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Old Sep 13th 2017, 05:27 AM   #1
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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 x-y 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
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Old Sep 13th 2017, 08:04 AM   #2
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(a) consider a rigid mass with mass elements, $m_i$, in the x-y plane, each with position $(x_i,y_i)$ ...

$\displaystyle I_x = \sum m_i y_i^2$

$\displaystyle I_y = \sum m_i x_i^2$

Let $r_i$ be the distance of each $m_i$ from the origin. Note $r_i^2 = x_i^2+y_i^2$

Moment of inertia about the origin (the $z$ axis) ...

$\displaystyle I_z = \sum m_i r_i^2 = \sum m_i (x_i^2+y_i^2) = \sum m_i x_i^2 + \sum m_i y_i^2 = I _y + I _x$
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Old Sep 14th 2017, 05:05 AM   #3
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Hi Skeeter
Thank you for taking the time to help me understand this one. I hadn't realized that the approach to the solution was so straight forward. I see it exactly now that you have shown me what approach I should have taken.
I think got confused as I had thought that I had firstly to decide on a shape to be analyzed with respect to Moment of Inertia and then apply the appropriate expression for that shape in order to solve the thing.
I think that once I had decided on this then allowed myself to take a blinkered view of the problem and therefore a blinkered view on how to approach the solution. I suppose should have tried to look at the problem from a different perspective once I saw that my attempts were on the wrong track but I didn't. Another lesson for me.
Anyway thanks again for explaining what was required so clearly, much appreciated.

Regards,
Jackthehat
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