Moment of Inertia for 2D Rectangle: Does it Depend on Both Sides?

[peripatein]

Hi,

Homework Statement

Will the moment of inertia of a two dimensional rectangle (with sides a, b) whose axis is parallel to one of its sides and passes through its center of mass, be (1/12)M(a²+b²)?

Homework Equations

The Attempt at a Solution

 

[haruspex]

Homework Statement

Will the moment of inertia of a two dimensional rectangle (with sides a, b) whose axis is parallel to one of its sides and passes through its center of mass, be (1/12)M(a²+b²)?

No. If the axis is, say, the y axis, think of the lamina as a set of parallel thin strips in the x direction. They'll all have the same moment. I think the formula you quoted would be right for an axis perpendicular to the lamina.

 

[peripatein]

But then wouldn't I be getting a moment of inertia equal to (Ma^2)/12 (supposing axis is parallel to a)?

 

[haruspex]

And what's wrong with that?

 

[peripatein]

It's not that something's wrong with that, simply that it seemed a bit strange it would not depend on the other side too.

 

[haruspex]

It's not that something's wrong with that, simply that it seemed a bit strange it would not depend on the other side too.

That's why I said to think of it as parallel strips across the axis. Adding more only increases the moment in proportion to the mass.