Engineering Mechanics

(^726) „„„„„ A Textbook of Engineering Mechanics
gravity of the immersed surface and towards the lower edge of the figure. The point, through which
this resultant pressure acts, is known as centre of pressure, and is always expressed in terms of depth
from the liquid surface.
35.11.CENTRE OF PRESSURE OF A VERTICALLY IMMERSED SURFACE
Fig. 35.9. Centre of pressure on a vertically immersed surface.
Consider a plane surface immersed vertically in a liquid as shown in Fig. 35.9.
Let w= Specific weight of the liquid,
A= Area of the immersed surface
x = Depth of centre of gravity of the immersed surface from the
liquid surface.
Divide the whole surface into a no. of small parallel strips as shown in the figure. Now let us
consider a strip of thickness dx, width b and at a depth of x from the free surface of the
liquid as shown in Fig. 35.9.
We know that the intensity of pressure on strip = wx
and area of the strip = b dx
∴ Pressure on the strip = Intensity of pressure × Area = wxb dx
and moment of this pressure about the liquid surface,
=(wxb dx)x = wx^2 b dx
Now the sum of moments of all such pressures about the liquid surface
M*=∫wx b dx^2

• The sum of moments of liquid pressure about the liquid surface may also be found out by dividing
  the whole surface into a no. of small parallel strips.
  Let a 1 , a 2 , a 3 ... = Areas of the strips, and
  x 1 , x 2 , x 3 ... = Depths of the corresponding strips from the liquid surface.
  ∴ Pressure on the first strip =wa 1 x 1
  and moment of this pressure about the liquid surface
  =wa 1 x 1 × x 1 = wa 1 x 12
  Similarly, moment of pressure on second strip about the liquid surface
  =wa 2 x 22
  and moment of pressure on the third strip about the liquid surface
  =wa 3 x 32
  ∴ Sum of moments of all such pressure about the liquid surface,
  M= wa 1 x 12 + wa 2 x 22 + wa 3 x 32 + ...
  =w (a 1 x 12 + a 2 x 22 + a 3 x 32 + ...) ...(i)
  =w I 0
  where I 0 =(a 1 x 12 + a 2 x 22 + a 3 x 32 + ...)
  = Moment of inertia of the surface about the liquid surface (also
  known as second moment of area).
  Now proceed further from equation (i).