1. Initial energy stored in the capacitor

Since the capacitor is first charged to potential difference $V=12\,\text{V}$ and then disconnected from the battery, the initial electrostatic energy is

$U_i=\frac12 CV^2$

Given:

$C=14\,\text{pF}=14\times 10^{-12}\,\text{F}$ $V=12\,\text{V}$

So,

$U_i=\frac12\times 14\times 10^{-12}\times (12)^2$

$U_i=7\times 10^{-12}\times 144$

$U_i=1008\times 10^{-12}\,\text{J}=1008\,\text{pJ}$

2. After inserting the dielectric slab

The battery is disconnected, so charge remains constant.

If a dielectric of constant $k=7$ completely fills the space between the plates, the new capacitance becomes

$C'=kC=7C$

For constant charge, the energy becomes

$U_f=\frac{Q^2}{2C'}$

But initially,

$U_i=\frac{Q^2}{2C}$

Hence,

$U_f=\frac{U_i}{k}$

So,

$U_f=\frac{1008}{7}=144\,\text{pJ}$

3. Mechanical energy gained by the porcelain plate

Because there is no friction, the decrease in electrostatic energy is converted into mechanical energy of oscillation of the plate.

Thus,

$E_{\text{mech}}=U_i-U_f$

$E_{\text{mech}}=1008-144=864\,\text{pJ}$

4. Final answer

The constant mechanical energy of oscillation is

$\boxed{864\,\text{pJ}}$