Moment of Inertia of a thin uniform wire

[kelslee28]

Homework Statement

A thin uniform wire is bent into a rectangle. The short, vertical sides are of length a, and the long, horizontal sides are of length b. If the total mass is 31.00 grams, a = 25.00 cm and b = 44.30 cm, what is the moment of rotational inertia about an axis through one of the vertical wires?

Homework Equations

I = 1/3mL^2

The Attempt at a Solution

I tried using the above equation, which is the moment of inertia for a solid rectangle. I knew it wasn't going to be right but I don't have an equation for a hollow rectangle. I tried modifying it by using 1/4 or 1/6 because I think the moment of inertia would be smaller for a hollow rectangle than a solid one.

 

[Doc Al]

Think of the object as composed of four thin rods.

 

[kelslee28]

So the top and bottom would be I= 1/3 mL² but it's only a portion of the mass, right? The right side would be I = mh² and the h would be the length b. The only thing I can think of for the left side is the formula for a solid cylinder, but for that you need a radius.

 

[ideasrule]

Because the wires are assumed to be very thin, the left side has nearly no moment of inertia. You can assume I=0.

 

[Doc Al]

So the top and bottom would be I= 1/3 mL² but it's only a portion of the mass, right?

Right.

The right side would be I = mh² and the h would be the length b.

Right.

The only thing I can think of for the left side is the formula for a solid cylinder, but for that you need a radius.

Think of it as being very thin.

 

[kelslee28]

Yes, I got it! Thanks a lot. I'm home sick with the flu so this means a lot that someone would help me.

 

[workinghard]

I'm having trouble with this problem, too. I don't understand how to get the masses of the individual parts of the rectangle.

 

[Doc Al]

I don't understand how to get the masses of the individual parts of the rectangle.

You have the total mass. Figure out the mass of each side, realizing that the mass is proportional to the length.