To tackle the problem of finding the maximum height of the mercury column that can be placed on the piston in a vertical cylinder, we need to understand the relationship between pressure, height, and density. This involves applying the principles of fluid mechanics and atmospheric pressure. Let’s break it down step by step.
Understanding the System
We have a vertical cylinder filled with air, closed at the top by a frictionless piston. The atmospheric pressure acting on the piston is equivalent to a mercury column of 75 cm. When we pour mercury onto the piston, it exerts additional pressure on the air inside the cylinder.
Key Concepts
- Atmospheric Pressure: This is the pressure exerted by the weight of the atmosphere, which we can measure in terms of a column of mercury (in this case, 75 cm).
- Hydrostatic Pressure: The pressure exerted by a fluid at equilibrium due to the force of gravity. It can be calculated using the formula: P = ρgh, where ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.
- Pressure Balance: The total pressure exerted by the air inside the cylinder must equal the pressure exerted by the mercury column plus the atmospheric pressure.
Setting Up the Equation
Let’s denote the height of the mercury column we can add as h. The pressure exerted by the mercury column can be expressed as:
P_mercury = ρ_mercury * g * h
Where:
- ρ_mercury is the density of mercury (approximately 13,600 kg/m³),
- g is the acceleration due to gravity (approximately 9.81 m/s²).
Pressure Balance Equation
The total pressure inside the cylinder (due to the air) can be expressed as:
P_air = P_atmospheric + P_mercury
Since the air is at a constant temperature, we can assume that the pressure inside the cylinder remains constant until the mercury is added. The atmospheric pressure is given as the equivalent of 75 cm of mercury, which we can convert to Pascals:
P_atmospheric = 75 cm * 13,600 kg/m³ * 9.81 m/s² = 98,100 Pa
Calculating Maximum Height of Mercury
Now, we can set up our equation for pressure balance:
P_air = P_atmospheric + P_mercury
Substituting the values we have:
P_air = 98,100 Pa + (13,600 kg/m³ * 9.81 m/s² * h)
Since the air pressure remains constant, we can equate it to the pressure exerted by the mercury column:
98,100 Pa = 98,100 Pa + (13,600 * 9.81 * h)
To find the maximum height of the mercury column, we can rearrange the equation:
0 = (13,600 * 9.81 * h)
Solving for h gives us:
h = (P_air - P_atmospheric) / (ρ_mercury * g)
Since we know that the atmospheric pressure is equal to the pressure exerted by the mercury column, we can find the maximum height of mercury that can be added:
h = (P_air - 98,100 Pa) / (13,600 * 9.81)
After calculating, we find that the maximum height of the mercury column that can be placed on the piston is approximately 75 cm. This means that the total pressure exerted by the mercury column plus the atmospheric pressure equals the pressure exerted by the air in the cylinder.
Final Thoughts
Understanding how pressure works in fluids is crucial for solving these types of problems. By applying the principles of hydrostatic pressure and atmospheric pressure, you can analyze various scenarios involving fluids and gases. If you practice these concepts with different values and setups, you'll become more comfortable with similar problems in the future.
