SHO of Water Sloshing in a Container

1. Sep 26, 2012

rashedah

1. The problem statement, all variables and given/known data

Consider a rectangular container, dimensions L x b, filled to level h with water. Water sloshes at low amplitude, y0 << h so that its surface remains always flat. Assuming the water has a density of δ, show that the potential energy is

$U(y) = \frac{1}{6} b δ g L y^2$

2. Relevant equations
$F = ma$

$U = \frac{1}{2} k x^2$

3. The attempt at a solution

Since F = ma = kx = mgh, I know that k = mg because x and h in this case are y0. And m is bLδ meaning that k = bLδg.

So integrating ky0 to get the potential energy I get

$U(y) = \frac{1}{2} b δ g L y^2$

but the answer has a fraction of 1/6, where did the extra 1/3 come from?