Pascal's law and effect of gravity

Pascal's law

One of the most important facts about fluid pressure is that a change in pressure at one part of the liquid will be transmitted without any change to other parts. This was put forward by Blaise Pascal (1623 - 1662), a French mathematician and physicist. This rule is known as Pascal's law.

Pascal's law states that if the effect of gravity can be neglected then the pressure in a fluid in equilibrium is the same everywhere.

Consider any two points A and B inside the fluid. Imagine a cylinder such that points A and B lie at the centre of the circular surfaces at the top and bottom of the cylinder (Fig.). Let the fluid inside this cylinder be in equilibrium under the action of forces from outside the fluid. These forces act everywhere perpendicular to the surface of the cylinder. The forces acting on the circular, top and bottom surfaces are perpendicular to the forces acting on the cylindrical surface. Therefore the forces acting on the faces at A and B are equal and opposite and hence add to zero. As the areas of these two faces are equal, we can conclude that pressure at A is equal to pressure at B. This is the proof of Pascal's law when the effect of gravity is not taken into account.

Pascal's law and effect of gravity

When gravity is taken into account, Pascal's law is to be modified. Consider a cylindrical liquid column of height h and density ρ in a vessel as shown in the Fig..

If the effect of gravity is neglected, then pressure at M will be equal to pressure at N. But, if force due to gravity is taken into account, then they are not equal.

As the liquid column is in equilibrium, the forces acting on it are balanced. The vertical forces acting are

(i) Force P1A acting vertically down on the top surface.

(ii) Weight mg of the liquid column acting vertically downwards.

(iii) Force P2A at the bottom surface acting vertically upwards. where P1 and P2 are the pressures at the top and bottom faces, A is the area of cross section of the circular face and m is the mass of the

cylindrical liquid column.

At equilibrium, P1A + mg - P2A = 0 or P1A + mg = P2A

P2 = P1 + mg/A

m = Ahρ

P2 = P1 + Ah g/A

P2 = P1 + hρg

This equation proves that the pressure is the same at all points at the same depth. This results in another statement of Pascal's law which can be stated as change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid and act in all directions.

Applications of Pascal's law

(i) Hydraulic lift

An important application of Pascal's law is the hydraulic lift used to lift heavy objects. A schematic diagram of a hydraulic lift is shown in the Fig.. It consists of a liquid container which has pistons fitted into the small and large opening cylinders. If a₁ and a₂ are the areas of the pistons A and B respectively, F is the force applied on A and W is the load on B, then

F/ a₁ = W/a₁

W = Fa₂/a₁

This is the load that can be lifted by applying a force F on A. In the above equation 2 a₁/a₂  is called mechanical advantage of the hydraulic lift. One can see such a lift in many automobile service stations

(ii) Hydraulic brake

When brakes are applied suddenly in a moving vehicle, there is every chance of the vehicle to skid because the wheels are not retarded uniformly. In order to avoid this danger of skidding when the brakes are applied, the brake mechanism must be such that each wheel is equally and simultaneously retarded. A hydraulic brake serves this purpose. It works on the principle of Pascal's law.

Fig.  shows the schematic diagram of a hydraulic brake system. The brake system has a main cylinder filled with brake oil. The main cylinder is provided with a piston P which is connected to the brake pedal through a lever assembly. A T shaped tube is provided at the other end of the main cylinder. The wheel cylinder having two pistons P₁ and P₂ is connected to the T tube. The pistons P₁ and P₂ are connected to the brake shoes S₁ and S₂ respectively.

When the brake pedal is pressed, piston P is pushed due to the lever assembly operation. The pressure in the main cylinder is transmitted to P₁ and P₂. The pistons P₁ and P₂ push the brake shoes away, which in turn press against the inner rim of the wheel. Thus the motion of the wheel is arrested. The area of the pistons P₁ and P₂ is greater than that of P. Therefore a small force applied to the brake pedal produces a large thrust on the wheel rim.

The main cylinder is connected to all the wheels of the automobile through pipe line for applying equal pressure to all the wheels .