The ratio of the time periods of small oscillation of the insulated spring and mass system before and after charging the masses is

= 1

In equilibrium of the charged small bodies $\frac{1}{4 \pi \varepsilon_0} . \frac{q^2}{\left(\ell_0+x_0\right)^2}=k x_0$ where x₀ is the elongration in the spring in equilibrium.

Let a further small elongation of x is made in the spring.

Then net restoring force on any of the charged particle is given by, $F=-\left[k\left(x_0+x\right)-\frac{1}{4 \pi \varepsilon_0} \frac{q^2}{\left(\ell_0+x_0+x\right)^2}\right]$

= −kx.                        Since x << x₀ from (1)

$\Rightarrow a=-\frac{2 k}{m} x$

As $F=\mu a$ where $\mu=\frac{m \times m}{m+m} \Rightarrow a=-\omega^2 x$,

Hence $\omega=\sqrt{\frac{2 k}{m}} \Rightarrow T=2 \pi \sqrt{\frac{m}{2 k}}$

In absence of charge is $T_0=2 \pi \sqrt{\frac{m}{2 k}}$.

Therefore $\frac{T}{T_0}=1$

∴ (D)