# Solving Fluid Statics Problem: Centre of Pressure

• AppleBite
In summary, the question asks for the depth at which the centre of pressure of a submerged semicircular plane is located. It is found that the answer is s = (3*pi*d) / 32, using a specific equation from a book on fluid statics. This approach does not involve integration.
AppleBite
Ok, I've come across this problem in fluid statics, but seem to be getting the integration wrong:

"A semicircular plane is submerged vertically in a homogeneous liquid with its diameter d at the free surface. At what depth s is the centre of pressure?"

s = (3*pi*d) / 32

Any ideas?

The solution to this problem involves using the principles of fluid statics and the concept of the centre of pressure. The centre of pressure is the point at which the total force acting on a submerged surface is equal to the weight of the fluid above that surface. In this case, the semicircular plane is submerged vertically, so we can use the formula for the centre of pressure on a vertical surface:

s = (3*pi*d) / 32

where s is the depth of the centre of pressure, d is the diameter of the semicircular plane, and pi is the mathematical constant pi.

To understand this formula, we need to first understand the concept of the centroid of a semicircle. The centroid is the point at which the area of a semicircle is evenly distributed. In this case, the centroid of the semicircular plane is located at a distance of d/4 from the base of the semicircle.

Now, let's consider the forces acting on the semicircular plane. The weight of the fluid above the plane is equal to the weight of the fluid displaced by the plane. This can be calculated using the formula for the volume of a cylinder (since the semicircular plane is essentially a half-cylinder):

V = (pi*d^2*s) / 8

where V is the volume of the fluid displaced, d is the diameter of the semicircular plane, and s is the depth of the centre of pressure. So, the weight of the fluid above the plane is:

F = ρgV = (pi*d^2*s*ρg) / 8

where ρ is the density of the fluid and g is the acceleration due to gravity.

Now, the total force acting on the plane is equal to the weight of the fluid above the plane plus the weight of the plane itself. The weight of the plane can be calculated using its surface area and the density of the material it is made of:

F' = ρgA = (pi*d^2*ρg) / 4

where A is the surface area of the semicircular plane.

To find the depth of the centre of pressure, we need to set these two forces equal to each other and solve for s:

F = F'

(pi*d^2*s*ρg) / 8 = (pi*d^2*ρg) / 4

s = (3*pi*d) /

## 1. What is fluid statics?

Fluid statics is a branch of fluid mechanics that deals with the study of fluids at rest. It involves the analysis of the behavior of fluids, such as liquids and gases, when they are not in motion.

## 2. What is the centre of pressure?

The centre of pressure is the point at which the total sum of the hydrostatic force acting on a submerged surface is considered to act. It is the point at which the resultant hydrostatic force is assumed to be acting on a submerged body.

## 3. How is the centre of pressure calculated?

The centre of pressure can be calculated by dividing the moment of the hydrostatic force about a chosen reference point by the total hydrostatic force acting on the submerged surface.

## 4. Why is it important to determine the centre of pressure?

Determining the centre of pressure is important in engineering and design applications, as it helps in predicting the stability and balance of submerged bodies. It is also crucial in calculating the forces acting on submerged structures, such as ships and dams.

## 5. What are some common methods for solving fluid statics problems?

Some common methods for solving fluid statics problems include using the principles of hydrostatics, such as Pascal's law and Archimedes' principle, and applying the equations of fluid statics, such as the hydrostatic equation and the equation of continuity. Other methods include using graphical and numerical techniques, such as the use of pressure diagrams and numerical integration.

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