A bicycle odometer (which measures distance traveled) is attac uw8t+kyx303zp 5oaa.yf m ah1hed near the wheel hub and is designed for 27-inch wheels. What ha y8uo a0.xk3p atfyzhw 1+53mappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocitizp7agz iq ) 1obg35y4y. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformliya7g g)p15 ibz4 3ozqy, 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 5r l;,0tdk hc3f(/pt2)f+ck jnmxrii0s ;hje single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a3eyd5pe6;e fc2d t( kp 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 te7, nn3/ig yq-jj xyqj649zarlndr 0 :o do a sit-up with your hands behind your head than whenzy 3dg7r n xn -eji:,4rq/a 0jj9qly6n your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprl7iy7ll+6afu e2mz4c*om , , z l5z90ezjlo kiockets at the rear wheel and three at the pedal cranks.f lkeeoz5 c *l0 o6m2,z7m4u, jyilli 7l9za+z 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 sprocket or a large front sprocket? Why?

Mammals that depend on being able7:b ak bjh6:u t7l,4 rabibe(i to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of ro77th ub: a6ira4 j( ,ilbbekb:tational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a rj 07 q59ilm.zlir ,gb7bo c-vlong, narrow beam?

If the net force on a system is zero, ip r5zlt(mbcux :g)w i:kij.74s the net torque also zero? If the net torque on a systeml .k g5 pibu4:c7w:xm(it jzr) is zero, is the net force zero?

Two inclines have the same height but mh:s1mt9a+z/r:gaug make different angles with the horizontal. The same steel ball is rolled sg/ rhz:9 u mg:1maa+tdown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously st3sk.n:rfpup - art rolling (from rest) down an incline. One sphere has twice theskn3.: -rufp p 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 bottom?

A sphere and a cylinder have the same radius and the same mass. They start from v,v4 t4.pe mxarest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which 4xmatv p4.,ve has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momkq her *mt7 m,n-1r4.p fmupd.entum are conserved. Yet most moving or rotating objects eventually slow down r7dhpeq .,mu 4p.tm-kmr1nf* and stop. Explain.

If there were a great +;k1ksiqp6i ok-fh5wx8c el, migration of people toward the Earth’s equator, how would this affect the length lh, 8k1+qe fw5 oks-i c6i;pkxof the day?

Can the diver of Fig. 8–29 do a somersault without having any initial pn9ho-4x4z h i62+u)e .iwp cwrmu9tsrotation whephwnme rpcu 4tu2 x ow)sh-+.z9i6 49in she leaves the board?

The moment of inertia of a rotating solid disk ab*ltm .: v,el9 xg/fxdh 6omgq i4(cyr+l( rax/out an axis through its center of massgr/+ mahrv(6tmf , o/x d:(ex4l* lg 9qylx.ic 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 holdi8tjn cin7x .9ang a 2-kg mass in each outstretche8 ji.9tnan7x cd hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is h;x) rk9(vimni4z:p pg ollow and the other is solid. Describe an experi)pp9gx:i n(4mivzk r;ment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p.qzf1-5 3u kfw+xy a c)tjo,jxoints west. In what-,fz.yk 3ta+xq1jxu)c5 ojw f direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large fre,s0b2- v wqzxgfhi9q.r3f dx.ely rotating turntable. What happ2z3,x90 v.qghiq s- .x dfbfrwens if you walk toward the center?

A shortstop may leap into the air to catch aq )o+,myb2 cxwpt *hc+ ball and throw 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 rotate in the oppomwco),qhxcb*2p t+ y +site direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, df 0m7qln 0wx:w7crn9aiscuss why a helicopter must have more than one rotor (or propeller). Dilwqwac7x0f 79:mn nr 0scuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following svce884v5ezo lj diskuj h ,63wy a:i4s+i9(y angles 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 becaus m5za 5urc+awll9,k3kht 27ale of an amazing coincidence. Calculate, using the information inside the Front Cover, the anguaakktc5 m3+,hzra75llwul29 lar 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.oy6 / -byqus;ua w-h1p The beam diverges at ana- bu;o-y/upshy6q 1w 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 the motor 0*;pf 682sulfq+noyv.sm o4qz -thi mis turned off during operation, the blades slow to rest in 3.0 s. What isumv pnf .ozi4q6oy*;tf02sh sqm8+l- 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. yut*/ojp4.yodms k-:If the ball makes 15.0 revolutions, what is i o*yy-js m :ku/pto4.dts diameter?    m

  A bicycle with tires 68 cm in dia7pdr93ffdifn ; v*rbmokx- c6* 3wu1emeter travels 8.0 km. How many revolutions do the whee 3fx3r-6de fwdpfu*i r ;m*91nkbovc7 ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atwln+ tj 4*i0)rf*, tfzqz7yd58 yrvns 2500 rpm. Calculate its angular velocity in ffy d*)+zv7w0inqrryz * ,jn5ts48lt $rad/s$ $\omega$ =    $rad/sec$
(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 comiige.( 6ox4 uaplete revolution in 4.0 s (Fig. 8–38). (a) What .xiei6uoga(4 is the linear speed of a child seated 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 its orbit around the Su;-)*xkr mfrsrp*zwa4rqea 33 n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee 4jy0dxjz( (/4 zziaxcd 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 r-lcuu 1 bgg4n+otate if a particle 7.0 cm from the axis of u+b l-ngu 4g1crotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about iy gcdqqfo7);3 ts center from 130 rpm to 280 rgqfy37 d;)qc opm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusq qddfq3- (5ffg ;wbs+ $R_1$ 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 spacrez y;f1znh3,:- wx nfecraft 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 spacecraft ca nh3zx;wf :-y z1enr,fn 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 unif 0ffk4br+re9gzf r zc/+;n5brz /k8jjormly from rest to 15,000 rpm in 220 s. Through how manb/z/jfrc f 4ke+f9gb rrk5zz ;0 +n8rjy revolutions did it turn in this time?    $rev$

  An automobile engine slows down fromhrs1fm m 39zle3*h mg3 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 jets in a:nmth:rt( hun) tjjog w0:4s. whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rwhj(j ng:hn .tu4)o: t:r0tmseaching 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 acceler9av806 nijwfi fk z+/pates uniformly from 240 rpm to 360 rpm in 6.5 s. /8zjfw+ipik0 af 9v6nHow far will a 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 Ir kue0ss; u 49v.2uz 2.jqnw j2kyq;vnt turns 1500 revolutions before it come kun;qqk .uw2 4s 0uyjjs9;.2zen vvr2s 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 speea(kq :gjrw0o+*;vz( j )8n 2tbonvciyd uniformly from 95km/h to 45k2azk;w+bov ivoqjr0( gn)( jcnt:8 y *m/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlyuj)n/fccslp 7z3 57vx from 95km/h to 45km/h The tires have a diameter of 0.80/x5sufzn) 7pc 3j7cl v 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 + d-: jez3fhsyweight on each pedal when climbing a hill. The pedalsd f+yzehsj :3- 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 o am;jx0 kii(c94 r)tb(k-xve xn the end of a door 74 cm wide. What is the magnitude of the torque if thav i k)(b mi0k;(j xt4rxxe9-ce 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 tozftna3pn5ll 1ahr9 i hczh2c( tu2//4rque about the axle of the wheel shown in Fig. 8–39. Assume that a friction t2znl34rua( p //an9 t2lfhcitc5h h1zorque 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 tt.mzptd/ 3q 3mhe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direct/ pmtt33zmdq.ion of the net torque on this system.

  The bolts on the cylinder rym8.7us;p yah9xbcds-a7b+head of an engine require tightening to a torque of 38d-m s8upsr h bx ayac79;+b7y. $m \cdot N$ 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 sphqhbhg u, 98w lzs.jw/*ere of radius 0.648 m when the axis of r*/l.j,w9q8uzbwg hhsotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel xs3 cm34 yq0fc66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why3qc ycs0m3fx 4?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light roj3gl46tq 9n( f no3obld is rotated in a horizontal circle of radj n4fotn3q (gb l9ol63ius 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 wheel rota(7gl 0jf *hsyw7r:gf uting at constant angular speed (Fig*w yff(7j:r u l7hsgg0. 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 i m,/ budbwl :-y4c0hvpn Fig. 8–43 aboblw-cmdyb: hu /4 p,v0ut
(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 condn5 5:wvj1 ldfsists of two oxygen 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 satellichj2u1c/rxr 5 te spinning at the correct rate, 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 for1 /jh5ruxc2rcce 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 cylir7q -x( n0;oevur cxnh;h*mb + ;lh6fifg+/ gvnder with a radius of 8.50 cm and a mass of 0.580 kg. Calcuq l +nx0-;vfbe(hvmfnucri;gx*/ +6hh7g;or late
(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 swinhhm z,6ys,k(cjjyc 7.f1fw4 8 ib q6apgs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangc/.8hw4wvxn yt t p1*sentially 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 opposite eaw* s h.tyx4wvtcn8 1p/ch 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 is eventually broug g wcegewmuwb 99f8(48ht unif(94u89wc wg b8feg ewmormly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a ev//yma)a- z0nco,qy 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action ,tww br9;jmgmh4s : if;4 arh7of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fj m:mr,4b f h7t w;w9h4rsia;grom rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in 3 j k/6l0d:;pg lrgvanFig. 8–46.3g 0 djrl: kv/a;gpl6n
(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 consists of two tev8j 6puc5 c(*5lmxlmasses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer f4bqugb /yh3jy sm,b(j96q kb /rom rest within four full turns (revolutions bb4yb/u /ykhsqq j(mb96 3gj,) 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 of69wke r2f m:ng $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque 0(d6or k ld:f,vi*peeven0o *itc5m2of 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)c)r-qvwg 83: kfpyyn without slipping down a lane a)q-g8 v cfpn:) kyw3yrt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of :l9ys-bwun.pjn ck8zw23b- athe Earth with respect to the Sun as the sum of two te l-s n9awn kz:jp3u8y-.bw2bcrms,
(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 much nrt . 0u0w5:envf4hmu4cn etf6 et work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s?5mhu eu0c6ve4nft4fr w:n t.0 Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg stjy2: ujo-caxrp nlh7h+d: a 35arts from rest and rolls without slip +jol ahxy5-:j:h2pna73 rcudping 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+t1dkzq c1ym t np*/: wp6pi;m balanced vertically on 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 t*;dpzw6k :i1qc pyn/p +mtm1the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mon39(eq +v,caf d dfvs1mentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radi9fd3sn(d c,e+1vf a qvus 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of, **zw:t rd;j.zqs3 ks zjqqg, a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1r*g sj*3j:zz.;tkw qzqd,s q,500 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 a rate of gr ;25qbs;nx i1.3rev/s If he ;gb5 nx2r;qisraises his 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 shog.xc,a -fz)r8xtl pl0 wn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuckpg- la f),tc0xlz r8.x 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 init xd;tb gyl,0j)t,i(-5vjxoukial rate of 1.0 rev every 2.0 s to a final ,-di j)vg( 5;ktl,ox jybx0turate 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 os518p 9( cizzwa *yveaxis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of i9a p*vywce 1sozz85( 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 momentl*ba( 37(qww)ewbpm hum 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 of the Ea/9 5e.neqawbtjv h3r,rth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an i5:vj5 c6ppf1n6t:efakov w o.denticalej5a ncp.fp:f 5 vvo:t1k66wo 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 azc5f1h r3q*u dlc7-havd3h(v h5c+xyt 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 merry-go-round plats :n+o+v8xxk3 *gc askform of radius 3.0 m and moment of n8s s* o+x:k+vax 3gkcinertia 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 rotati bznl:zu+s6c,xu/ 0f gng freely with an angular velocity of 0. s:gn cfu+bzuz6l /,0x8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its m oc vtd9/r0*fvass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momen/* 9tfcovvd0rtum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 kkz.oa +6t *xi4 ,stwl$km/h$ 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 vmwa. u cte n2xmm5g) uu;o4-5:kclc+ $ 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 wv )q2gs 2i:z2y zj6cpplatform, initially at rest, that can rotate freely without friction. The moment of inertia of the persyipscw zjvqz) 622g:2on plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round tu vlwb-+gbwsq j /g,28mh6 zy(vrntable that is mounted on frictionless bearings and has aww6y z( qg-vsv+bg8b2j /h, lm 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 groundj. 0fnm q+xsr- with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behim jn+f 0-r.sxqnd the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earthy;9(i5a7kl1qxpw el o. Determine the rat lo71i e;al kyw9qx(p5io of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate r3*,fwzpw f n6of 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(qg me pl ba w( 7bawv44bvq y5yyhjl2v5j4*,5 cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate thepeb4a bw a5hq4y(g m54,lyj q*w5l27vyvbv(j torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrxiw: fhj .0kib)otc :4ical disks, each of mass 0.050 kg and dia4k tjbco).hi0fi:w:x meter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-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 of2y79d*lso/ib(zh txr8a 3huq 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 8.0e/x( jgewo9s 6 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed wee(6js9xo g/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 for it to use energy stored i 3kqefdkg*( x:l(m9 or/r k9dme5(ghx n a heavy rotating flywheel. Suppose such a cd*h (xrd m(o3ke gfx/ :l9 eq5krg(9kmar 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 an jd(ep2. rcgp1ee ,7w q$H_2O$ 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 -.ap4 mjdhfna5y88xt5/ 8 kzkjuyd c1rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of massn1r 8*x0 jhq g5lssqm- M and length L can pivot freely (i.e., we ignore friction) about a hqr*l 58gm 0h-n jqxss1inge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, andah1 w7,vo5 kgvw7s8tu we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has he1g78v7 vawth 5 ,wuoskight h, where h

  A bicyclist traveling with 1jo,njyzak:km 3 tr7g2 6, hrrsi.9wcspeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the ogkw. j1 69cr 2 i:mhzarnkjy 7t,rs3,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 . +zjwqvoy/bx*9 8a qy8mw5svand 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,w*vwyzoj.mxqaqb s58v y98 /+ and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rdl1yv1t90mh;z m3sl+ ou6kirods, making reasonable estimates for the dimerytvum1kl9 +6ds ; 1mo0z i3hlnsions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which ik v +lkxy934 /8nsikrconsists of two cylindrical plates, of massxkn r43k 9i v8y+il/ks $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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+merm 70 p ;7wqxotnqo 8t98zo rolls along the looped rough track of Fig. 8–58. What iso; mr9e pqm +87o7z t0q8wxotn 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 doknk/2 pgp. qd4 not assume $r\ll R$ h =    (R-r)

  The tires of a car mf*iz5-a2 4wr )gckk hi *440pdmxg+ eexj q-vcake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angulari)r0h 2c efkzx4xje4avwk4p5-*id qg *-g+cm 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