To solve the problem of how much the mercury column will be displaced when the tube is held vertically, 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
Initially, you have a horizontal tube that is 1 meter long, with a mercury column of 10 cm in the middle. Each half of the tube contains air at a pressure equivalent to 76 cm of mercury. This means that the air pressure on both sides of the mercury column is equal, and the mercury remains stationary in the horizontal position.
Transitioning to a Vertical Position
When the tube is turned vertical, the dynamics change. The air pressure on both sides of the mercury column will still be 76 cm of Hg, but now the weight of the mercury column will also play a significant role in how the mercury behaves. The pressure exerted by the mercury column will depend on its height and the density of mercury.
Calculating the Displacement
To find out how far the mercury column will be displaced, we can use the following steps:
- Determine the pressure exerted by the mercury column: The height of the mercury column is 10 cm, which exerts a pressure of 10 cm of Hg.
- Calculate the net pressure difference: When the tube is vertical, the pressure from the air on both sides remains at 76 cm of Hg. However, the mercury will try to balance out the pressure exerted by the air and its own weight.
- Apply the principle of equilibrium: The mercury will displace until the pressure exerted by the height of the mercury column equals the air pressure on either side. The total pressure on one side will be the air pressure minus the pressure due to the mercury column.
In this case, the air pressure is 76 cm of Hg, and the mercury column exerts a pressure of 10 cm of Hg. Thus, the effective pressure difference that causes the displacement is:
Effective Pressure = Air Pressure - Mercury Pressure = 76 cm - 10 cm = 66 cm of Hg.
Final Displacement Calculation
Since the mercury column will adjust to balance the pressure, it will move up on one side and down on the other side. The total displacement of the mercury column will be half of the effective pressure difference because the mercury will rise on one side and fall on the other side equally.
Displacement = Effective Pressure / 2 = 66 cm / 2 = 33 cm.
Therefore, when the tube is held vertically, the mercury column will be displaced by a distance of 33 cm. This means that one side of the mercury column will rise by 33 cm, while the other side will drop by the same amount, maintaining equilibrium.
Summary
In summary, the displacement of the mercury column when the tube is held vertically is 33 cm. This result illustrates the balance of forces and pressures acting on the fluid, showcasing the principles of hydrostatics and gas laws in action.
