A bicycle odometer (which measures distance traveled) is attached near the wh(gi;n5pd,q 2 )rnn cox0 + y5arfcdh;ceel hub and is designed for 27-inch wheels. What happens if 50c nhd y),p 2ccf rox+n(gr;nqdia;5 you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a vhsi)(0ra;zqn b9y r*point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either compone9hr ;z(v qr b*)isy0annt of linear acceleration change?

Could a nonrigid body be described by a single value d-p/ 2+ c knnb- v7x(obp,ddalof the angular velocity $\omega$ Explain.

Can a small force ever ex 8f )(nxj4emhp2 tt-ucert a greater 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 with your hands bh 8k3v ;orx.orehind your head than when your arms are stretched out in front of you? A diagram may help you to answer to83 rkx;vroh.his.

A 21-speed bicycle has seven sprocl3gn+sr+qa m*f32 ytw kets at the rear wheel and three at the pedal cranks. In which gear i2 l3rm+ qw a3*+ytsfngs it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fles 4in7s6.rded/zp: nvq h and muscle concentrated high, close to the body (Fi7r/ p d 4n:s.v6iqnedzg. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkemnp.rk2du db,hrgr g932q 5 v)rs (Fig. 8–35) carry a long, narrow beam?

If the net force on a sl1gx7b6pxkmjj 7 o4l7ystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net forj7l6opbx7 k1jml74 xgce zero?

Two inclines have the same height but6o as:mv cghw5a l172z make different angles with the horizontal. The same steel ball is rollsm6:7zhac2 ova1g l 5wed down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously4 e9por 1lkn.geid86c start rolling (from rest) 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 koc6erg9 lnd.41i8 ep greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same m3co0 /avw- h;x xrlty;r5v)loass. 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 yc-;/r0orv l;xav5th) x wlo3greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and an:kbak) 4u1u7f1i lqc xgular momentum are conserved. Yet most moving or rotating object7kc:b1if1uq )a4uxkl s eventually slow down and stop. Explain.

If there were a great mi ,6qdp2k6fpf;w9w maigration of people toward the Earth’s equator, how would pwfq m2;6,6kd ia 9wpfthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without havd3li11tcv47u ;t tnxding any initial rotation when she leaves t4 1tn1i7udtc3x tlv; dhe board?

The moment of inertia of a rotating solid disk a9 npvjc(/5hjmbout an axis through its center of mash5(npm 9cjj/ vs 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 sittl:ln5x.lsy/q+g5r c a ing on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the say+q5rna:5s g. lllcx/me? Explain.

Two spheres look identical and have the same mass. However,f6co jh,f(q, s one is hollow and the other is solid. Describe an experiment to determine which ( , jhf,sc6qfois which.

The angular velocity of a wheel rotating on a horizontal axle poin6rs kxy-8v/ ytts west. In what direction is the linear velocity of a point on the top of the wheel? If trt/xs 6-vy8ky he angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a lar1nqw57 gzx ev3ge freely rotating turntable. What happens if you walk towa1 new q35x7vgzrd the center?

A shortstop may leap into the air to catch a ball and t cuas q x04ymbc),w,s3hrow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs roumyw) c xs4q,c3a0 ,sbtate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angula4z:vbtgtg,ogn l7 +h 6r 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 heli,7 bzgg nv6thtl4g: o+copter stable.

  Express the following angles in r6x s/i; bbt t0s8xiyi8adians: (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 ac 1gg5c (+gzfapwmr 0.mazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of thr cwag.m5+c1 ggp( 0fze Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directkhez/btd;(005ekq/grgw+7zv ry : d xed at the Moon, 380,000 km from Earth. The beam diverges at an agw+;vb q zrdet /ky0z:5rd (/h7kx 0egngle $\theta$ (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. Wm,0ki, e ;3k5 akvgqakhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angularqk;,kva3 a05 miekk, g acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. 3c ,dgx 1g v7,6d q29xal56gm.cemnl svzrih2If the ball makes 15.0 revolutions, what is its di1e7sg6xl vgv3ia .lx 6,rg52,cd d2zn9mmqhcameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho .,u pisx20fzsw many revolutions do t ,0ps2uzix sf.he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 etek6nxz+4 ;+ia7mmqm ng;lyl; y) k9rpm. Calculate its angular vellz49+mq x geeky; ;mlnk7y n;+ i)mta6ocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s c1l 4jb--1sgju mcfl/(Fig. 8–38). (a) What is the linear speed of a child seated 1bucc s j/-1ml-1lgfj 4.2 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 vel*z0ppb l.v sur+zco/aq, u) c8ocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe +wshti3f ck 1xb,r.m;ed 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 particlew;cvxra5nvl a4tz 7bs4c 4n/ ; 7.0 cm from the axis of rotation is to experi s4a4/xlbn75cvw z;4t n rcva;ence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unif/ ,qv+g vnoas1ormly about its center from 130 rpm to 280 rpm in 4.0+ vsa1, vg/qon s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusw8n2i6trm4 n i $R_1$ 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 (x8ss otb :8dfaboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated dfs( 8:8 xbsotfrom 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 ixl9wzs1f y)m: n 220 s. Through how many revols y:1xfz lmw9)utions did it turn in this time?    $rev$

  An automobile engine slows down fro6n*-y :mzld zvz7gmn: m 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jethy 9hoh-os9 4rs in a whirling “human centrh4hyh-o s9o r9ifuge,” which 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 uniformly from 240 rpm to 360 rpoy6 5vnxhqo) 6;6z aldm in 6.5 s. How far will a point on the edge of the wheex66vn )qa6yloo5d z h;l have traveled in this time?    m

  A cooling fan is turned off wr-3 46q)q ivq yjqbmj,hen it is running at 850rev/min It turns 1500 revolutions before it46qqq b,rjji)v yq3-m 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 speed bu*de oyy 560sxpe)u ,uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m., 5)b* pxodseuy0 ye6u
(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 )sa+cl):hng naml;ftxkt 9 (,as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dca, gt(9k) xhtl;l:n s) mfa+niameter 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 eachalzsyc7am9 0 / pedal when climbing a hill. The pedals rotate in a circle of radius 1mza 0 cs97/yal7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Wha( q1l(qe- anpw(cba7et is the magnitude of aec(-q1 napebq( 7l(wthe torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle ofw :kib-a4 tt)o ;l;pu zg6sbd3w1ag/l the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4-kds/tatllg1z 6;ai; wug :3)b4 opwb $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of ma 3 zpc:mgq s(6nsak(h9o )0chzss 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 c9cq(po6sn3ksz0h ) gh(m:zareleased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an enuo(2z4zlh8(qzzdqkmt74 5 eqk) v1i /jfn w9ggine require tightening to a torque of 38 28ztkqdez7no u 4g19i v(kf( jh45m/qq zw) lz$m \cdot N$ 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 1wno8 /qgvn bv r;6. xoo-w.54cu7mv xm0.8-kg sphere of radius 0.648 m when the axis of rotation is through v.v 5;/6woq8wc7mgnroun .xv- ox4 mb its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi xc/xl( hokj;h1yo0(mx + ad*scycle wheel 66.7 cm in diameter. The rim an 0x s1o*ox(jdxka+hy;(lc h /md tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light r 6 qxnc3sdae/d we f,llcy:+5*dz1xl7od is rotated in a horizontal circle of radius 1.2 m. Calculat6+/ 3zsdx:c75n lcla w1,e*xdde lfyqe
(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 constaoi(ig ,c ojz60nt angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totalgo0(o ,6 cizij.
(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 po e iawv;vc qm z.)bi*sah3 rfuyk)w3j:1io .+8int objects shown in Fig. 8–43*.y3vh;1:)3 8.)+f eicwr ivm wu baazjiqoks about
(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 oxyfox6b,yv lt:3v*7z x:4eng eogen atoms whose 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 the correct rabrpna 593uu :cqqx8(r te, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to 8bp5q 9ucnrx ua q(:r3reach 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 c6ncos1l o wv(/9 e1nrfm and a mass of 0.580 k 9of1wn/vo6l(nrsec1 g. 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 from rest 0qtrdstmd 5e1/4tg9xoso7 h2 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 mqby+.5 y:-jgdej tf/gerry-go-round and is able to accele5q+/ g -j:dtfj.b egyyrate 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, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and ir s)*y:u0tt;rhga3f-,-qw g3:rwt jq7 iaxbms eventually brought uniformly to rest by a frictional torque of :h mt-7bj-tqyf0w,gt)rr3xau* g as:; ri 3qw1.2 $m \cdot N$ 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. 8–45p7b1akkz ):ld accelerates 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 -q6qw6o*tf0 xx nld ,rat6x1eby the action of the forearm, which rx1n w*d a tqto66rq,xx0-lf6eotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be cons1;ecqxf*k*ameb- s q0idered a long thin rod, as shown in Fig. 8–4*1eqes c f-;ak*0bmq x6.
(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 cn8qy)-5n+ ozp w3jf hhonsists 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 ;1tvtkpx ,e0pthe hammer from rest within four full turns (revolutions) and releases it at a speed o ,exp0;tvtk p1f 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 mnzb( u.9el .dyoment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop))g4msa7n - 0tcryyf ss 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 wxskbnshf)w o 9+5 9,uyithout slipping down a lane,sybk xh s9u9nwof+)5 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resobq+ an68yhke44i6df pect to the Sun as the sum of twokf 6 h4e8yd bqi4n6+oa 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 f kn hp vp(l-r33gt ,qfm i8cyq,6f1l4of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolutn,g 3kphmr8q,y1 ftv6 p43fqlic l(-fion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg st/9jfhx .,i+v.:jnh my6yfe;dienu6l arts from rest and rolls without slipn hv9.:jh en/l6mdfi xey.j yi,;fu6 +ping 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 is balanced vertically on its tip. It starts to fawtcwbt0;g r0 :l(vxm.ll and its lower end does not slip. Wha tl0wtw.v xr:; 0(bgcmt 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 rotatinuimi3mf6o g ud0 d/q0.g on the end of a thin string in a circle ofq 6m03imog0.duu / idf radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentlym,y m-d 2-2o3u6ll* fafo1r zslgj.um of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm*2l-y 3,ufdz.2lr-os1 a f6mll myjog?    $kg \cdot m^2$
(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 rotati t)tuu s09lz6mng at a rate of 1.3rev/s If he raises his arms to a horizltz9s0 6u tu)montal 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 moment of inertia by1s)(un bh: j5u mnkfy1 a factor of about 3.5 when changing from thes m:yun)1 1nbf5jhk(u 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 frb:orfwbu4t 91vq zh ,)om an initial rate of 1.0 rev every 2.0 s to a final rate of htqbr wobf 94),u:1zv3 $rev/s$ 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 9mekh /x1r xi;w38ewla 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 flk;xwrh e3li /1 xwe8m9at 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 momv6.xb l1pgto* entum 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 o;fne(aae3pb vk1jy8wlc:r ,9k 9ly mu5 4:rvk f 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.n. wrv3qwd- r3s*kge of inertia I is dropped onto an identical disk rotating at angulaeqn.3krgvr-wd* s. w3 r speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns,gvsot4l: wi:vok1pf+0d6m fnh8 *p a 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 stan/1hm3d 9rwcun nn(b-ab. eft) ds at the center of a rotating merry-go-round platform of radius 3.0 m and moment of r-n cd9nb/eah) tfnb u.1wm(3 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 withuk ,m+g5a;bxk an angular velocity of 0.8+;, ag bkxmk5u $rad/s$ 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 co / 2cfbzbmf q7)ncc4 o2tbo(v+ 9q:tghllapses 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 momentum, what would the Sun’s new rotation rate be?(round to the nearest integehvf :bom /qf 4 9z+nt2c2bq7 to(ccgb)r)$\approx$    $rad/s$ (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 involv4b68xalb1mtdf.qq * pddam)1:d emx,e winds in excess 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 h hecw3cn*q6 e o1g4sd(3yl 5l$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate r b2ozq+ik1ni3f2u b ,freely without frictii1,oq+ 2 burfz3k2bnion. The moment 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 standsno-m+gaqzs2 ( en7*g a at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment ofa2n- n*egsa7+zg mo(q 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 lying 16b g-yum:zz lon 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 length of rope unwinds from the spool? How far does the spool’s center b1 m-g6ul:z yzof mass move?

  The Moon orbits the Earth such that the same side always faces the Epc;f3tt4e dp i/ygk ge960;jtarth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latt6ekpyc30tg4ipjdf g;t t 9;e/ er case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ra*,u ud -q 7udpjgq(1jwte 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 k0f)vq-- lza ynfim 78a uniform cylinder of radius 0.20 m acquires a rotational rate of from res87-a0niq-)m z vy lkfft over 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, each of mass 0.050 kg and d52pu w 0mz.pf*-d :og)dalazxiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservatiogdz pa 5.mwflpdux02o)-za* : n 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 of the reaus:tio ws)y7n/pm--cc kh*q 2r 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 greaoztj qvu*+ 6 pwb/es1ovh ,-t8t, 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 the lost mass carries off nwth+poj u /6qvzev-,s1*b8too angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy j7w d ayqt2ik)32cn-o stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical dic2-t3 )oq 7nw2ykajflywheel 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 illustrater;+jrmntl;e4 ec2 f1yp qgf9* s 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 horizonta3b10 8 fvoaoopl 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+sgh 9r)uavi tnw //h*,:ml fi (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 ai*uatil h9,rvg/m nw:f+ )sh/cceleration 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, ;-yilm4 rc li)*lss63n ajosouu0v *)and we want to exert a horizontal force F at its axlu);y4va)rlins* i 3l6ls *jsouco-m0e 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 flat road is making a turn h(/3jiuc5q e ih8vi9qo7tww : with a radius The forces acting on the cyclist and cy hiwt9jiieqvqh/(w: 58c3u 7 ocle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins gno+)z7n-y:n nh mbm60u*gpz whirling the rock in a nearly horizontal circle gngzo+6 bn0 pnn: m*m)hyzu- 7above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid:/v;ibwi(x,gca o u5b0( srhs cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angulahugw:c( o vs0i /bia(s;,rbx5 r speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cons (uk hlc*i..: :thhxx nw7o bf/o20nimists of two cylindrical plates, of mcwht0ux2*o7/inhoix :hk(b.. l :mf nass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 js8bi2rwvx4) 6zdkqvn )20dnrolls 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 leavinb2d w8zdnq4)i j)sv026r xkvng the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not acil5g5axc 8x p f(+.i -. ak1kfswg2osssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolut gr;z*zpkg4hvmv + 38hions as the car reduces its speed uniformly from 90km/h to 60km/h The tires hav rpvgh;vgh+3z4kz m*8e 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