For a simple harmonic oscillator, the velocity $v$ as a function of displacement $x$ and amplitude $x_0$ is given by the equation:

$$v=\omega \sqrt{x_0^2-x^2}$$

where $\omega$ is the angular frequency. Show that this equation is homogeneous by reducing both sides to S.I. base units.

A glass U-tube is partially filled with oil. One end of the U-tube is connected to a gas supply of constant pressure and the other end is open to the atmosphere. The displacement of the oil from its equilibrium position is $x$. The density $\rho$ of the oil is $800{kg\; m}^{-3}$.

The pressure from the gas supply raises the oil on one side of the U-tube. The vertical distance between the two levels of oil in the two vertical sections of the U-tube is $15.0cm$ ($x=7.5cm$). $\Delta p$ is the difference between the gas pressure and atmospheric pressure. Calculate $\Delta p$.

When the gas supply is disconnected, the oil levels in the U-tube oscillate with simple harmonic motion. The acceleration $a$ of the oil level is given by the equation:

$$a=-\frac{2\rho gA}{m}x$$

where $m$ is the mass of the oil in the U-tube, $A$ is the internal cross-sectional area of the U-tube, $\rho$ is the density of the oil, $g$ is the acceleration of free fall ($9.81{m\; s}^{-2}$), and $x$ is the displacement.

For this U-tube, $A=1.2\times 10^{-4}{m}^2$ and $m=0.075kg$. Show that the period $T$ of the oscillations is approximately $1.3s$.