Question

Moment of inertia of equilateral triangle about centroidal axis

Answer 1

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The problem is to find the moment of inertia for a solid equilateral triangle about an axis at the triangle's vertex which is perpendicular to the plane of the triangle. Mass is given as M. I was wanting to treat it like 3 point masses at each vertex with 1/3 mass. Then I could use the sum definition of moment of inertia instead of the integral. The distance from axis at center of the equilateral triangle to each vertex is L/√3. Ʃ mr2= 3(1/3 M * L2/9) = ML2/9 Then by the parallel axis theorem I can shift this out to a vertex using that same distance for a final result of 2ML2/9 I know it can be done by treating dA like a thin rod and using calculus, I was just curious why the above doesn't get the same answer. Isn't it dynamically the same to treat it like 3 1/3 masses on each vertex?

Answer 2

Answer:

The problem is to find the moment of inertia for a solid equilateral triangle about an axis at the triangle's vertex which is perpendicular to the plane of the triangle. Mass is given as M. I was wanting to treat it like 3 point masses at each vertex with 1/3 mass. Then I could use the sum definition of moment of inertia instead of the integral. The distance from axis at center of the equilateral triangle to each vertex is L/√3. Ʃ mr2= 3(1/3 M * L2/9) = ML2/9 Then by the parallel axis theorem I can shift this out to a vertex using that same distance for a final result of 2ML2/9 I know it can be done by treating dA like a thin rod and using calculus, I was just curious why the above doesn't get the same answer. Isn't it dynamically the same to treat it like 3 1/3 masses on each vertex?