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How do you explain point 44 of the attached pdf document on surface tension? Here's the link.

How is the direction of surface tension found out? (I know it tangential but in which direction along the tangent?)

Also why is surface tension for solid-air surface considered when balancing the forces on liquid? Any help will be appreciated.

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The problem is azimuthally-symmetric, so the tangential direction is the one with no azimuthal component, i.e. the one "straight up the side" if you were a small mountain climber climbing from the surface to the top of the drop.

All three surface tensions are required because all three exert forces. For example, if the solid-air surface tension were extremely high, the system would try to minimize the area of contact between the surface and the air, which corresponds to spreading the drop out flat. If the solid-air surface tension were very low, the opposite would occur and the drop would be a sphere touching the surface in a very small area.

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