(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A triangular prism of mass M, whose two ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis on the z axis. Find its moment of inertia for rotation about the z axis. Without doing any integrals write down and explain its two products of inertia for rotation about the z axis.

2. Relevant equations

3. The attempt at a solution

I've arrived at a soultion, but I'd like to see what other people think, if it is right or not.

I know that the moment of inertia (here, just I), is this:

I = integral of [ (x^2 + y^2) dm]

and

p = M/V, so

dm = pdV, so

I = p * integral of [ (x^2 + y^2) dV], so

I = p * triple integral of [ (x^2 + y^2) dx dy dz]

This will need to be done as the sum of two integrals, since there is a discontinuity in the change in y.

For the first integral:

The limits of the x integral are -a to 0.

The limits of the y integral are -a/√3 to 2a/√3.

The limits of the z integral are, let's say, 0 to h.

I evaluated this to be p*h*(-4/3)(a^4)/(√3)

Since p = M/V and V = (h*a^2)/(√3), the first integral equals

M*(-4/3)(a^2)

For the second integral:

The limits of the x integral are 0 to a.

The limits of the y integral are 2a/√3 to -a/√3.

The limits of the z integral are 0 to h.

I evaluated this to be, of course, the same thing,

M*(-4/3)(a^2)

So,

I = (-8/3)(a^2)M

That should be the moment of inertia for rotation about the z axis.

The products of inertia should both be zero (because of symmetry).

Does this seem right? More importantly, is it right? Thank you for looking at it.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Moment of Inertia of a Triangular Prism

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**