Two cylindrical vessels of equal cross-sectional area $16\ \text{cm}^2$ contain water upto heights $100$ cm and $150$ cm respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take density of water $= 10^3\ \text{kg m}^{-3}$ and $g = 10\ \text{m s}^{-2}$]
Answer: (B) $1$ J
A water column of height $h$ has potential energy $\rho A g\dfrac{h^2}{2}$. The final level is $\dfrac{1 + 1.5}{2} = 1.25$ m in each vessel.
$$U_i = \frac{\rho Ag}{2}(1^2 + 1.5^2) = \frac{\rho Ag}{2}(3.25), \qquad U_f = \frac{\rho Ag}{2}(2\times1.25^2) = \frac{\rho Ag}{2}(3.125)$$
$$W_g = U_i - U_f = \frac{10^3\times16\times10^{-4}\times10}{2}\times0.125 = 8\times0.125 = 1\ \text{J}$$
Solution by Sreeraj P, M.Sc Physics