Question

A block having charge Q and mass m is connected to spring and kept at smooth surface. Find time period for oscillations.

Answer 1

The time period for oscillations is $2\pi\sqrt{\frac{m}{K}}$.

Answer 2

The net force on the block is the sum of the spring force ($-Kx$) and the constant electric force ($QE_0$). The equation of motion is $m\ddot{x} = QE_0 - Kx$. By defining a new coordinate $x' = x - x_{eq}$ where $x_{eq} = QE_0/K$ is the equilibrium position, the equation of motion becomes $m\ddot{x'} = -Kx'$. This is the standard equation for Simple Harmonic Motion with angular frequency $\omega = \sqrt{K/m}$. The time period of oscillation is $T = 2\pi/\omega$.