Tuesday, 6 February 2018

CENTER OF PRESSURE DERIVATION

CENTER OF PRESSURE DERIVATION

We were discussing the basic definition and derivation of total pressure in our previous post. Today we will understand here the basic concept of "Centre of pressure", in the subject of fluid mechanics, with the help of this post. We will also find out here the expression for centre of pressure with the help of this post.

Centre of pressure

Centre of pressure is basically defined as a single point through which or at which total pressure or total hydrostatic force will act.

 
Let us consider that we have one tank filled with liquid e.g. water. Let us consider that there is one object of arbitrary shape immersed inside the water as displayed here in following figure.



Let us consider G is the centre of gravity and P is the centre of pressure. ħ is the height of C.G from free surface of liquid and h* is the height of centre of pressure from free surface of liquid.

Centre of pressure will be determined with the help of following formula as mentioned here.

Derivation of centre of pressure

In order to determine the centre of pressure, we will consider the object in terms of small strips as displayed here in above figure. We will use the concept of “principle of moments” to determine the centre of pressure.

According to the principle of moments, moment of the resultant force about an axis will be equal to the sum of the moments of components about the same axis.

As we have shown above in figure, total hydrostatic force F is applied at centre of pressure P which is at height of h* from the free surface of liquid.

Therefore, let us determine the moment of resultant force F about the free surface of liquid and it will be determined as F x h*.

 
As we have considered here the object in terms of small strips as displayed here in above figure and hence we will determine the moment of force dF acting on small strip about the free surface of liquid.
Moment of force dF = dF x h
Moment of force dF = ρ g h x b dh x h

Let us sum of all moments of such small forces about the free surface of liquid and it will be written as mentioned here.

We have already shown above that moment of resultant force F about the free surface of liquid i.e. F x h* and sum of moments about free surface i.e. ρ g I0
We can also write here 


We will discuss the total pressure and centre of pressure for inclined plane surface submerged in liquid, in the subject of fluid mechanics, in our next post.

Do you have any suggestions? Please write in comment box.

 

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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