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Force is a vector and it is the external influence that acts on a body that causes a body to move. It is like a push.  

or stress is defined as magnitude of force / surface area on which force is acting. Pressure is a scalar quantity.

Suppose we have a steel needle and a piece of paper, both of same weight. Now fold the paper to make a boat. Put the paper boat and needle on water surface in a tub. Needle sinks. Boat floats.   Here weight of a needle is same as boat, so force exerted by needle and boat are same on water surface. But pressure or stress due to the needle is more, so it sinks. The stress or pressure is more because it is force / surface area, or weight / surface area.  

The needle can pierce through our body but not a boat or a rod. Pressure or stress exerted by needle is much more than the stress or pressure exerted by a rod. Even though we push both against our body with same force, rod cannot achieve what needle achieves.  

We take a water tube connected to a pipe. Water through pipe comes out in the shape of a parabola. Now press against the nozzle of the pipe. Water stream goes higher and farther.  This is explained by pressure = force/area.  Area of nozzle decreases so pressure increases. Here the force is same on water in both cases.

Different things happen due to variation of force and variation of pressure. So we need different quantities to be defined.


P = F / A
        Here surface area is also taken as a vector.

In a liquid at a point P at a depth d, the water particles experience a force equal to the weight of a water column above that point P per unit area.   Pressure at a point inside a liquid is same in all directions from 0 to 360 degrees.

Pressure does not follow the vector addition law.  Force follows vector laws.
 Force exerted in all directions is not same. But pressure inside water is same in all directions.

There are some phenomenon that are explained by pressure or stress = force / unit cross sectional area.

Like expansion of a rod when heated or pulled by a force. The extension is proportional to force / area = stress. But not related to force alone.  

When air is filled inside a rubber tube inside a tyre or a baloon, the more relevant quantity is pressure and not force. So force and pressure are required to be defined and used separately.
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