plz give me the whole derivation of pascal's law - Physics - Mechanical Properties Of Fluids

Question

plz give me the whole derivation of pascal's law

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Consider a very small right angled triangular element ABC of a liquid as shown in figure 23547/pascal png

Let: $p_{x}$ = Intensity of horizontal pressure on the element of the liquid $p_{y}$ = Intensity of vertical pressure on the element of the liquid $p_{z}$ = Intensity of pressure on the diagonal of the triangular element of the liquid

= Angle of the triangular element of the liquid Now total pressure on the vertical side AC of the liquid, $P_{x} = p_{x}times AC$ (1) Similarly,total pressure on the horizontal side BC of the liquid, $P_{y} = p_{y}times BC$ (2) and total pressure on the diagonal side AB of the liquid, $P_{z} = p_{z}times AB$ (3) Since the element of the liquid is at rest, therefore sum of the horizontal and vertical components of the liquid pressure must be equal to zero Now using eqilibrium condition for horizontal pressure, $P_{z}sin theta = P_{x}$ $Rightarrow p_{z} AB sin theta = p_{x} AC$ From the geometry of the figure, we find that,

$therefore p_{z} AC = p_{x} AC$ $Rightarrow p_{z} = p_{x}$ (4) Now using equilibrium condition for vertical pressure, i e , $p_{z} cos theta = p_{y}-W$ (where W = Weight of the liquid) As the triangular element is very small, the weight of the liquid W is neglected, so, $p_{z} cos theta = p_{y}$ $therefore p_{z} AB cos theta = p_{y} BC$ From the geometry of the figure, we find that

$therefore p_{z} BC = p_{y} BC$ $Rightarrow p_{z} = p_{y}$ (5) Now from equation (4) and (5), we find that $p_{x} = p_{y} = p_{z}$ Thus the intensity of pressure at any point in a fluid, at rest, is the same in all direction

plz give me the whole derivation of pascal's law...