Question

Two cylindrical vessels of equal cross sectional area $A$, contain liquid of density $\rho$, upto the heights $h_1$ and $h_2$. After inter connection the levels become equal then work done by force of gravity in this process is:

• A. ${\rho A\left(\frac{h_1-h_2}{2}\right)}^{2}g$
• B. $\rho A\left(\frac{h_1-h_2}{2}\right)g$
• C. $\rho A{\left(\frac{h_1+h_2}{2}\right)}^{2}g$
• D. $\rho A(h_1-h_2)g$

Answer 1

The correct option is A ${\rho A\left(\frac{h_1-h_2}{2}\right)}^{2}g$

Total volume of water will remain same.

Height of water in one container will decrease and in other it will increase.

For equilibrium, height of water in both the vessels must be, $h=\frac{h_1+h_2}{2}$

Volume $=$ Area $\times$ height

Mass of water transfer $=$ Density $\times$ Volume

$\Rightarrow m=\rho A(h_1-\frac{h_1+h_2}{2})$

Work done $=$ Force $\times$ Displacement

$\Rightarrow W=$ weight $\times$ Displacement

$\Rightarrow W=mg(h_1-\frac{h_1+h_2}{2})$

$\Rightarrow W=\rho Ag{(h_1-\frac{h_1+h_2}{2})}^2$

$\Rightarrow W=\rho Ag{\left(\frac{h_1-h_2}{2}\right)}^2$