Question

Show that the moment of inertia of a uniform rod about an axis through the middle and at
the end of a rod are given by MI/12 and ML/3 respectively?

Answer 1

We can use the formula for the moment of inertia of a uniform rod about its center of mass, which is:

I = (1/12) * M * L^2

where I is the moment of inertia, M is the mass of the rod, and L is its length.

To find the moment of inertia about an axis through the end of the rod, we can use the parallel axis theorem, which states that:

I = I_cm + M * d^2

where I_cm is the moment of inertia about the center of mass, M is the mass of the rod, and d is the distance between the axis of rotation and the center of mass.

For the axis through the middle of the rod, d = L/2, so we get:

I = (1/12) * M * L^2 + M * (L/2)^2
I = (1/12) * M * L^2 + (1/4) * M * L^2
I = (1/3) * M * L^2

which is the desired result.

For the axis through the end of the rod, d = L, so we get:

I = (1/12) * M * L^2 + M * L^2
I = (1/12 + 1) * M * L^2
I = (1/3) * M * L^2

which is also the desired result.