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77 (c). Hydrostatic Equilibrium in Spherical Fluid Mass We have to show that in a homogeneous spherical fluid mass
of radius R and density
For the Earth taking
|
,
pressure at a distance r from the centre of the sphere is 
and
, we
have to find the pressure at the centre.|
Solution: Click For PDF Version
Let us consider a cylindrical surface of unit cross-section width
The inward gravitational pull on this fluid element will be the force due to the fluid mass contained in the sphere of radius r and acting from its centre. That is
Net pressure force acting in the outward radial direction will be
Condition of equilibrium
gives the equation for pressure variation,
Integrating this equation, we get
Requiring that at the boundary of the fluid mass pressure is zero,
Radial variation of pressure is, therefore, given by the function
We next calculate the pressure at the centre of the Earth considering it to
be a homogeneous spherical fluid mass of radius
Substituting the values, we find that
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at a distance r from the centre of the spherical fluid system in
hydrostatic equilibrium. Condition of equilibrium for this fluid element is that
the inward gravitational pull on it has to be balanced by the pressure force
acting on its opposite faces, each of which is assumed to be unit
cross-sectional area. Let the pressure inside the fluid be denoted by the
function
.
,
we find
and density
.
