1.

State and prove perpendicular axis theorem.

Answer» <html><body><p></p>Solution :Perpendicularaxistheorem : theperpendicularaxistheoremholdsgoodforplanelaminarstartsthattheis equalto thesumof momentsof inertia about twoperpendicularaxeslyingin theplaneof thebodysuch thatall thethreeaxesmutuallyperpendicularand havea commonpoint <br/>Let theX and Y- axesliein the planeandZ-axisperpendicularto theplaneof thelaminarobject . ifthe momentsof inertiaof thebodyaboutX and Y-axesare ` I_(X) andI_Y`respendicularand havea common <a href="https://interviewquestions.tuteehub.com/tag/point-1157106" style="font-weight:bold;" target="_blank" title="Click to know more about POINT">POINT</a>. <br/>Let theX <a href="https://interviewquestions.tuteehub.com/tag/andy-874984" style="font-weight:bold;" target="_blank" title="Click to know more about ANDY">ANDY</a> axesliein theplaneand <a href="https://interviewquestions.tuteehub.com/tag/z-750254" style="font-weight:bold;" target="_blank" title="Click to know more about Z">Z</a> - axis,thenthe perpendicularaxistheoremcouldbe expressedas , <br/>`I_(Z)= I_(X )+I_(Y)` <br/> to provethistheorem. let usconsidera planelaminer objectof negligiblethicknesson whichliesthe origin(O ).The Xand Y- axes lieon theplaneand Z-<a href="https://interviewquestions.tuteehub.com/tag/axisis-889956" style="font-weight:bold;" target="_blank" title="Click to know more about AXISIS">AXISIS</a> perpendicularto itas shownin figure.Thelamina isconsidered tothemadeup ofa largenumberofparticlesof massm . Letuschooseone suchpartivle at apointP whichhascoordinates(x, y)at adistancer from O . <br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/FM_PHY_XI_SP_03_E01_037_S01.png" width="80%"/> <br/>Themomentof inertiaof theparticleaboutZ- axisis `mr^2 `<br/>thesummationof hteaboveexpressiongivesthe momentofinertiaof thelaminaaboutZ- axisas , `L_(Z)= summr^2` <br/> Here,`r^2=x^2+y^2` <br/>then ` I_(z)= summ(x^2+y^2)` <br/> ` I_(Z)= summx^2+ summy^2` <br/>in theaboveexpression , theterm` sum mx^2` is the moment ofinertiaof thebodyabouttheY -axisand similary the term` summy^2` is themomentof inertiaaboutX - axisthus<br/> ` I_(X) = summy^2and I_(Y)= sum mx^2` <br/> Substitutingin theequationof `I_Z`gives <br/> ` I_(Z )= I_(X ) +I_(Y)` <br/>Thusthe perpendicularaxistheoremis proved .</body></html>


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