SureDenPascal Law and its Applications
Pascal’s Law
“A change in pressure applied to an enclosed ( incompressible ) fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.” This statement is known as Pascal’s law.
Pascal’s law is also given as “If the effect of gravitation is neglected, the pressure at every point in an incompressible liquid, in equilibrium, is the same.”
Applications of Pascal’s Law:
The figure shows the principle of a hydraulic lift used to raise heavy loads. This device has two vertical cylinders of different diameters connected by a horizontal tube. A liquid is filled in this vessel. Air- tight pistons having cross-sectional areas A1 andA2 (A1 <A2)are fitted touching the liquid surface in both the cylinders. According to Pascal’s law, in equilibrium, the pressure on liquid in both the arms is the same. Hence,
Thus, a large force, F2, is generated using a small force, F1, as the magnitude of F2 is (A2/A1)times the magnitude of F1. Using Pascal’s law, devices like hydraulic lift, hydraulic
jack, hydraulic brake and hydraulic press are developed.
Pressure due to a fluid column:
For liquid of density ρ in a static equilibrium in a container, pressure at all points at the same depth ( or in other words, at the same horizontal layer ) is the same.
Consider an imaginary cylindrical volume element of height dy and cross-sectional area A at the depth y from the surface of liquid as shown in the figure.
The weight of liquid in this volume element is dW = 
g A dy
If P and P + dP are the pressures on the upper and lower faces of the element, then PA and ( P + dP ) A are the forces acting on them respectively. In equilibrium,
PA + dW = (P+dP)A
∴ PA + gAdy = PA + AdP
dP/dy = g
This equation is valid for any fluid ( liquid or gas ). It shows that the pressure increases with increase in the depth. Here, ρ g is the weight density, i.e., weight per unit volume of the fluid. Its value for water is 9800 N / m3. Pressure P at the depth y = h can be obtained by integration as under.
As is independent of pressure and constant for liquid, the above integration gives P - Pa = gh ∴ P = Pa + gh
This equation is valid only for incompressible fluid, i.e., liquid and gives the pressure at depth h in a liquid of density ρ.
Here, P (= Pa + gh) is the absolute pressure whereas P - Pa (= gh) is the gauge pressure also known as the hydrostatic pressure.
The pressure at any point in a liquid does not depend on the shape or cross-sectional area of its container. This is known as hydrostatic paradox.
When liquid is filled in the containers of different shapes and sizes, joined at the bottom as shown in the figure, the height of liquid columns in all the containers is found to be the same.