A parallel plate capacitor whose capacitance C is 14 pF is charged by a battery to a potential difference  $\text{V}=12V$ between its plates. The charging battery is now disconnected and a porcelain plate with k=7 is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of ____ pJ. (Assume no friction)

The energy stored in capacitor
 $U=\frac{Q^2}{2C}=\frac{1}{2}C{V}^2=\frac{1}{2}\times 14\times 12\times 12$$=7\times 144pJ$  
When a dielectric plate is introduced the energy will change keeping charge constant as it is disconnected with battery
The new energy of capacitor  :   $U^1=\frac{Q^2}{2C^1}=\frac{Q^2}{2CK}=\frac{U}{K}=\frac{7\times 144}{7}$$=144pJ$
The change in energy will impose to the plate and it oscillates with energy of 
   $U-U^1=7\times 144-144=6\times 144=864pJ$     
[Note: Assume thickness of porcelain plate same as gap between capacitor plates]