Compute the centroidal moment of inertia:

Compute the centroidal moment of inertia of a thin homogeneous bar AB of length l and having a mass of m kg

Solution

The mass centre of the bar is at mid point. Further, it is supposed that the z_o axis coincides with the axis of the bar. The whole mass of the bar is m.

The elementary mass of the body of length dz may be written as

dm = (m/ l) dz

The coordinates of the point where this elementary mass is situated is (0, 0, z_o).

Thus, from Eq. we have

                          =(1/12)ml²

I z_o = I x_o y_o = I y_o z_o = I z_o x_o   = 0

The moment of inertia around the axis x passing through the end A may be calculated by substituting z_c  = (½) l , in the parallel axis theorem.

Then from Eq. we have

I_x  = I_y  = ml²/12   + m ((1/2) l )² = (1/3) ml ²