Askiitians Tutor Team
Last Activity: 5 Months ago
To solve this problem, we need to understand how the pressure of the air and the mercury column interact when the tube is held vertically. Let's break it down step by step.
Understanding the Initial Setup
Initially, we 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 (Hg). This means that the air pressure in both halves is equal, and the mercury column is balanced in this horizontal position.
Transitioning to a Vertical Position
When the tube is turned vertical, the dynamics change. The mercury column will experience gravitational force acting downward, and the air pressure in the tube will exert an upward force on the mercury. The key to finding the displacement of the mercury column lies in understanding how these forces balance out.
Calculating the Forces
In the vertical position, the pressure exerted by the air in the tube can be calculated using the height of the mercury column. The atmospheric pressure is given as 76 cm of Hg, which is equivalent to 76 cm of mercury pushing down on the column. When the tube is vertical, the mercury column will adjust until the pressures on both sides are equal.
Pressure Equilibrium
The pressure exerted by the mercury column can be calculated using the formula:
- Pressure = Height × Density × g
Where:
- Height = height of the mercury column (in cm)
- Density of mercury = approximately 13.6 g/cm³
- g = acceleration due to gravity (approximately 980 cm/s²)
Finding the Displacement
Initially, the mercury column is 10 cm long. When the tube is vertical, the air pressure will push up against the mercury column. The total pressure exerted by the air is 76 cm of Hg. Therefore, we can set up the equation:
Let x be the displacement of the mercury column. The new height of the mercury column on one side will be (10 cm - x), and on the other side, it will be (x). The pressure balance can be expressed as:
Pressure from the air = Pressure from the mercury column
76 cm of Hg = (10 cm - x) + x
Solving for x:
This indicates that the mercury column will not be displaced at all, as the pressures balance out perfectly. Therefore, the displacement of the mercury column when the tube is held vertically is:
Final Answer
The column of mercury will not be displaced; it will remain at its original height of 10 cm.
This scenario illustrates the principles of pressure equilibrium and how fluids behave under different orientations. The balance of forces is crucial in determining the behavior of the mercury column in response to changes in orientation.
