A

To determine the weight required in the other pan to balance the mercury barometer tube, we need to consider the force due to atmospheric pressure acting on the surface area of the mercury column.
1. Atmospheric pressure is given by the formula: \( P = \rho g h \), where \( P \) is the pressure, \( \rho \) is the density of mercury (approximately 13,600 kg/m³), \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height of the mercury column (typically around 760 mm or 0.760 m).
2. Rearranging gives us the weight of the mercury column: \( F = P A = \rho g h A \), where \( A \) is the cross-sectional area of the tube.
3. Since the weight of mercury creates a force, the mass of the weight required in the other pan must equal \( F \) to keep the system in equilibrium.
4. The gravitational force on the weight can be given as: \( W = mg \), where \( W \) is the weight, and \( m \) is the mass of the weight placed in the other pan.
5. Therefore, to balance the mercury barometer, the required weight must equal the force exerted by atmospheric pressure on the mercury column, which is specific to its height and the resulting pressure. The exact calculations depend on the provided heights in the diagram, which is not explicitly detailed in this question.
Given standard sea level pressure, the weight required will typically be around 760 g for equilibrium under standard conditions.
Therefore, based on typical measurements and the above reasoning, the correct answer is A.