Consider a water tank shown in the figure. It has one wall at x = L and can be taken to be very wide in the z direction. When filled with a liquid of surface tension S and density , the liquid surface makes angle 0 ( 0 1) with the x-axis at x = L. If y(x) is the height of the surface then the equation for y(x) is: [figure] (take (x) = (x) = (x) = dydx, g is the acceleration due to gravity)
- d2ydx2 = gS
- dydx = gSx
- d2ydx2 = gSx
- d2ydx2 = gSy
Answer
(4) d2ydx2 = gSy
Solution
Take a thin strip of the surface between x and x + dx (unit width along z). Surface tension pulls along the surface at both ends; the net vertical pull is
S (x + dx) - S (x) = S ddx ( dydx )dx = S d2ydx2 dx
This supports the liquid column raised to height y under the strip (pressure difference g y across the surface):
S d2ydx2 dx = g y dx
d2ydx2 = gSy
Answer: (4)