A thin plate of area $\backslash ($ A $\backslash )$ and mass $\text{'}\backslash ($ m $\backslash )\text{'}$ is attached at the end of a spring and is allowed to oscillate in the vertical plane. The plate oscillates in air with an undamped time period of $\text{'}\backslash (\backslash$tau$\_\{$n$\}\backslash )\text{'}$ When the plate is made to oscillate in a tub containing viscous fluid, its time period has increased to $\text{'}\backslash (\backslash$tau$\_\{\backslash$mathrm$\{$d$\}\}\backslash )\text{'}.$ Derive the expression for the viscosity of the oil. Assume that the damping force in the system is proportional to the product of total area of the plate and its velocity, the proportionality constant being the viscosity of the oil.