Moment of Inertia: Calculation Method

[Benjamin_harsh]

Homework Statement

Ratio of moment of inertia of a circle and that of a square having same area about their centroidal axis is:

Relevant Equations

Why they calculated $\large\frac{d}{a}$ in first step?

Ratio of moment of inertia of a circle and that of a square having same area about their centroidal axis is:

Sol: both area and square have same area:

$a^2 = \large\frac{π}{4}\normalsize d^{2}; \large\frac{d}{a} =\large \sqrt\frac{4}{p}$

Ratio of moment of inertia of a circle and that of a square about their centroidal axis is:

$\large\frac{I_{c}}{I_{s}} = \large\frac{\frac{π}{64}d^4}{\frac{a^{4}}{12}} = \large\frac{12π}{64}\frac{d}{a}^{4}$

$\large\frac{3π}{16}(\sqrt\frac{4}{p})^{4} = \frac{3}{π}$

 

[PhanthomJay]

It could have been calculated later, but that ratio is needed for the equal areas
is

 

[haruspex]

moment of inertia of a circle and that of a square having same area about their centroidal axis

These questions you are working through keep referring, ambiguously, to "the" centroidal axis. I have tended to assume they mean the one normal to the plane. In the present question it doesn’t matter whether it is that axis or any axis in the plane as long as the choice is consistent, but looking at the detailed calculation they seem to be referring to an axis in the plane. If they meant normal to the plane then both values would be doubled.
Interesting...

 

[Benjamin_harsh]

Correction in last step: $\large\frac{3π}{16}(\sqrt\frac{4}{π})^{4} = \large \frac{3}{π}$.