Given data:
Diameter of the circular plate (D) = 2.5 m
Greatest depth below the free surface (h1) = 3 m
Least depth below the free surface (h2) = 1 m

Step 1: Calculate the pressure at the greatest depth:
The pressure at the greatest depth (P1) can be calculated using the formula:
P1 = ρgh1
Where,
ρ = density of water (assumed to be 1000 kg/m³)
g = acceleration due to gravity (assumed to be 9.81 m/s²)

P1 = 1000 * 9.81 * 3 = 29430 Pa

Step 2: Calculate the pressure at the least depth:
The pressure at the least depth (P2) can be calculated using the formula:
P2 = ρgh2
Where,
h2 = 1 m

P2 = 1000 * 9.81 * 1 = 9810 Pa

Step 3: Calculate the total pressure on the side of the plate:
The total pressure on the side of the plate can be calculated by integrating the pressure over the entire side of the plate.

The pressure at any point on the side of the plate can be calculated using the formula:
P = ρgz
Where,
z = distance of the point below the free surface

To calculate the total pressure, we divide the side of the plate into small vertical strips and integrate the pressure over each strip.

Let's divide the side of the plate into small vertical strips of width Δz.

The pressure on each strip can be approximated as:
ΔP = ρgΔz

The total pressure (Ptotal) can be calculated by integrating the pressure over the entire height of the plate:
Ptotal = ∫ΔP = ∫ρgΔz = ρg∫Δz
Where,
∫ denotes integration

The integration of Δz will give us the total height of the plate (H):
H = h1 - h2 = 3 - 1 = 2 m

Therefore, the total pressure on the side of the plate (Ptotal) is given by:
Ptotal = ρg∫Δz = ρgHz
Where,
Hz = height of the point above the least depth

Ptotal = ρgHz = 1000 * 9.81 * 2 = 19620 Pa

Step 4: Calculate the position of the centre of pressure:
The position of the centre of pressure can be calculated using the formula:
Xcp = ∫z * ΔP / Ptotal
Where,
z = distance of the point below the free surface

To calculate the position of the centre of pressure, we divide the side of the plate into small vertical strips and integrate the product of z and ΔP over each strip.

The position of the centre of pressure (Xcp) can be calculated by integrating the product of z and ΔP over the entire height of the plate:
Xcp = ∫z * ΔP / Ptotal = ∫z * (ρgΔz) / Ptotal = ρg / Ptotal * ∫z * Δz
Where,
∫ denotes