 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 another child. If the ball ma3 z jiu3xrgl22i.fc,pkes 15.0 revolutions, what i2 pxz.clfui,jr 2 33igs its diameter?    m

  A bicycle with tires 68 cm in diamete-fk mns7*r6*g4l khjjr travels 8.0 km. How many revolutions do the ws g k*kf *hrm4l-jjn76heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itsxclb /zd-ir n0 n99 6ukbrau79 angular veloci9lucixb67 bnu9 kan0dr/z 9-rty 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 compl6c3nd0e s r4otete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thr4c63stde0o ne 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 j: m7o0-hobuhrun6idqudwp:n k 0)1 *) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi9f,-qabx 3d ahnt
(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(aptl t3rqd)8 a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration oprd 8tq3 )alt(f 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheegzu y l vkos04dd-xsw5r.j:o3;0x +zi l accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determineyoido s0s;+l3w04 z-:dxkuv5 zrjg . x
(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 radiu2olge8lv mq( 5s $R_1$ 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 Ms hp/m sah0-u/lf 9dpku d7w.f.as01soon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecrafth 1fwsa9a k 0u sf/ hp/7mdl.sps0-u.d 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 tq uyff ;l+3r7ho 15,000 rpm in 220 s. Through how many revolutions did iu ;lqf3+hr7fyt turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calc;k -e u o9a m(l4gfre4nkm9mf aw94e7xulate
(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 s gp /xu,fh6j;highspeed jets in a whirling “human centrifuge,” which takes 1pj h,gs;/fu 6x.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 accef1jc81mh oux 7zq/0hu lerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a hj18u/ fh17u z0xqmco point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500iya5e7 a,oc dt,90pup. s5oa g revolutions before i05e,g ac 5oo9, aia7dspytpu.t 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 revolutio;n2 u5 +hoayjp4weyg6ns as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0wnh6 j yg4a5;e2up +yo.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 thelcg d2,7 yaq4n car reduces its speed uniformly from 95km/h to 45km/h The tires haveya7l g2 dcq4n, 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 pedal when climbin(31+l;ogcn3pq g(v/mbnq m,ewi9 fflg a hill. The pef 1wbc9/n(i; lq3n3(v molqp ,+ fggemdals 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 7-9h7 7rbkx6v x8kxzpq t,q0io4 cm wide. What is the magnitude of the torque if the forc,k p7qxk-rith6 xzvxb9q 807oe is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of thq zk,ge3pap 53e wheel shown in Fig. 8–39. Assume that a friction torp ,eakzqp53 3gque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a.i9irb0 jnfh (ttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially ti n0f j.i(hrb9he rod is held in 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 tightening to/zs 8 vs(bjwzur*a f+30axgg/ a torque of 38 z a3xvg /wb8ss0 j(fa+zg/ r*u$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when yhp.10ry.b la4awq9 uthe axis of rotation is through its centbwhurq.9l.y4 0y a 1aper.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy4 b* acd*pv)trcle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mas4c*a) d prbtv*s of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball oldwfio/rz 7a.2 ntptjv1 s2 d,d7m f5*n the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. /o dsnim*wl,dz t j1rtf d2vp257af7. 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 wheel rotating at constant anz4;u oy /j yql-yubo6:gular speedyjb qyo4 y;ou6-:u/ zl (Fig. 8–42). 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 of inertia of the array of point objects shown;gbkbj6 b8ueg1)3 )t xwfw9 bhe)(vvy in Fig. 8–43 jk( ge6b)3weyfg ub;t x)vbw8)9hvb1about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass is 0/.v n,g6 wnf3uzog0tlc:n nk$5.3 \times10^{ -26}$ 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 coh f.(hr opywqz *8/:dmrrect rate, engineers fire four tangential rockets as shown in:8zm /(fyoqwph .dh*r 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-. 8ux8 xn lf+w2v3rc0iqcjiwt hxf*6 radius of 8.50 cm and a mass of 0.580 kg+-u lf.h x 8nwxj f*8cx 6itiv2w3q0rc. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it fd52 *(emr8p+ohn b2f wodi /khrom 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 u35t khdo)ig2;dgvz9 /jlu k(mhx7i7 on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit oppx5h gth7li(ozgmdv u7uk kd3;)ij/ 2 9osite 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 rot;v6a;a yjarh bfg lmxj3x 43+x5vm0i0ating at 10,300 rpm is shut off and is eventually brought uniformly t aji; x lx30v5 ;axf6a3 mmh4y+gvrb0jo 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 3.6o(a*urm .sl 6n-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 tutjb- r/5(a9b 35mztm: xnyemhrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is ac-j ye9:mtm3tbm/5b 5nzu(a r xcelerated 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 :fq;y)tir; sy long thin rod, as shown in Fig. 8–46.tqf:y ;;yisr)
(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 consi4j+mwk55bge z sts 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 hl8f:2oe/xeo7jc gm g ;ammer from rest within four full turns (revol/loxcge2m fj ;8e o7:gutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia o/ dv (z2+ilszmf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develo*xq4ny z;r1nzx y :fg-ps 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 cx85f6r:t x u(w lxm.au:+ z 7qqyvgk( e2yy:sslipping down a lane a (su5cryq8w:x x6g fe 27mtx (:zlaqv+:uy.ky t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of t8e;j7hnciqf7m +e3 qjwo+jieqqe3m8h 7jcn f7 ; 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 and2 jeq-*hpnx*h a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.0*j*pn -eqx2h h0 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kjqyg s1r k*:kxz- 2x9kg starts from rest and rolls without slipping down 2zx1 -qkg*s:r 9xjkk ya 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 on;lwtt0h6.p-cqh3/l f5 rmr l88i hpcm its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservatpt5h8mc6q r0 ltp;-l mh rl8./3ifwchion of energy.]    $m/s$

  What is the angular momvw m4r xclok/rm( 7:d8entum of a 0.210-kg ball rotating on the end of a thin string i:xo84rm vm d/kl rwc(7n a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cyf fr(nc6yqw2*vs /7 qolindrical grinding wheel of radius 18 cm when rotating at 1500 rpmvq n(fr/62 7socw qyf*?    $kg \cdot m^2$
(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 n5b 6+ .tlifa qe(kr:wrotating at a rate of 1.3rev/s If he raises his arms to a horizonbew 6:q r5tl ka.+f(nital 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 moiiib lx.,i3i3 ment 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 position, what is hili,i .ii3 x3ber angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rat7 ev0e7 r5n -jcyq8as6opzmm ,e from an initial rate of 1.0 rev every 6jepvn8emy smza-c 57o07,rq2.0 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 bm: vx1 )8psy w/o+ccg ipjtsxy:/44 vthrough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk ofx cs8p/4m1j )sw:pvg+btycx v:y/i 4 o 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 momenta3a bes7ae oyx1) 5yc5um 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 oin;xs(wyxz c)q o 0+n:no u-/b:+5hrm2t kfdjf 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 i9bbyuh 7cc*vx4r:/o) r /nbpos dropped onto an identical d7 bryhonx )4ub b/r9cv/*p oc:isk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 4b+v0 dp-f v jev /vu,o7tceo,at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating me8 lq9rh/1fzu3w. crtw rry-go-round platform of radius 3.0 m and moment ofc8q 9rwr.z/1w3 thl uf 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 angulag5qq alld/:8tr velocity of 0.8 q d5t:/qall8 g$rad/s$ 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 rz ,3r8 wz 4lpspk4at/cm6h2k white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Asr 8,a4hc2r kwp6z4tzsl3p/m k suming 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 m wvvtic92ki (k7x;e hqky605 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 . g ud tlmiq u0l;x/.b9jf+lu7$ 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:ydz6n )3pmz uest, that can rotate freely without friction. The moment of inertia of the person plus thpmu6zd:3yz n) e platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m( 90 kxus-b,q 1h;t9wxgv srya diameter merry-go-round turntable that is mounted on frictionless bearings and ha,vgt09 qxh1k;x(ws a 9u sy-rbs 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 rope5k(xigf* yj m9 lying on the top edge of the spool. A person grabs the end of the rope and wy9 *mf(x 5igkjalks 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 tjhp+mw d:, cd-he same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular hj+cd,p d :-wmmomentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a g5cri th, e 1i9 c4t-lnfm9q+erate 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+fif1we v1y )d96q jhs cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculafd9w)yvhj eq6s +1if1te the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylida*mld a.,1ozndrical 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 enerdl oa.,1admz*gy 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 of the rear wheel (8 gxpp2u p0lwt85(*t5wxjqu u$\omega_R$) 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 mass68p dq.bcn*2m gech 0q 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 speedb d2 .* gm0pnqceqch68 be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is fod*zows 6gd-m 2r it to use energy stored in a zo wg -s*6dm2dheavy 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 s g4j5 k-r1 nui-gz(.ai,sgoman $H_2O$ 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 surf8qj(dv84av hvace 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 pivh/-q-us hm lv(zv p/n49kv5qwot 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 angu9hzs -(nw puqvvh/k v l4mq5/-lar 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 floor,ycri// uvccjs8-,lh*4 rz t3o and we want to exert a horizontal force F at its axlercsz3/8,v* jrl 4 h-cyo uic/t so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flaqvy4gq -w/mv 7jaut62t road is making a turn with a radius The forces acting on thetmjyq-v4gqw7 /vu2 6a cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and ki5yup0q*d uiu 80g0cbegins 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 tory *5i80u0u ug kdq0icpque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arm(es a9-4(gyhr u*r tgms as thin rods, making reasonable estimates for the dimensions. Thar4(u(g*gmes y9t - rhen 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 consists of +h9m fmbym; +xh5a/0rb8tk xe two cylindrical plates, of x+e8 arh5bmf0mb ;y9hmk /+x tmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r t7 h:pg1,zy k-ttq3 39jdynevough track of Fig. 8–58. t9y nkh3jzvd 3ygp7e- t,1:tq 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, but do not assuqxee5 9r jkk i(2(wec:me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduceswc-x n4 .hyb*q its speed uniformly frocq-nx. yb w*h4m 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s