Moment of Inertia of a Rectangular Picture Frame

[kepherax]

Homework Statement

A picture frame consists of 4 thin pieces of wood glued together. Each wooden piece has the same mass per unit length lambda. The dimensions of each piece is given below. Determine the rotational inertia of the frame about the dashed axis.

Relevant Equations

I = mr^2, for rod at edge I = 1/3 mr^2
I total = I1+I2+I3....+In

https://www.physicsforums.com/attachments/250905

I know the answer, but am not certain how they got Lsin(angle) for R?

 

[haruspex]

Homework Statement: A picture frame consists of 4 thin pieces of wood glued together. Each wooden piece has the same mass per unit length lambda. The dimensions of each piece is given below. Determine the rotational inertia of the frame about the dashed axis.
Homework Equations: I = mr^2, for rod at edge I = 1/3 mr^2
I total = I1+I2+I3...+In

https://www.physicsforums.com/attachments/250905

I know the answer, but am not certain how they got Lsin(angle) for R?

The link does not work for me. Please post an attempt, per forum rules.

 

[kepherax]

I don't know why it won't let me edit this, but here is the problem and my attempt.

 

[haruspex]

how they got Lsin(angle) for R?

Consider a small element dx of one of the spars length L, distance x from the end at the axis.
Its distance from the axis is $x\sin(\theta)$, so its MoI about the axis is $\lambda dx (x\sin(\theta))^2$. Then integrate, or observe that this is the same as if the spar were the same mass but length $L\sin(\theta)$ and normal to the axis.