Equation of Surface of Horizontal Membrane Distorted by a Vertical Force at the Origin
A membrane is stretched in the xy plane and pushed upby a force at the origin. Let
bethe surface tension, force per unit length, in the membrane. If a cutis made in the surface, the edges of the cut will tend to move awayfrom each other as a result of this.
Consider a piece of the membrane with length
andwidth
Theforces are acting along each edge. The force acting along edge 1is
atan angle
tothe xy plane and along edge 2 the force is
atan angle
to the surface.
The total vertical force along these edges is
If%phi is smaill
sowe can write
Also, for small
Hence
The contribution from edges 3 and 4 is
Hence the total force is
If the membrane is in equilibrium then the externalforce per unit area is balanced by this net force per unit area hence
Ifisconstant then