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Blazing goIITian

Joined:	8 Dec 2006
Post:	351

9 Jun 2007 15:20:36 IST

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408

a fundamental query

Engineering Entrance , Medical Entrance , AIPMT , JEE Main , AIIMS , JEE Advanced , Physics , Mechanics

while finding the moment of inertia of a body frm an axis why do we integrate r^2 dm and not any other value of r ?why is the distance from the axis of rotation chosen as r^2?

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Priyesh

Blazing goIITian

Joined: 18 Feb 2007

Posts: 1042

9 Jun 2007 16:55:16 IST

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see

first of all let us see the definition of moment of inertia

now i will do this with the help of an example

for eg: we have to find the kinetic energy of a circular body rotating about it's centre of mass with angular velocity w(omega)

now total kinetic energy = sum of kinetic energy of all particles

= 1/2m₁v₁² + 1/2m₂v₂² + 1/2m₃v₃²..............

= 1/2m₁r₁²w² + 1/2m₂r₂²w²+ 1/2m₃r₃²w²........... (as v = rw where r is the distance of the particle from axis of rotation(centre of mass in this case).

=1/2(m₁r₁2 + m₂r₂²+m₃r₃² ...........) * w²

1/2 * I * w²

therefore we see that I i.e moment of inertia about an axis is the sum of products of the mass of the particles and square of thier distance from that axis (centre of mass in this case) (definition)

So when we integrating we take a small particle of mass dm & measure it's distance from the axis chosen & then square it therefore the integral becomes r²dm

therefore it is now clear from definition that r is always the distance of the particle of mass dm from the axis and not some other arbitrary r

I HOPE I'VE CLEARED YOUR DOUBT

CHEERS

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