A bicycle odometer (which measures distance traveled) iesdx)3 ky+w+ vs attached near the wheel hub and is designed for 27-inch wheels. What happens if yoy+ed k)sv+w3xu use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point 5e t6 koh5zj qs8qix03on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accele5jxk6 oh38eq5 z i0stqration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describeq k au5fcmb-28l 9u6ykd by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.ck-*wjz5b jla 14sa0ww:zlh(

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 h-d 6io7i lmx-rwlvk/:tp6 x0behind your head than when your arms are stretched out in front of you?dp/ vlhx-i6mt-il 0ro7w:6 xk A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at-+ 6ly7bmvh4 kl;lu ry 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 sproc 6k+l-7llhu; yb4mvryket or a large front sprocket? Why?

Mammals that depend on being able to run fast*4o1ir da bmy)e ys2+ pw;dl1i have slender lower legs with flesh and muscle concentraoyi*l2 1bp rdw)+y1a;id4e sm ted high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35b 93z+r fme9cs) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net tomfc) 62p8rrs vrque on a system is zero, is thep)26s8mrvr c f net force zero?

Two inclines have the same height but maoh g-hs44;jr zke different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be grez44- ohrs;gh jater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One i + hjk5+g*o9mcipg8w / oobo:sphere has twice the radiuo/h8g gbpoi:okiw+o* 59 +cjms 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 samjbbgpivx -),(i/7i)sy0q mkh .ln;qx e 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 thes0,j/qi( )vi)- h 7y.b bmq pgkxxil;n bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mo 8wax h6bgl/h:9yulm, mentum are conserved. Yet most moving or rotating objects eventually slow down and stop. m l aw/:,6yx9g8 ulbhhExplain.

If there were a great bqeo+ bloh 3j1,aflo+po,lo:nef- *, migration of people toward the Earth’s equator, how would this afa,,+hf:fepob3 l+lonooj * ,bo1 qle-fect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initiphhipg85*umri5q ,d( al rotation whenu5*8i,i5rm( hqpg dhp she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its cen q6hyjs//p zz*ter of ma sjzq 6*phz/y/ss 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 rotatingll: cwt/p,c-(b p/ l +gs-viox stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreab /p p/(i+vl,gl:xco- slw-tcse, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol tf h7l7w0bpij7p7cka8w8 yf5low and the other is solid. Describe an experiment to determine which iswfp0 8ijp5a7 b7cf8ltw77yk h which.

The angular velocity of a whee:v/nce2w 3uh:hc p hc4l rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the 3cc e c2 :4puw:hvhn/htop 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 rm, tg6+) +fjrj ruhe(the edge of a large freely rotating turntable. What happens if you walk toward the centm e6 r)hrr,gjf(j++u ter?

A shortstop may leap into the air to catch a ball and throw it quickpkbtbcr3r 0h h)a1b o;w2 vk:yn.)r* -uclj)b ly. As he throws the ball, the uppr )2akyb;r )hb)t3 .lk:vr -bhcp1oc n0wubj*er part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a; etqvb5;frjct* 7 ub) helicoptercer 5t uqf7 ;vb*tjb); must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: lym dcpict7 )m 6aj-(/x2em( e(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 coincid29ev( lugz 3kbc3 q+jience. Calculate, using the information inside the Front Cover,guqk z3(2bli+vc 9e j3 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,xb;p3 ) pml5rh 380,000 km from Earth. The beam diverges at a;bm h3p5plxr)n 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 is turnedaoo m:x2./b avawx p49 off during operation,/ :x2p .aob mvwx9aoa4 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 c4)c 9rfpm1y +ady m**g 7unqv:)hapxbhild. If the ball makes 15.0 revolutions, what igmqxync 7yb19)4m aaphp ) * +r*fd:vus its diameter?    m

  A bicycle with tires 68 cm in diameter trm:jz ea.w rdut m+xe m4v69+/uavels 8.0 km. How many revolutions do the wheels. +u4wud9tjz mxmr6m/:+e vae make?    $rev$

  (a) A grinding wheel 4a0/(a ts+x hb g daj0z3stw 7q;ha7nb0.35 m in diameter rotates at 2500 rpm. Calculate its angular ve/ja h+an(tt aw 3ba7h z7s0dsb;q0x 4glocity 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 makes9 :q x0jitj6j qv+m5p86dohrs one complete revolution in 4.0 s (Fig. 8–38). (a)xpv9jrj0io+656t :q h qsm dj8 What 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 Ea0ps5g3rpl3 ut5nqxc 3cnd1e3 rth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed c2zp,-x2 uwh5 zj fp 0ia0v;enof 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 ,6gja -e tsrlvcl)h: 0bg6sr9 particle 7.0 cm from the axis of rotation is to experience an acceleration of 1t -h9se 6s:0l,j)rgr6bvgc al00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpgi*j,2tiqcsz 0q,:6z:vy fai m to 280 rpm in 4.t i*,0: vf: qjygq i,2cza6zsi0 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 radius vp mii7 .ffkcmf0)u8;$R_1$ 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 Ap vha4k;n4o qv:7 qb2 fxu9bx(wollo spacecraft put themselves into a slow roavfn2794; x:vqwhub qx b4 (kotation 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 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 uniform/205vaxau: 9za6kmqgg jb nb/ly from rest to 15,000 rpm in 220 s. Through how many re9uakb/m a6g ng:q02jba 5v z/xvolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 12udz,,p gc:ka0mhz,vhk),x) .n6v y in00 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 stres-gnd z ,9onj 9aqswntmt/k/ 7bfx1l0 9ses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachb 9d m9 kto ng90-f a/q/lzwxtnj1s,7ning 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 accelerae/j9fz pvy60ates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a0vep 69yj z/fa 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/mins; errdy* h odh0xk4sk16d) n* It turns 1500 revolutions before it cd *n6oxh yk kd1d *)rres40h;somes 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 redu99:f jb0t yf9rq(.u3)jhgej0/pf3 x on wwubo ces its speed uniformly from 95kmuh.nwqwy99t/brjo (90:je03gfb o )u xff3j p /h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uv0hg-(a3pmszct ct84 niformly from 95km/h to 45km/h The tires have a diameter of pz-t0c3 mgts va4ch8( 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 d6jp 3v9u 6bh5c aio-i86 sanlall her weight on each pedal when climbing a hill. ip-uv bc69li5 8j6ha a63dosnThe pedals 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 oocw+y(3.v gg end of a door 74 cm wide. What is the magnitude of the torque if the force i.o(g+gvw3ocy s 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 the wheevm18y6ms:4ts*d e87kydbi jtm.rz r) l shown in Fig. 8–39. Assume that a friction48j s mt m v1rdim)8kybed 6t.z*:y7rs torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whicyd6 2o5 35pvsa)mcthakt+ tb +h pivots as shown in Fig. 8–40. Initially the rod is held in the horizontvyt+s2p a )d53ck tbt6aho +m5al 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 4 wrofc+.)ahxtightening to a torque of 38 c w) hf4orax.+$m \cdot N$ 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 2p0nzq*y ju;fof a 10.8-kg sphere of radius 0.648 m when the axis of rotation is t20qyn;* z fupjhrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamet dcv 77,e*0mcb;nzr9wz+bf q uer. The rim and tire have a combined mass of 1.25 kg. The mass cq ubfz z+9 *e;nr 0wb77,cdvmof the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rkckx7gls*wvbe nz-+bn1qt;e3 dv0vo e0q1- 3od is rotated in a horizontal circle of radiuv 0t b7zlex3nvcqg3;1keq + ndvw-o1*-skb e0 s 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 whee(z h0zukud5dt6 vh); rqh3b(f3mof r0f7fia.l rotating at constant angular speed (Fig. 8–42);zm(7dufor.fi 6tvz 50ffb)d h 3ka3hrhq0u(. 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 e1 0z 2g:ob; 1xdyroybobjects shown in Fig. 8–43:1brzgy2o; o0ye1 dxb 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 consis v l (48 al(qimnhtvz,t6dmt;1ts 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 cylindricas6 tug 9vwe7l: hjko20l satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 26hok7e sg0uvtw9lj :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* .m.0xi vztsx uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kgz*0xivx .mst .. 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 iyrfpy ;udhtny, q;3 *,t from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drub7e djicd. 0ag+ie 3x+q*ba+c7 j x/diven 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 7+db+*caj +xed37e 0 u/bx .cijqad 7ig60 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 is eventually)w2a( )r 6e0kifwtlar brought unifa(6ta wrl)ir )0kefw2ormly 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 3.6-kg ball at 7 bqjvw5dxd+0e /a8-sq $m/s^2$ 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-k1s:rzhy 2,rrmek.gpg-uhj hlec9981g ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the acurh91ske- : z,.lhr8p h2 jeyr19g mgction 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 considered a long thin rod, as shown in Fig.jg in3 s58 m,8xad0trv 8–46.88vn5gjr,amx d t3i0s
(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 consis1ouq1f -,z)dna 7buun:5;nf bectvh- ts 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 thejb8hvh5 jg, h*we:yd9 hammer from rest within four full turns (revolutions) and releasejd8y:wh eh,5 *gvb9hjs 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 ie /xa1g,yrgw.tmhg xyxee: ;4i4)a +xnertia 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 develops a tou4 f 8sjmka/*ij5 iwp7rque 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 r s /s xqf:6ljlql8h2o8z3 gt,jolls without slipping down a lane atotfjsqsxl hz8jg /q6l: l2 83, 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the s+ulajt/ 6 v,is;9,sy p,iyhup um of twvt 9l u,jasyih,ipu;s6py + /,o 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 kjv1 leouc,rd(y7 k)f-g and a radius of 7.50 m. How much net work is required to accelerate it from rest tov-ckr 7ud1) fljy,o( e a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from restbr.sl0(k:kb ,vwg+ bp and rolls without slipping down a bl(b s.kb:+ wk0gpr ,v30.0 $^{\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 staj 0qia2x7ib7 2ix1xe wrts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before72iwx ia0 7bqxxji21e it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating z2e .k/0m 1hpw/qk9. aaqqdq gon the end of a thin string in a circle of radius 1.10 m at gqkp2w9.edakzhq .1 q/0maq/an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular moment;lj/ xy:xi 1s)scx (fwum of a 2.8-kg uniform cylindrical grinding wheel of rai /wx)cjyfxs s:l1 ;(xdius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ri6.uk ,6 dmxfuate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation dkix6u f6u. ,mdecreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one showj*17jvuff3ep:m0je d p ir 0x3n in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the 71xp03r*: fejvej fdmju 0 pi3straight 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 incss zb.sy 1+gz8rease her spin rotation rate from an initial rate of 1.0 rev everysgbzs+ syz8 .1 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cen24yg+j kpvea*tdonfcyy4 +2( ter at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg andf(ec d ygtakj +vy2 *p24o+ny4 diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of imos-g:yl+9 ,4;hvv dknvt:* hed:hv 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 angularluk t:ahz a30h58e d6a momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto aew85pop y ) a36w.3egzd2u3zbssa1zs n identical dis) a ysd ezbz3gpau3.wezw6s28 31sop 5k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4cl.+ ibbhw4d k z:dff jb +34oij:f6+b $rev/s$ 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-ro3 xr ry;j7j,4 5rw8g6/sly8jjwyeyal und platforjlg w5yyr68 7y ,r8eyjawl3/j4s;jxrm of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular veloc)sdu- e )td4d*lmwsd*ity of 0d wsdds4*)-lu )tde *m.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing abor 4(4 7gcdnpjr4e ycx-ut half its mass in the pg4 cdr 4ryxpjc(4 -7enrocess, 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 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 exces6ezyoz*y2hi h8.0jk xs 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 6kgs9gyrkn 1gx*yjr2 :xw9 fapa9l86 $ 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 pkf+;* z3umrh alatform, initially at rest, that can rotate freely without friction. The moment of inertia of the p*+rkf; ahz 3muerson 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 merry3 e2 bli.k7w a04gfltka0us8f-go-round turntable that is mounted on frictionless bearings and has a moment of inerti40gtb0 fkaksi.ue 8wf3la2l 7a 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 grou)t-n6, bt8jro-vk,zcc 8ttmk dx1m 3knd 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. Themdn-k z3c8)tto- ,rc kjt m18,t kbxv6 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 65z amtflx0 /jthe same side always faces the Earth. Determine the 6fl0xm at5/zjratio 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 fr /s mow6kc7d -rx7n.q8li lv9np8q+cgom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cyl xk2l3:kjp z5qinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate th2lkp5j xz :k3qe torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0y0 b qb+ron)s tygp6keeyat/)4:; +wb if )t,q.050 kg and diameter 0.075 m,y e 6fi,wq )kgybqb;nyp +t/)rabt:ose+0t 4) 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 a.a/h wnt/0b u cj++xbty, gpb,ngular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, werjz; o5fbw wvo.yp9 -j)s0a;nka hr/k0 e rotating-;0 w5afkj;rv0ob/os )9ay.hwpk zn j 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 no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution a /ok n+hxsy(aw/4.ps: wz 2vwiutomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a fyzxs4 o kv (iww/sph.:nw2/a+lywheel “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 illustratf u 6klze4:4jn+bd9w zes 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 horizons ,g7z4luj q-4kkv 86ffah tu:tal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e.;8flqh:t7 7v5fa kwd l2z0d:xcpht vk /ut3 i4, we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rot ;/k lclz 04vfdhuit2kphw: 7tdaqx87f3v :5d. 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, and we wtd.wisa61l9u2o otti44 l 1vrant to exert a horizontal force F at its axle so that it will climbt4ltr9 it1a.o2 1iuv w4osl6 d 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 turns r+su dh;5wg) with a radius The forces acting on the cy)5s +wgr;dhsu clist 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 intorpbw y emh6fi xp*x2o23*d+9+j)tju u a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque requi)e+t6jyo2 r*b uux jwpfdp29i+*xmh3red 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 arms as thin rods, x *.3 npsqg/akf )hmw1w,k4 2n gv2wzamaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wnnkap34*g 22)wx /s1 h,a wwv.mfgzqkith arms held tightly against her body.   

  You are designing a clutj te) k5e51:jp a;jntlch assembly which consists of two cylindrical plates, oflj; tnjpjt ekea55)1 : mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track oa*c+p7u :hm,t9b i nxvf Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hig xcpab79ht:v,mn* ui+ hest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do n0yylbvl u-dw08y7l 1hay d2 c/vr(k3kot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed ukw3hy aeavku 3:(--a2yaxx z 3niformly from 90km/h to 60km/h The tires have a diame xhuz3v-2k a:-w 3xa3keya ya(ter 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