A bicycle odometer (which measures distance traveled) is attacxw ns 5.hzmwh45si -e-hed near the wheel hub and zh. s5m-eiwn5 -hwsx4 is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocita0htv,e*vud-dk w 8m9 9)wtso y. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear a9h 9vt0semk ad,-tw v)uw do8*cceleration change?

Could a nonrigid body be described r(tv,*hhcnmpr4qt 85 aqda/vw9/ rea6w: e m7by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than +c 9gz76r9 .p 2yofg1u) zu dwok/ihhma 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 -u,f, gwl a6qwto do a sit-up with your hands behind your head than when youraqg -l6w w,u,f arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thre8oq( 55 ergbh / g-kab+wp6jrde at the pedal cranks. In which gear iskb5rqg w5+ (egad6b/ o-rjh8p 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 flesj1wr q8:6 ub-dij4z(pu qoo*th and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamiwdqqj zi-o:t48*(op6 rubu1 jcs, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lonz3 ;8/l. l xycuv z7wb*aoy.-gqt 1fzig, narrow beam?

If the net force on a system is zero, is the net torq+fs i6a2 nun2jue also zero? If the net torque on a system is zeron6sin+f2 u j2a, is the net force zero?

Two inclines have the same z;j34,vvuu3wpz6; a xpr1nxx height but make different angles with the horizontal. The same steel ball is rolled down each incw 6axuu3 v xp xn34j;1vzpz;r,line. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollowry/; my+ts9kb 9. mwing (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the im.9+/y9ot ; wbrmsky wncline 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 w*xrs+vf p-:y3z;xh n and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater sp*vz:3fspx +;rw- x nyheed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momdn6n ) juzm*z)entum are conserved. Yet most moving or rotating objects eventuazd n)m*z)un6jlly slow down and stop. Explain.

If there were a great migration of people toward the Ea8p*fzlj mk6) orth’s equator, how would this affect the length *m zko )lj8p6fof the day?

Can the diver of Fig. 8–29 do a somersault without having+ 7mxg)rwp.hp 8sfmq.fj 0w, d any initial rotation when f.dgw+. mp0r8x, )qm7hfjp wsshe leaves the board?

The moment of inertia of a rotating solid disk about an axis through auh1jsg s 8rx44mz8 s3its center ofrs44 sj g13uxs8hz8a m mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass i )glf 3pf5kykg7, ywi ,5xr(nln each outstretched hand. ) l3y5ykk5r wfig,ln,px 7(fgIf 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. y zh)mw90jpy(zqmhe ( jl381lHowever, one is hollow and the other is solid. Describe an experiment to zljeq(m(yw18)9 3y pzlh0mhjdetermine which is which.

The angular velocity of a wheed ,aqbhrajsy*i-yz::z sv :k(;e2 g-kl rotating on a horizontal axle points west. In what direction is y,drh(: sva* qzej b-:k;a: sgzi2k-y 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 freelyyb5mg8w8 xt()* t cnko rotating turntable. What ht kt 8*bxomg8)wycn 5(appens if you walk toward the center?

A shortstop may leap into the v x 6n4iyw63lbair to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If youy6 4vx36wi lbn 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 momejo)-fr8 5a+taf2w ct1x .l. eqmm0u ryntum, discuss why a helicopter must haver2y te.fm5x81o) tlu c+.-aqa0 wfmjr 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: (a) oma /nk n8,mnwgq( *i*30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate,y ;s(6eur 9wou using the information inside the Front Cover, the u 9eo;r6wy us(angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam:,yfztyse3+pn0u 7n lb,4 orknqv- ,c diverges atlp,t-o e4kq, rybc0s 3nnyn:+z fvu7, an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When t l5rcq+n41(aem3tg8ho ejxx1he motor is turned off during operation, the blades slow to rest in 3.0 s. Whe8r 3gxcn+5m1t oxqahj le 1(4at is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child./vin al c 67 ul00ftybe/t:pcd/zl+m 1vp6i9f If the ball makes 15p n u6ecfz/y+pft/ imatv0 i: l6 1/bl790ldcv.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revol - 4lb 0tnkvhcei9;e4d bc:mg4utions do the wheels make?tk94de:4- gv ic0;l4hbbcn m e    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcul; jy.whj ak.:jj d,r3gate its angular velocirw,:3 g; jk.jdh. yjajty 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 (,w)ie2sn 7,;aq7m wo tiaw2sgFig. 8–38). (a) What is the linear speed a,s)no mis7tw qg,wwe7i22a;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 Ea1wdlwo0 h6ho) l9 -ebzj18kcrrth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinya/j zh3;mny ygy-*fg h cr564t
(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 centrk rmamxzvtr+qzy79t, k1i4 - :ifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accelera:,zt 1km t7qairxk -v9r4+ myztion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 6 mwzko )kjl+npi2-- f7la0dlrpm to 280 rpm in 4.0z6n p-laiklo0f m)7 2jlwk-d + 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 4 ;tls5uo* mgr$R_1$ 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 spaceyg/da -lzfj9+qf4cz0sbe n47 craft 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-m 04fgneblqz+dy z 4cjasf -79/in 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 t1v41m *rf8vk 3 l3da.vk xfixko 15,000 rpm in 220 s. Through how many revolutions did it turn in this tixak f3k1 km *vr.d4flx3v 18ivme?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. 0fe(x( ilzb+o t.cmo 9Calculate
(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 theslh c., m4 .y/c,xizio7cpo6ap j; q9e stresses of flying highspeed jets in a whirling “human centrix69qci y,,ao cpsleihz c pjo47.. ;/mfuge,” 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 accele.t :/f r6oisyzs -lj9srates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travelf/9 z.jt6lyrs- :sosi ed in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Itu+ 8fwma./qdj6 zm7. l4xozgg turns 1500 revolutions before it comes to a stop. 4dgm7 .uozqm8x6 wgfz+a/lj .
(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 reducoaj1( ;yrogc*/e18z jd sswdo69v6 fces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m. zrs9 (o6ofdjd/6o c ajse1v1; * wgc8y
(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 rt8jtx nhr 0z*-l 6h,sp0aaceu2 2uuw/evolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diam2p cjn*z hursh ,06 xl-at0eu8/ u2twaeter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each ng ( ot,e/hl/0pqjlh+ pedal when climbing a hill. The pedals rotate in a circle of,np/lghqht +/el j0(o radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cmg7(s rv(mxua5 ea0qzs:)p/ l)9 l tksn wide. What is the magnitude of th0 gks5lauxl (:) )apnres /t( z9s7qmve 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 of the wheel shown d) se eyya0).kin Fig. 8–39. Assume that a friction torque kyy ))0eesad.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 en it tb6ld, +k(/o:)gxkwmywi/ds of a massless rod which pivots as shown in Fig. 8–40. Initt o:ktwwy, blg/+ x iid(6mk)/ially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a tornu:ex/1j; dqrtvfdb+ i78 uj(que of 38 ru1d/7dif+njqe8( ;xt jvu b:$m \cdot N$ 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 o5cc+ensew, 2-t2 aq noef83iw f a 10.8-kg sphere of radius 0.648 m when the axises+c,i35f c owwnte2q- 28ae n of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of34 anl4hxhw:j*f jr;d*:grk b inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The rjhfk ad*:l 434rgw;bjn*x :h mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod ljt3;*ydvv 08 mne)3zkdu7ha *r fa-ris rotated in a horizontal circle of radiu 3fh dvz*0)7un *ymj;aat-v8drk3rl es 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 ax k77a yhb99bxo67wnz potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction fo6bkoy7wn7h9 z9b a7xxrce 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 l. csvyf60rgp7),4nv bt8ak ain Fig. 8–43tyvgk 7p6slf,r)0b aa48 .cvn 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 oxygen atoms whose total man *hd1l,d pd;h ca;cesr/f pma75*z2hss 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 s,b:qa8o kwdku1:ng d4 pinning 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 isag8kn 4qdkb:u 1d,:wo 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 i3if1lo ,v7fiyl,i0 m uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cal7mii0y3i,lf, v1 fl oiculate
(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, acceleratiazhljhum0hc (ne 6n)51 :vh dvl;l5+q ng 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 tangentially on a small hand-driven merry-go-n 2lkghkzdra1/3 m,8qround and is able to aznh2krgq8k1 ,3dla /mccelerate 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 is eveo x5t3 uics+3f ndsvt5m7v*.sntually brought uniformly to rest by a frictional torqt+*3x5sts v5fuins v7 .modc3ue 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 bal4g5k,k luv +s.e0rm tyl 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 bytmt068j /;t6rxq jtb m the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniform/ rtq68j tmmtxj;6 0tbly 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 atvn+w8 6prs p70r3cip;e vs2,ky m( xp long thin rod, as shown in Fig. 8–4xe;tp2(6v 8w 7+03 p cpips,mrvrkyns6.
(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 massesq(f+rit; f/ pq, $m_1$ 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 from rest withinyt.m19j ng4 bk four full turns (revolutiob.mkn4tj y 19gns) 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 inert0 e,0pvuks, ye oiv0,qia 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 torqutjb78pg5k71 lb8srp g ja)tjgz 4yl(3e 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 andi)scex b3 .h;+ u- 7ovup6fnkf q0bg2m radius 9.0 cm rolls without slipping down a lane at 3.3 ovexu 637+b)cgu;q - kfnh2 mip.b0fs $m/s$ Calculate its total kinetic energy.    J

  Estimate the kineticknh..7 exxg tz2;po+0xgnk4x energy of the Earth with respect to the Sun as the sum of two termsnkg ;xgtx7 z.ox.e4x 2 +kphn0,
(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 e o,ok+7g,8 hs3v1/ ,g u: kqgjp1iua: vhedemkg and a radius of 7.50 m. How much net work ,pegoduq ,j :ah1 kvoskv/, mi3he+:egug718is required to accelerate it from rest to 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 starhkyqr,l),5l*ivw c6v td p6,q ts from rest and rolls without slipping down* 6vhiq,,6dv5rtk,ll ywqp c) 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 fall and itsutwd l1n8sm5 - lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [-wd1nt58sulm Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rwc:kp i 11(dwcbn5bl+otating on the end of a thin string in a circle of r1bk w b:pcci(5lnwd1+adius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniformdcrnsd s/)m1e,5ht0 e7t(/ tqnkrh+o cylindrical grinding wheel of +emd7/tn 5)thr/ (sk,t 1sqncdhr eo0 radius 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, oy z fvhlx2xmh g34(co6agri;: f6on,8n a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. hz6xy8gv alfnm:r4fgoc hx 2o,;3(6 i8–48, 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. sqzq7k5- pkf0 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotationsf-5qpz7qk 0sk in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.04n7i xwok. wy. rev every 2.0 s to a final rate oy4i.koxn w7 .wf 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 center at a t 5k8nhh2 a5vxtjl;v* v+7l ypfrequency of 1.5rev/s The wheel can be considered a uniform diskvvtt h*n kyp5+2 7j5;8 xvlhal of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skat1o*g vx);5 gp3fs;n8bj)pfu+nc ix vier 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 teo2x5l0: groy he 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 Ic/.fy ev9bo:t is dropped onto an identical disk rotating at angulay v9cb/.:tofer speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.dih8on9xzo j-)+ etdj; *4f /unk+wsd4 $rev/s$ 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 mexs m;ic4(gb+viem+ .ai g:5gyrry-go-round platform of radius 3.0 m and moment of inertia 920 g+ x cvmi5gbi(+s4y;i.a:gme $kg \cdot m^2$ 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 fre)ifu1c8noya6ee j +k2gv*e c.ely with an angular velocity of 0. c a.key6n *ovc 1u2f)j+e8ige8 $rad/s$ 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 collapsesbyn, +xpc.3rz9v3fu z -d:ran 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 ryp3zr9-acx.nvf :ubz+,3 r n dotation 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 winmg) 3/kxg 8,bh g4ibk/y o6zxxyzpde 4+g-8pjds 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 y( /nqpqvcav282 /ojd$ 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 plm(au-/maqnjnc,30 6 bc kmp+latform, initially at rest, that can rotate freely without friction. The momenu bm(+,- 6/p3lqc m mnjn0acakt 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 stands at the edge of a 6.5-m :(qiq.c.o)v,xnlytc z/ d8acdiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170.a,q/(clzy8n ic dx:cvqo. )t0 $kg \cdot m^2$ 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 wc8ubsu87xtmy 60:j yjoyp8 ; the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a 8j6 u8s0u jp:b8 7yoyy ;cxmtwdistance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Dek:a)5 7mqhcs1wi,osn termine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treas7)1k,so m5:nhciqwa t the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from n e6u9 hka6wq9-wc j4s :kl2gvrest 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 cylinder of radius 0.2)vxys- h)cyd+*nh i 9w0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accehcwdy)yh x*sn-9+i)v leration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of tw d w(mia1y:7dlo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to m: iyd(1wad7lcalculate 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 isnt (h+vx hlmapw;j *+: the angular 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, were rot x gq1 hgez5d3b+pwhhcv0 ,2)-ygp0 /wnl1y sxating 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 unp/gh5s,dcepgy+ n g0h1b)lv32hx x 0wwq-zy1iform 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+enn1cc5g0 an 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 2cnanc15 + gn0e40 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 illustrat3jua:sg)g n .+nm0ihdes 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 horizontal surface)3rb08s 5jp36xkgsfnwgqy *s 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 andf w1 l9nyq5ba- length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acnay 5b9q-1 wflceleration 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 radiuk zhg u;lmtclwh9.*e9v3 g.q :s R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–;lql . tvhew:*.9ukm gghcz 3955). The step has height h, where h

  A bicyclist traveling y*jjn mdim4 -+/q 9mohwith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle m9oqjn+d4 yi/hmjm -* 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 ad/d ild qu3-f8j8hquo 1g8i1jnd begins whirling the rock in a nearly horizontal circle above his head, accelerati3/i1u8h8 l8dq ujd-d1o qj gifng 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 cylinder and her arms as thin rods fqd 9zbxk/2g/ 43womn, making reasonable estimates for the dimensions. Then cwgxdzn94q f/3 2o/mbkalculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pl,*pz/f82um gv zy 4nhimk(l* cates, of n,fgh 8cplz* 2yi ( mk/vm*zu4mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alh3ti,amk*bau p ec09cz5i:u/ong 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 leaving the track? Asb3c/c0: ,mptuau*5k izeh 9iasume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumessf)xj .,2 1o iga.tcx $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its ( iws3.y5v +kp zwntq*3 ghuz)speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What w3uszz* k tpny+h3qi) 5(w.wgvas 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