# Moments of inertia of a rectangular plate

1. Jul 10, 2015

### Karol

1. The problem statement, all variables and given/known data
Show that the moment of inertia of a rectangular plate round it's diagonal, line B-B is equal to the one round a line parallel to one of the edges and passing through the center, line C-C

2. Relevant equations
Moment of inertia: $I=mr^2$

3. The attempt at a solution
For the first case, round the diagonal: the MOI of a straight angle and isosceles triangle round one of the 2 identical edges is $I=\frac{1}{6}ma^2$, and if i divide the plate into 4 such rectangles and calculate MOI it equals the one round line C-C.
But is there an easier way?

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2. Jul 10, 2015

### Orodruin

Staff Emeritus
Depending on your background knowledge, the symmetry of the square will tell you that they have to be equal.

3. Jul 11, 2015

### Karol

I have little background knowledge, please hint the subject that i have to study, is it principal axes? as far as i (think) i know i can't know anything about the MOI itself around principal axis, especially if they aren't orthogonal, but i am not sure.
Please guide me to a book or publication that explains this topic, but, if possible, without the complexity of determinants, tensors etc.

Last edited: Jul 11, 2015
4. Jul 11, 2015

### Orodruin

Staff Emeritus
You can do it with principal axes, but all that is required is noting that an axis through the corners is a linear combination of the axes through the center and the center of the sides, which must have the same moment of inertia due to symmetry.

5. Jul 11, 2015

### Karol

So i just have to know the formula for modifying MOI with the changing of the angle of the axis?

6. Jul 11, 2015

### Orodruin

Staff Emeritus
If you want to call it a formula, fine. The point is that any axis in the plane of the square has the same moment of inertia because the axes spanning the plane do.