For a simple harmonic oscillator, the potential energy $E_p$ as a function of displacement $x$ is given by the equation:

$$E_p=\frac{1}{2}m{\omega }^2{x}^2$$

where $m$ is the mass of the oscillator and $\omega$ is its angular frequency. Show that this equation is homogeneous by reducing both sides to S.I. base units.

A vertical cylindrical buoy of cross-sectional area $A=0.035m^2$ floats in seawater of density $\rho =1020{kg\; m}^{-3}$.

A weight of mass $M$ is placed on top of the buoy, depressing it vertically by a distance $x=15.0cm$ to a new equilibrium position. Calculate $M$.

When the weight is suddenly removed, the buoy oscillates vertically with simple harmonic motion. The acceleration $a$ of the buoy is given by the equation:

$$a=-\frac{\rho gA}{m}x$$

where $m$ is the mass of the buoy, $A$ is its cross-sectional area, $\rho$ is the density of the seawater, $g$ is the acceleration of free fall ($9.81{m\; s}^{-2}$), and $x$ is the displacement from the equilibrium position.

For this buoy, $m=12.5kg$ and $A=0.035m^2$. Show that the period $T$ of the oscillations is approximately $1.2s$.