# Surface Tension: Why Tangential and Not Downward?

• Frigus

#### Frigus

Why direction of surface tension is tangential to the surface and not perpendicular downwards since it is caused by molecules in the bulk?

Another question,
We say a system tends to be in lowest energy state that's why a water drop tends to be sphere as it reduces it's surface area the drop will have less surface energy but the energy it reduces by reducing surface area have got converted into internal energy so how it's aim to reach lowest energy state have been reached by becaming a sphere.

To answer your first question, it might be worthwhile to look at an analogous but simpler system=a rope that is wrapped around a cylinder. The tension ## T ## in the rope is along the tangential path, but it exerts a force on the cylinder that is perpendicular to the path, because of the curvature. The normal force on the cylinder per unit length is ## f_l=\frac{T}{r} ##, where ##r ## is the radius of the cylinder.

To answer your first question, it might be worthwhile to look at an analogous but simpler system=a rope that is wrapped around a cylinder. The tension ## T ## in the rope is along the tangential path, but it exerts a force on the cylinder that is perpendicular to the path, because of the curvature. The normal force on the cylinder per unit length is ## f_l=\frac{T}{r} ##, where ##r ## is the radius of the cylinder.

Sir in the rope case the tangential component can be resolved inside but in case of liquid how will the inside force can be resolved into the tangetial component

The surface tension will create a pressure ##P ## inside a spherical droplet. The book "Equilibrium Thermodynamics" by Adkins pp. 39-40 treats this particular case very well. Defining the surface tension ## \gamma=\frac{dW}{dA} ##, we can write ## dW=\gamma \, dA =8 \pi \gamma r \, dr##. We also have ## dW=P \, dV=P \, 4 \pi r^2 \, dr ##. Equating these two expressions for ## dW ## gives ## P=\frac{2 \gamma}{r} ##. ## \\ ## This expression for pressure ## P ## is similar to the expression of the force per unit length in the rope, where, in both cases, the radius ## r ## appears in the denominator.

The surface tension will create pressure ##P ## inside a spherical droplet. The book "Equilibrium Thermodynamics" by Adkins pp. 39-40 treats this particular case very well. Defining the surface tension ## \gamma=\frac{dW}{dA} ##, we can write ## dW=\gamma \, dA =8 \pi \gamma r \, dr##. We also have ## dW=P \, dV=P \, 4 \pi r^2 \, dr ##. Equating these two expressions for ## dW ## gives ## P=\frac{2 \gamma}{r} ##. ## \\ ## This expression for pressure ## P ## is similar to the expression of the force per unit length in the rope, where, in both cases, the radius ## r ## appears in the denominator.

Sir can you explain it verbally because I want to understand core idea behind it.

Pressure is a force per unit area that acts normal to a surface. The ## W ## above is work or energy. Surface tension ## \gamma ## is defined as the work required to increase the area of the surface by one unit, thereby ## \gamma=\frac{dW}{dA} ##. ## \\ ## Meanwhile, the above area ## A=4 \pi r^2 ##, thereby ## dA=8 \pi r \, dr ##.

Pressure is a force per unit area that acts normal to a surface. The ## W ## above is work or energy. Surface tension ## \gamma ## is defined as the work required to increase the area of the surface by one unit, thereby ## \gamma=\frac{dW}{dA} ##.
Sir but how can I understand that
Pressure is a force per unit area that acts normal to a surface. The ## W ## above is work or energy. Surface tension ## \gamma ## is defined as the work required to increase the area of the surface by one unit, thereby ## \gamma=\frac{dW}{dA} ##. ## \\ ## Meanwhile, the above area ## A=4 \pi r^2 ##, thereby ## dA=8 \pi r \, dr ##.
Sir but from these equations how can I understand that surface tension is tangetial.

For the case of a rope, the tension ## T ## is along the rope, and for many cases, the rope is in a straight line, and not wrapped around a cylinder... Similarly with surface tension=it works for a planar geometry, and then the theory is applied to a spherical droplet.

Perhaps it would help to think of the liquid in a planar geometry (thickness ## d ## in a plane) as consisting of the liquid in bulk plus a (fictitious) rubber-like membrane on the two flat surfaces=top and bottom. Materials (such as water), that exhibit surface tension, behave like this=it takes extra energy to increase their surface area, as if there were an elastic membrane at the surface. (The area of the plane is increased by stretching the elastic membranes in the plane=with a tangential stretching).