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)ml^{2}
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} = ml^{2}/12 + m ((1/2) l )^{2} = (1/3) ml ^{2}