Simple Harmonic (2).

A uniform rod of mass $2m$ and length $2 \ell$ was bent at its mid-point to form an angle of $\dfrac{2\pi}{3}$ radians. It was then balanced on a very sharp spike in the shape of a cone (whose apex angle is assumed to be negligible) by the virtue of its mid-point. The rod is given a small angular displacement on one side and it executes SHM. If the time-period of this SHM can be represented as

$T = a \pi \sqrt{\dfrac{b \ell}{cg}}$

where $a,b,c$ are positive co-prime integers less than $10$ . Find $a+b+c$ .

Details: $g = 9.8 \text{ms}^{-2}$ is the acceleration due to gravity.