In a U-tube, the restoring force is provided by gravity acting on the unbalanced liquid column. The “stiffness” of the system (restoring force per unit displacement) remains constant because it depends just on the heavy liquid (mercury) displacement, so the period of oscillation depends only on the total mass being moved. \[ T = 2\pi \sqrt{\frac{M_{total}}{k}} \]

Let mass of mercury be \(M\). Period \(T_1 = 2.0\) s. \[ T_1^2 \propto M \implies k T_1^2 = 4\pi^2 M \quad \text{— (1)} \]

Mass \(m = 100\) g of water is added. Total oscillating mass is \(M + m\). Period \(T_2 = 3.0\) s. \[ T_2^2 \propto (M + m) \implies k T_2^2 = 4\pi^2 (M + m) \quad \text{— (2)} \]

Dividing equation (2) by (1): \[ \frac{T_2^2}{T_1^2} = \frac{M + m}{M} = 1 + \frac{m}{M} \] \[ \frac{9}{4} = 1 + \frac{100}{M} \] \[ \frac{5}{4} = \frac{100}{M} \] \[ M = \frac{400}{5} = 80 \text{ g} \]

Formula used: \( M = \frac{m T_1^2}{T_2^2 – T_1^2} \)
Answer: 80 g