Momentum and Impulse
In this post I try to explain how the moment of inertia of an arbitrary triangle may be derived (full disclosure: this is based entirely on this excellent video here).
Also it is recommended that before reading this post, the post here which (again, based on the same Youtube channel) covers how the centroid of an arbitrary triangle may be calculated, since the techniques used are essentially the same.
Triangle consisting of 3 points: a, b, and c
Consider a triangle consisting of points a, b and c, as follows. The goal is to calculate the moment of inertia around the axis passing through vertex a (which is also assumed to be the origin), perpendicular to the plane, .
Start with the stanard formula for the mmoi.
where:
r is the vector pointing to dA from the origin (= vertex a), and ρ is the density of our "lamina".
r can of course be expressed as being some units (scalar u) along axis u and some units (scalar v) along axis v.
where:
As in the calculation of the centroid, we make use of the following relationship.
where A is the total area of the triangle.
We can express the mmoi equation as follows.
As with the centroid calculation, we make use of the Green Theorem to convert the above integral over an area to an integral over a line, to make solving the above equation easier.
Green's Therem
We will consider the elements in converting the area integral into a line integral.
Taking the first element in the brackets, we can come up with the following by Green Theorem equivalent (assume that M is zero).
We need to perform the above integral over the line = outline of the triangle anti-clockwise, ie vertex a -> vertex b -> vertex c -> vertex a.
For a->b, v = 0 and dv = 0, hence the line integral is zero.
For c->a, u = 0, and du = 0, hence the line integral is zero.
So we only are left with vertex b = vertex c. The line from vertex b to vertex c can be expressed as the following (just think of the equation of a line joining (1,0) and (0,1) in a normal cartesian coordinate system).
It follows that du = -dv. Hence:
Similarly, the second element would be:
For the third element below, we turn to the Green's theorem to convert the area integral into a line integral.
Referring back to the Green's theorem, assume that L = 0 and partial derivative of M against u is the 2uv, arrive at the following:
Now we need to perform the line integral against the outline of the triangle. Similar to the other elements, paths a->b and c->a turns out to be zero, so we only need to worry about path b->c (which is expressed as u+v = 1).
Now we can add the elements together to arrive at the equation for the mmoi.
where M is the product of the density and the area, ie mass of the triangular lamina.