Aditya's challenges in Mechanics 1

Three similar rods of length $l=1\text{ m}$ and mass $m=1\text{ kg}$ are welded to form a structure similar to the positive sides of 3D coordinate axes, as shown in figure above. Now it is rested on a very sharp pin about the intersection of the rods so that they remain in equilibrium facing downwards. The system is in equilibrium. Now the system is shaken a bit horizontally so that it performs SHM. Find the time period of its SHM.

The answer is of the form $\dfrac {A\pi} {B} \sqrt { \dfrac {2 \sqrt { C } }{ D } },$ where$A$ and $B$ are coprime positive integers and $C$ and $D$ are square-free.

Find the value of $A+B+C+D.$

$$
Details and Assumptions:

• $g=10\text{ m/s}^2$.

• Neglect air resistance.