# Oscillations of a Charged Arc

An arc with radius R has a uniform positive charge density $\lambda$ exists as shown.The arc of mass M is initially in equilibrium due to its weight and electrostatic force of interaction between a fixed charge at its centre [mass m charge Q] The arc is then displaced from the mean position a very small distance as compared to the radius R, along the symmetrical axis of the arc.It undergoes Simple harmonic motion under certain approximations. Consider gravity constant everywhere .Find the $M\frac{\frac{arccos(\frac{1}{\sqrt{3}})}{2}+\sqrt{2}}{\frac{arccos(\frac{1}{\sqrt{3}})}{2}+\frac{1}{\sqrt{2}}}$ upto two decimal in SI units of the arc given that the time period of oscillations has a minimum value of$2\pi$ seconds.Given that $Q=\frac{\lambda}{R^{2}}=\sqrt{(2\pi\varepsilon)}$ Consider all forces of gravitation constant throughout considering only values in SI and not the units.

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