Consider pressure inside a drop of water. I have seen a formula for it where
$$\text{Pressure Inside}=\text{Pressure Outside}+2\frac{S}{R}.$$
I want to know how pressure exists inside the liquid. Is it due to weight of the liquid or molecular interaction or something? What is the reason that there is a pressure inside the drop of water?
The pressure outside a droplet arises from molecules bouncing against the droplet's surface, delivering an inward momentum "kick." Since the droplet doesn't shrink to nothing, there must be an internal pressure at least equal to the outward pressure to provide an equal and opposite momentum kick from internal particle collision with the surface. This provides one term, as mediated by the surrounding pressure.
In addition, there's an energy penalty associated with any surface, as bonds are poorly satisfied at surfaces relative to the bulk. (There are fewer like particles in the vicinity of surfaces.) This energy penalty is equivalent to a driving force for the surface to reduce its area, known as surface tension. This provides the second pressure term, as mediated by the surface tension and the surface area.
The sum of the two terms corresponds to the expression you give in the question.
The atmosphere and the surface tension forces both squeeze the water droplet which generates a pressure more than the atmospheric pressure inside the drop
the reason is that the surface of the droplet acts as if it were a stretched rubber membrane which applies a squeezing force to the water inside. The stretched rubber effect is due to something called surface tension. The Young-Laplace equation gives the excess pressure inside the droplet in terms of the surface tension and the radius of curvature of the droplet.
The pressure exists inside a drop of water due to the cohesive forces between water molecules. These forces cause the molecules to be attracted to each other, creating tension within the drop.
This tension results in a pressure that pushes outward on the surface of the drop, trying to minimize the surface area of the drop and maintain equilibrium. This pressure is known as surface tension and is what allows water droplets to maintain their spherical shape.
