Step 1: Identifying the Energy Components

The system consists of a mechanical oscillator (mass on a spring) and an electrical component due to the motion of the conductor in the magnetic field. When the conducting disc moves with velocity $v$ in a magnetic field $B$, an electric field $E$ is induced inside the conductor due to charge separation (polarization).

The induced electric field is given by magnitude $E = vB$.

Step 2: Energy Stored in the Electric Field

Since the disc acts as a dielectric (in the sense of permitting field formation) or simply considering the energy density of the induced electric field in the volume $V$ of the disc:

Energy density $u_E = \frac{1}{2}\varepsilon_0 E^2$.

Total electrical energy $U_E = \text{Volume} \times u_E = V \left( \frac{1}{2}\varepsilon_0 (vB)^2 \right) = \frac{1}{2} (\varepsilon_0 V B^2) v^2$.

Step 3: Effective Mass and Period

The total energy of the system is the sum of kinetic energy (mechanical + electrical) and potential energy (spring).

$$ E_{total} = \frac{1}{2}mv^2 + \frac{1}{2}(\varepsilon_0 V B^2)v^2 + \frac{1}{2}kx^2 $$ $$ E_{total} = \frac{1}{2}(m + \varepsilon_0 V B^2)v^2 + \frac{1}{2}kx^2 $$

This equation resembles the energy of a simple harmonic oscillator with an effective mass $m_{eff} = m + \varepsilon_0 V B^2$.

The period of oscillation for a spring-mass system is $T = 2\pi \sqrt{\frac{m_{eff}}{k}}$.