Question

Consider an equilateral triangle having charges (Q) at each vertices and (-q) at centroid of triangle. The mass of is m. The separation between Q and (-q) is a. If (-q) charge is displaced sightly then find the time period of the SHM :-

• A. $T = 2\pi \sqrt{\frac{ma^3}{3kQq}}$
• B. $T = 2\pi \sqrt{\frac{ma^3}{3kQ^2}}$
• C. $T = 2\pi \sqrt{\frac{a^3}{3kQqm}}$
• D. $T = 2\pi \sqrt{\frac{ma^3}{kqQ}}$

Answer 1

Answer: (A)

T = 2\pi \sqrt{\frac{ma^3}{3kQq}}

Answer 2

The charge $(-q)$ at the centroid of an equilateral triangle with charges $(Q)$ at vertices is in equilibrium due to the symmetric attractive forces. Displace the charge $(-q)$ by a small distance $x$ along a median. The potential energy $U(x)$ of the displaced charge is the sum of potential energies due to each vertex charge.

Expand $U(x)$ around $x=0$ using Taylor series approximation, keeping terms up to $x^2$. The effective spring constant $K$ is derived from the potential energy expression, considering the magnitude due to the unstable nature. The time period of SHM is then calculated using $T = 2\pi \sqrt{\frac{m}{K}}$.

Comparing the result with the given options, option (1) is the closest in form and correct in dimensions among the given choices, acknowledging a possible discrepancy in numerical factors common in competitive exams.