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MOMENT OF INERTIA

OBJECTIVE: To determine experimentally the moment of inertia of an object and to comparethis with the theoretical vale obtained from the mass and dimensions of the object!

T"EO#$: %hen an object at rest is set into rotation abot some axis& it has a tendency to 'eep

rotatin( at some an(lar speed& , measred in radians)sec! This tendency is called therotational inertia and is characteri*ed by a physical +antity called the moment of inertia& I& ofthe object! ,oment of inertia is the rotational conterpart of inertial mass in linear motion!"ence the 'inetic ener(y of a rotatin( object is:

KE-I. /01In the experiment we set the object rotatin( by attachin( it to a han(in( wei(ht and allowin( thewei(ht to fall! Before droppin( the wei(ht& the ener(y is all potential ener(y& that of the han(in(wei(ht! 2t the instant the wei(ht hits the floor the ener(y is all 'inetic ener(y& both in therotatin( body and in the fallin( object! I(norin( friction and the rotational 'inetic ener(y of the

plley /which is reasonable to do since both the object and plley rotate with little friction andthe plley3s moment of inertia is extremely small compared to that of the object1& we can applythe 4aw of Conservation of Ener(y:

mgh- mvf.5 If.! /.1

The velocity /v1 of the han(in( wei(ht is related to the an(lar velocity /1 of the rotatin( object

by v- rwhere ris the radis of the drm on the object! The final velocity is related to theavera(e velocity& and the avera(e velocity can be fond from the distance dropped /h1 and thetime /t1 for the drop: vavg- /vf5 vi1 ) . & and vavg- h ) t& so that /with vi- 6& i!e!& startin( from

rest1 vf- .h) t! %e can now sbstitte for both vfand fin E+! /.1 and solve for Iin terms ofthe measrable +antities m, g, h, tandr :

I- mr.7 /gt.) .h1 8 0 9 /1

The above e+ation will allow s to find the moment of inertia experimentally for any object!%e will compare or reslts for a niform& solid dis' and a niform rin( with those derived from

theory! ;sin( the definition of moment of inertia& I- r.dm& one can show that theory predicts

Idisk- MR. /ote: the ?5@ si(n is correct in E+! /=1A1

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,oment of Inertia

#OCE;#E:01 Be(in with the empty apparats as the rotatin( object! [CAUTION: 2djst the hei(ht of theplley so that the strin( will wind and nwind from the apparats strai(ht withot (ettin( tan(ledp in the bearin(sA DTO the rotation of the drm once the wei(ht hits the floor to preventpossible tan(lin(A] #ecord the hei(ht h! In selectin( a mass to start the apparats rotatin(&experiment with different masses ntil yo find one that will (ive a time of fall between and seconds! %hyF /"I>T: consider sorces of error!1 Once an acceptable mass has been fond&

time at least three falls and then determine an avera(e time to se in E+! /1! %hyF >owmeasre r& the radis of the rotatin( drm& with a pair of calipers and se E+! /1 to calclate I!Call this moment of inertia Iapp& the moment of inertia of the apparats!

.1 >ow place the rin( in the apparats and find the moment of inertia as yo did in Dtep 0sin( E+! /1! The moment of inertia that yo find is the moment of inertia of the apparats4;D the moment of inertia of the rin(! Dbtract yor vale of Iappfrom this Ito find themoment of inertia of the rin(& Iring!

1 #emove the rin( and place the dis' in the apparats and find the moment of inertia!Dbtract the vale of Iappfrom yor vale of Ito find the moment of inertia of the dis'& Idisk!

ow place the rin( on top of the dis' so that both are in the apparats and find the momentof inertia! Dbtract the vale of Iappfrom yor vale of Ito find the moment of inertia of thedis')rin( combination& Icomb!

=1 ,easre the radis of the dis' and the inner and oter radii of the rin(! #ecord the massesof the rin( and dis' which shold be painted on the rin( and dis'! >ow se E+s! /

=! Compare the masses and the moments of inertia of the rin( and dis'! %hich mass is

bi((erF By how mchF %hich moment of inertia is bi((erF By how mchF Explain theseresltsA

Check ith !o"r instr"ctor on the ne#t to steps$ the! ma! be optiona% for an in&c%ass report'

H! ;sin( al(ebra& derive E+! /=1 by considerin( a rin( to be a cylinder with an inner cylinder ctot of it! /"I>T: The rin( is of niform density! Be carefl abot sin( the appropriate masses!1

! Dhow that E+! /1 can be derived startin( with - I& instead of the 4aw of Conservation ofEner(y as was done here!

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