To solve the problem of the mercury column displacement when the tube is turned from a horizontal to a vertical position, we need to consider the principles of fluid mechanics and the behavior of gases under pressure. Let's break it down step by step.
Understanding the Initial Setup
We have a tube that is 1 meter long, closed at both ends, with a mercury column of 0.1 meters in the middle. The remaining space in the tube is filled with air. When the tube is horizontal, the mercury column is supported by the air pressure on either side. The air pressure exerts a force on the mercury, keeping it in place.
Analyzing the Situation
When we turn the tube vertically, the dynamics change significantly. The mercury will want to settle due to gravity, and we need to determine how much it will displace. Here’s how we can approach this:
- Initial Volume of Mercury: The volume of the mercury column can be calculated using the formula for the volume of a cylinder, which is V = A × h, where A is the cross-sectional area and h is the height (or length) of the mercury column. In our case, the height is 0.1 m.
- Air Pressure Consideration: The air above the mercury column exerts pressure on it. When the tube is vertical, the air pressure will act on the mercury column differently than when it was horizontal.
Calculating the Displacement
When the tube is turned vertically, the mercury will settle at the bottom of the tube due to gravity. The total height of the tube is 1 meter, and the mercury column initially occupies 0.1 meters. Therefore, when the tube is vertical, the mercury will occupy the bottom 0.1 meters of the tube.
Final Position of Mercury
In the vertical position, the mercury will not change its volume but will change its position. The air above the mercury will compress slightly, but since the tube is closed, the total volume of the air plus the mercury must remain constant. The air pressure will adjust to accommodate the new position of the mercury.
Determining the Displacement
The displacement of the mercury column when the tube is turned vertical is essentially how far the mercury moves from its original position. Since the mercury was initially in the middle of the tube (0.5 meters from either end) and will now settle at the bottom (0 meters from the bottom), we can calculate the displacement:
- Initial Position: 0.5 m (middle of the tube)
- Final Position: 0 m (bottom of the tube)
- Displacement: 0.5 m - 0 m = 0.5 m
Conclusion
Therefore, when the tube is turned vertical, the mercury column will displace by 0.5 meters downward. This movement is due to the gravitational force acting on the mercury, which causes it to settle at the lowest point of the tube. The air pressure above the mercury will adjust accordingly, but the total volume remains unchanged. This example illustrates the principles of fluid mechanics and how fluids behave under different orientations and pressures.
