A vertical partition in a tank has a square aperture of side \(a\), the upper and lower edges of which are horizontal. The aperture is completely closed by a thin diaphragm. On one side on the diaphragm there is water with a free surface at a distance \(b(>a / 2)\) above the centre-line of the diaphragm. On the other side there is water in contact with the lower half of the diaphragm, and this is surmounted by a layer of oil of thickness \(c\) and relative density \(\sigma\). The free surfaces on each side of the partition are in contact with the atmosphere. If there is no net force on the diaphragm, determine the relation between \(b\) and \(c\), and the position of the axis of the couple on the diaphragm.

Forces in fluids arise due to the fluid's pressure. Pressure is a type of force exerted uniformly in all directions within a fluid. It acts perpendicularly on any surface in contact with the fluid.
In our given problem, the pressure on the diaphragm is caused by the weight of the water and oil, creating forces from both sides. To maintain equilibrium, these forces must balance out so the diaphragm experiences no net force.
The force exerted by a fluid is determined by its density, depth, and the gravitational acceleration. Understanding these relationships is key to solving any problems involving forces in fluids.

Fluid statics, also known as hydrostatics, deals with fluids at rest. When fluids are stationary, the primary factor affecting their behavior is pressure variation with depth.
In the described exercise, we have a static scenario where water and oil layers exert pressure on a diaphragm within a tank. We must consider how pressure increases with depth due to the weight (density) of the fluid above a given point.
Exploring fluid statics further, the pressure at any depth in a fluid can be calculated using the formula \( P = \rho gh \) where \( P \) is pressure, \( \rho \) is the fluid density, \( g \) is gravitational acceleration, and \( h \) is the depth from the surface.

Pressure equilibrium means that the pressures exerted on either side of a surface are equal, resulting in no net force on that surface. In practical terms, it indicates a balance in forces.
For our diaphragm problem, the pressure due to water on one side has to match the combined pressure of water and oil on the other side. If these pressures are not balanced, the diaphragm would move or deform.
Pressure equilibrium can be expressed as \( P_{water} = P_{total\text{ on the other side}} \). This condition leads us to find the relationship between the height \( b \) of the water and the thickness \( c \) of the oil layer to ensure no net force on the diaphragm.

Hydrostatics studies the properties of fluids at rest and the forces and pressures associated with them. A fundamental principle in hydrostatics is that the pressure at a given depth in a fluid is directly proportional to its density and the height of the fluid column above it.
In the problem, both water and oil contribute to the total pressure at the midpoint of the diaphragm. Hydrostatics helps determine how these pressures distribute and balance each other.
Using the hydrostatic pressure equation \( P = \rho gh \) to calculate pressures from the water and oil layers enables us to set up equations that must be satisfied for the diaphragm to remain balanced.

Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance, typically water. It provides a dimensionless quantity that compares how heavy a fluid is relative to water.
In this exercise, the relative density \( \sigma \) of oil is used to calculate the pressure contribution from the oil layer above the water.
If the relative density \( \sigma \) of the oil is given as, for example, 0.8, it implies that the oil is 80% as dense as water. This helps in calculating the hydrostatic pressure exerted by the oil as \ P_{oil} = \rho_{water} g \sigma c \.
Understanding relative density is crucial for determining how different fluids interact and balance each other in hydrostatic scenarios.