Perpendicular axis theorem problem
Hi everyone,
I have come across a problem that I am afraid I really don't know how to start to tackle. It is a problem of the type  to show that a theorem works for particular situation from first principals using generic elements and coordinate system and so prove the theorem in general terms, rather than being given actual values for certain variables needed in order to solve the problem.
The problem is as follows .. PROBLEM
Consider a rigid body that is a plane sheet of arbitrary shape. Take the body to lie in the xy plane and let the origin 'O' of the coordinates be located at any point within or outside the body.
Let 'I(x)' and 'I(x)' be Moments of Inertia about the 'X' and 'Y'axes, and let 'I(o)'
be the Moment of Inertia through 'O', perpendicular to the plane.
(a) By considering Mass Elements 'm(i)' with coordinates '(x(i), y(i))', show
that .. I(x) + I(y) = I(o) .. (This is called Perpendicular Axis Theorem)
[ Note : Point 'O' does not have to be the centre of mass]
Now as I say I was at a bit of a loss as to how to proceed. I did try at first to draw a diagram of the situation but was unsure of how the diagram should actually look. I tried a couple of times to come up with a suitable diagram but was, and still am, unsure of what I should be drawing.
I then tried to calculate I(x), I(y) and I(o) using equations for Moment of Inertia for a rectangular plate although I do not think that I can assume this unique shape for the body described.
I took I(x) and I(y) for axes along sidesof the plate in x and y directions and axis through centre of plate for I(o) as follows .. I(x) = 1/3 M x(sqrd) ; I(y) = 1/3 M y(sqrd) ; I(o) = 1/12[(x)sqrd + (y)sqrd)]
However this does not lead to the required solution and I don't know where to go from here ... can anyone help ?
Regards,
Jackthehat
