Q: 1 metre long tube, closed at both ends, is lying horizontally. A mercury column of length 0.1 m is filled in its middle and in rest two part air is filled at atmospheric pressure. What will be the displacement of the mercury column when the tube is turned to vertical position ? Atmospheric pressure = 0.76m of Hg.

To tackle this question, we need to consider how the behavior of the mercury column changes when the tube is rotated from a horizontal to a vertical position. The key concepts here involve atmospheric pressure, the properties of fluids, and how pressure is distributed in a closed system.

Understanding the Initial Setup

Initially, we have a tube that is 1 meter long, closed at both ends, with a mercury column of 0.1 meters in the center. The remaining space in the tube is filled with air at atmospheric pressure, which is given as 0.76 meters of mercury (Hg).

Pressure in the Horizontal Position

In the horizontal position, the mercury column is supported by the air pressure on either side. The pressure exerted by the air above the mercury column is equal to the atmospheric pressure, which is 0.76 m of Hg. The mercury column is balanced by the air pressure acting on it from both sides.

Transitioning to the Vertical Position

When the tube is turned vertically, the dynamics of the mercury column change significantly. In this position, the mercury will settle due to gravity, and we need to analyze how the pressures will adjust.

Calculating the New Position of Mercury

In the vertical position, the mercury will try to find a new equilibrium. The total height of the tube is 1 meter, and we have 0.1 meters of mercury. The remaining height is filled with air, which will exert pressure on the mercury column. The air pressure above the mercury column will still be 0.76 m of Hg, but now the mercury is influenced by gravity.

• The pressure at the bottom of the mercury column must equal the pressure exerted by the air above it.
• The height of the mercury column will adjust based on the balance of forces.

In the vertical position, the mercury column will be displaced upwards due to the atmospheric pressure acting on the air column above it. The pressure exerted by the air column can be calculated as follows:

Pressure Calculation

The pressure exerted by the air column above the mercury in the vertical position can be expressed as:

Pressure = Height of air column × Density of mercury × Gravitational acceleration

Given that the atmospheric pressure is 0.76 m of Hg, we can say that the pressure at the top of the mercury column in the vertical tube is also 0.76 m of Hg.

Final Displacement of the Mercury Column

In the vertical position, the mercury will rise until the pressure exerted by the column of mercury equals the atmospheric pressure. Since the initial height of the mercury column was 0.1 m, we can calculate the new height of the mercury column:

Let’s denote the new height of the mercury column as \( h \). The total height of the tube is 1 m, and the air column above the mercury will exert pressure equivalent to 0.76 m of Hg. Therefore, the mercury will rise until:

h = 0.76 m - 0.1 m = 0.66 m

This means that the mercury column will rise to a height of 0.66 m in the vertical position. The displacement of the mercury column can be calculated as:

Displacement = New height - Initial height = 0.66 m - 0.1 m = 0.56 m

Conclusion

Thus, when the tube is turned to a vertical position, the mercury column will be displaced upwards by 0.56 meters. This demonstrates how fluid dynamics and atmospheric pressure interact in a closed system, leading to changes in fluid levels based on orientation and pressure differences.