# Answer to Question #15601 in Mechanics | Relativity for ARUNANGSHU

Question #15601

About which axis of a body 'Moment of Inertia' is the least?

Expert's answer

First you need to understand the definition of moment of inertia of a body about "a-axis":

I(a-axis) = Integral [ rho * r² dv ] integrated over all the infinitesimal volume elements "dv" of the body which are "r" distance apart from the a-axis. (density = rho, assuming uniform body).

You can easily understand that minimum moment of inertia means choosing the axis "a-axis" here such that the integral is minimum, and the integral will be minimum when "r²" will be minimum, so cleverly if we can estimate for which axis the average of "r²" will be least then we can be sure that this particular axis for which average of r² or the "root mean square" is least then the moment of inertia will also be least.

An Example:

A Thin Rod: very easily you can see that average r² will be least if you take the axis to be passing through the center of the rod and along the length of the rod then M.I will be least, because maximum r² in this case will be radius² of the rod and as the rod is thin so radius is very small. Suppose one takes the axis to be perpendicular to the rod and at the edge of the rod, clearly if the length is "L" then r² can go up to "L²" which is much bigger than radius² of the rod.

I(a-axis) = Integral [ rho * r² dv ] integrated over all the infinitesimal volume elements "dv" of the body which are "r" distance apart from the a-axis. (density = rho, assuming uniform body).

You can easily understand that minimum moment of inertia means choosing the axis "a-axis" here such that the integral is minimum, and the integral will be minimum when "r²" will be minimum, so cleverly if we can estimate for which axis the average of "r²" will be least then we can be sure that this particular axis for which average of r² or the "root mean square" is least then the moment of inertia will also be least.

An Example:

A Thin Rod: very easily you can see that average r² will be least if you take the axis to be passing through the center of the rod and along the length of the rod then M.I will be least, because maximum r² in this case will be radius² of the rod and as the rod is thin so radius is very small. Suppose one takes the axis to be perpendicular to the rod and at the edge of the rod, clearly if the length is "L" then r² can go up to "L²" which is much bigger than radius² of the rod.

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