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?

Attached Files:

• Snap1.jpg
File size:
10.5 KB
Views:
106
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.