Chapter 35 : Hydrostatics 729
35.12. CENTRE OF PRESSURE OF AN INCLINED IMMERSED SURFACE
Fig. 35.12. Centre of pressure on an inclined immersed surface.
Consider a plane inclined surface immersed in a liquid as shown in Fig. 35.12.
Let w= Specific weight of the liquid in kN/m^3 ,
A= Area of immersed surface in m^2 ,
x = Depth of centre of gravity of the immersed surface from the
liquid surface in metres.
θ= Angle at which the immersed surface is inclined with 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 of dx, width b and at a distance l from O (the point on the liquid
surface where the immersed surface will meet, if produced).
We know that intensity of pressure on the strip
= wl sin θ
and area of the strip = b dx
∴ Pressure on the strip = Intensity of pressure × Area = wl sin θ × b dx
and moment of this pressure about O
=(wl sin θ b dx) l = wl^2 sin θ b dx
Now sum of moments of all such pressures about O,
M*=∫wl^2 sinθb dx
* The sum of moments of liquid pressure about O 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
l 1 , l 2 , l 3 ... = Distances of the corresponding strips from O.
∴ Pressure on the first strip =wa 1 l 1 sin θ
and moment of this pressure about O
=wa 1 l 1 sin θ × l 1 = wa 1 l 12 sin θ
Similarly, moment of second strip about O
=wa 2 l 22 sin θ
and moment of inertia of the third strip about O
=wa 3 l 32 sin θ
∴ Sum of moments of all such pressure about O
M=wa 1 l 12 sin θ + wa 2 l 22 sin θ + wa 3 l 32 sin θ + ...
=w sin θ (a 1 l 12 + a 2 l 22 + a 3 l 32 + ...) = w sin θ I 0
where I 0 =(a 1 l 12 + a 2 l 22 + a 3 l 32 +...)
= Moment of inertia of the surface about O
Now proceed further from equation (i).