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: ( take (x)= (x)= (x)= d y d x , g . is the acceleration due to gravity)
B. d ^2 y dx ^2 = g S y
For the given element, (consider length d in Z direction) Net force in upward direction = Weight (S ( +d )-S ) d=m g Angle is small d y d x = g S ...(1) aligned & = d y d x Differentiating wrt x \\ & ^2 d d x = d^2 y d x^2 ...(2) aligned Put d from (2) in (1) d take 1, we get d^2 y d^2 = g y S Alternative Solution : aligned & P_A=P_B=P_0 \\ & P_C=P_0- g y \\ & P_C=P_0- S R \\ & g y= S R \\ & g y=S d^2 y d x^2 \\ & d^2 y d^2 = g y S aligned R = \ 1+ ( dy dx )^2 \ ^ 3 / 2 ~d ^2 y dx ^2 dy / dx is very small aligned & R = 1 ~d ^2 y / dx ^2 \\ & ~d ^2 y dx ^2 = 1 R aligned
Related: Physics — Mechanical Properties of Fluids · All PYQ Banks