# Homework Help: Finding the moment of inertia

1. Apr 1, 2013

### hms.tech

1. The problem statement, all variables and given/known data

Two uniform square laminas are combined into a single body. One lamina ABCD has mass 5m and the other lamina PQRS has mass m. The lamina PQRS has side 2a, and its vertices are at the mid-points of the sides of ABCD, with P on AB and S on AD. The line PS meets AC at K, and the body rotates in a vertical plane about a horizontal axis k through K (see diagram). Find the moment of inertia of the body about k.

2. Relevant equations

parallel and the perpendicular axes theorems

3. The attempt at a solution

If i understand the question correctly, is the "dashed"(broken)line, the axis about which we need to find the moment of inertia ? If that is the case, here is what i did :

Using the perpendicular axes theorem : (let $I_{A}$ be the moment of inertia about the required axis for ABCD only)
2$I_{A}$ = $\frac{5m(2a^{2}+2a^{2})}{3}$
$I_{A}$ = $\frac{10ma^{2}}{3}$

Similarly, let $I_{B}$ be the moment of inertia about the required axis for PQRS only :
Using the perpendicular axes theorem,
$I_{B}$ = $\frac{ma^{2}}{3}$

The total moment of inertia = $I_{A}$ + $I_{B}$
Total = $\frac{11ma^{2}}{3}$

The problem is that the solution notes state the answer to be $\frac{40ma^{2}}{3}$

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2. Apr 1, 2013

### rude man

Since k was not indicated on the diagram it's a bit of a puzzle all right as to what the axis of rotation is. It's supposed to be a "horizontal" axis which would seem to eliminate the dashed line though. I think the intended axis is a "hozizontal" line passing thru point K. In other words, the axis of rotation is parallel to line AB, and also to a line drawn thru SQ, and situated inbetween those two and passing thru point K.

Happy integration!

3. Apr 1, 2013

### haruspex

I would interpret rotating "in a ... plane" as meaning the axis is perpendicular to the lamina.

4. Apr 2, 2013

### rude man

That sounds sound. I agree.