In the case of equilibrium of hemispherical drop, what i know is that the surface tension is responsible for holding the hemispherical drop with the other hemisphere thus forming a spherical drop and the surface tension also holds the liquid on the curved surface thus giving it the curved shape. Why is force due to surface tension measured only along periphery(T2πr) whereas the surface tension exists also on the surface(2πr²) ?

2 hours ago, Vivek98phyboy said:

In the case of equilibrium of hemispherical drop, what i know is that the surface tension is responsible for holding the hemispherical drop with the other hemisphere thus forming a spherical drop and the surface tension also holds the liquid on the curved surface thus giving it the curved shape. Why is force due to surface tension measured only along periphery(T2πr) whereas the surface tension exists also on the surface(2πr²) ?

There is no surface tension in the interior of a homogeneous liquid.
Surface tension acts at the surface or interface with a different phase, as the name implies.

The disk you refer to is in the interior of a homogeneous liquid.

The cohesive force you refer to acts at right angles to the surface ring (periphery) you refer to, all around the ring.

3 minutes ago, studiot said:

There is no surface tension in the interior of a homogeneous liquid.
Surface tension acts at the surface or interface with a different phase, as the name implies.

The disk you refer to is in the interior of a homogeneous liquid.

The cohesive force you refer to acts at right angles to the surface ring (periphery) you refer to, all around the ring.

So if i find that cohesive force along the periphery, does that give me the same force that is required to hold the molecules on the curved surface together?

6 minutes ago, Vivek98phyboy said:

So if i find that cohesive force along the periphery, does that give me the same force that is required to hold the molecules on the curved surface together?

Forces, not force, it is distributed like the pressure force, but around a line rather than over an area.

This sketch may help, it is rather like the intersecting lines of latitude and longitude on a globe.

You perform a sum (an integral) of all the longitudinal forces along each line of longitude intersecting an equatorial line of latidude.

By symmetry this happens all round each line.

 

 

3 hours ago, studiot said:

Forces, not force, it is distributed like the pressure force, but around a line rather than over an area.

This sketch may help, it is rather like the intersecting lines of latitude and longitude on a globe.

You perform a sum (an integral) of all the longitudinal forces along each line of longitude intersecting an equatorial line of latidude.

By symmetry this happens all round each line.

 

 

So all we need to do is to find that along just one of those longitude. Right?

 

11 hours ago, Vivek98phyboy said:

Why is force due to surface tension measured only along periphery(T2πr) whereas the surface tension exists also on the surface(2πr²) ?

Remember that the units of surface tension are newtons per metre ie force per unit length.

So the total hoop or tangential force acting all the way round one line of longitude is the surface tension times the length all the way round the line of longitude, ie T*2πr.

Here are 3 different ways to derive your formula.

Note how this applies to a soap bubble that has two surfaces, an inner and a outer surface, and a droplet which only has one.