# Area/2nd Moment of Inertia of an arc/ring

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1. Mar 11, 2015

### Wreak_Peace

1. The problem statement, all variables and given/known data
I'm trying to solve for the area moment of inertia of a curved arc. To visualize this, it would be like a bent piece of cardboard (two arcs with two lines connecting them at their end points). I'm modelling the differences in area MOI with an increasingly curved piece of acrylic which must be held up only by the bottom (nothing can be supporting it from the sides or top or back). The piece of acrylic is 5ft long, 3ft wide, and 3mm in thickness. The arc length will be 3 ft.

2. Relevant equations
There really are no equations out there for the 2nd moment of inertia of an arc portion of a ring. I'm trying to solve for the equation basically.

3. The attempt at a solution
To approximate this, at first I attempted using the area MOI of a rectangle enclosing the "arc ring", I used basic trig to find the chord length and height, and got area MOIs much much higher than a straight piece of acrylic would be, even with huge radii so there wasn't much curvature. Next I attempted essentially a Riemann sum, adding up the MOIs of many rectangles inside the arc ring, which has proved difficult. I'm wondering if there is any way I can find a formula for the area MOI of this. The wikipedia page on lists of moments of inertia has an equation for a semicircle's 2nd MOI from its centroid. Is there any way for this equation to be altered to change the degrees of the arc, so it can be smaller than a semicircle?

I know this is somewhat vague, I did my best at trying to explain. Please don't hesitate to ask me to clarify anything and thank you for your help!

2. Mar 11, 2015

### SteamKing

Staff Emeritus
Your verbal description of the figure is somewhat unclear to me. Can you provide a sketch of the figure?

Also, are you trying to find the second moment of area, or are you trying to find the mass moment of inertia of this piece?

It's not clear for what this information is going to be used.