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State and prove perpendicular axis theorem. |
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Answer» Solution :Perpendicularaxistheorem : theperpendicularaxistheoremholdsgoodforplanelaminarstartsthattheis equalto thesumof momentsof inertia about twoperpendicularaxeslyingin theplaneof thebodysuch thatall thethreeaxesmutuallyperpendicularand havea commonpoint Let theX and Y- axesliein the planeandZ-axisperpendicularto theplaneof thelaminarobject . ifthe momentsof inertiaof thebodyaboutX and Y-axesare ` I_(X) andI_Y`respendicularand havea common POINT. Let theX ANDY axesliein theplaneand Z - axis,thenthe perpendicularaxistheoremcouldbe expressedas , `I_(Z)= I_(X )+I_(Y)` to provethistheorem. let usconsidera planelaminer objectof negligiblethicknesson whichliesthe origin(O ).The Xand Y- axes lieon theplaneand Z-AXISIS perpendicularto itas shownin figure.Thelamina isconsidered tothemadeup ofa largenumberofparticlesof massm . Letuschooseone suchpartivle at apointP whichhascoordinates(x, y)at adistancer from O .  Themomentof inertiaof theparticleaboutZ- axisis `mr^2 ` thesummationof hteaboveexpressiongivesthe momentofinertiaof thelaminaaboutZ- axisas , `L_(Z)= summr^2` Here,`r^2=x^2+y^2` then ` I_(z)= summ(x^2+y^2)` ` I_(Z)= summx^2+ summy^2` in theaboveexpression , theterm` sum mx^2` is the moment ofinertiaof thebodyabouttheY -axisand similary the term` summy^2` is themomentof inertiaaboutX - axisthus ` I_(X) = summy^2and I_(Y)= sum mx^2` Substitutingin theequationof `I_Z`gives ` I_(Z )= I_(X ) +I_(Y)` Thusthe perpendicularaxistheoremis proved . |
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