Consider the cylindrical outer surface of the column of liquid in a tube of length \(\ell\) and radius \(R\). Imagine slicing this surface into two half cylinders along the long central axis of the pipe. Show, by integration, that Eq. 18. 52 is valid-that is, (a) that the magnitude of the force due to the pressure difference \(P_{\text {in }}-P_{\text {out }}\) across the right semicylindrical surface (balanced by the force due to the surface tension due to the left half) is equal to the pressure difference \(\left(P_{\text {in }}-P_{\text {out }}\right)\) times \(2 \ell R\), and \((b)\) that the direction of this force is perpendicular to the pipe length and bisects the semi-cylindrical surface.

Data from Eq. 18. 52

Transcribed Image Text:

Pin - Pout (cylindrical surface). R