PASCAL'S LAW 

In fluids under static conditions pressure is found to be independent of the orientation of the area. This concept explained by Pascal’s law which states that the pressure at a point in a fluid at rest is equal in magnitude in all directions. Tangential stress cannot exist if a fluid is to be rest. This is possible only if the pressure at a point in a fluid at rest is the same in all directions so that the resultant force at that point will be zero.

Fig.  Pascal's law demonstration

Consider a wedge shaped element in a volume of a fluid as shown in figure. Let the thickness perpendicular to the plane be dy . Let the pressure on the surface inclined at an angle θ to vertical be P_θ and its length be dl. Let the pressure in the x, y and z direction be P_x ,P_{y ,} P_z .

First considering the x direction. For the element to be in equilibrium,

P_θ × dl × dy × cosθ = P_x × dy × dz

But,

dl × cosθ = dz  So, P_θ = P_x

When considering the vertical components, the force due to specific weight should be considered.

P_z × dx × dy = P_θ × dl × dy × sinθ + 0.5 × ϒ × dx × dy × dz

The second term on RHS of the above equation is negligible, its magnitude is one order less compared to the other terms.

Also,

dl × sinθ = dx   So, P_z = P_θ

Hence,

P_x = P_z = P_θ

Note that the angle has been chosen arbitrarily and so this relationship should hold for all angles. By using an element in the other direction, it can be shown that,

P_y = P_θ   and  so   P_x = P_y = P_Z

Hence, the pressure at any point in a fluid at rest in the same in all directions.

     A hydraulic press has a ram of 20cm diameter and plunger of 3 cm diameter. It is used for lifting a weight of 30 KN. Find the force required at the plunger.