To tackle this problem, we need to understand the relationship between pressure, height, and density in a fluid system. The scenario involves a vertical cylinder with a piston at the top, where atmospheric pressure and the pressure exerted by the mercury column come into play. Let’s break it down step by step.
Understanding Pressure in Fluids
In fluid mechanics, pressure at a certain depth in a fluid is given by the equation:
P = P₀ + ρgh
Where:
- P is the total pressure at depth.
- P₀ is the atmospheric pressure above the fluid.
- ρ is the density of the fluid (in this case, mercury).
- g is the acceleration due to gravity (approximately 9.81 m/s²).
- h is the height of the fluid column.
Given Data
In your scenario, we have:
- Height of the cylinder (h₁) = 100 cm
- Atmospheric pressure (P₀) = 75 cm of mercury
- Density of mercury (ρ) ≈ 13,600 kg/m³
- Acceleration due to gravity (g) = 9.81 m/s²
Calculating the Maximum Height of the Mercury Column
To find the maximum height of the mercury column (h₂) that can be placed on the piston, we need to ensure that the total pressure exerted by the mercury does not exceed the atmospheric pressure plus the pressure due to the air in the cylinder.
The pressure exerted by the mercury column can be expressed as:
P₂ = ρgh₂
At equilibrium, the pressure exerted by the mercury column plus the atmospheric pressure must equal the pressure exerted by the air in the cylinder. Since the air is at a constant temperature, we can assume that the pressure of the air remains constant at 75 cm of mercury. Therefore, we can set up the equation:
P₀ + P₂ = P₁
Substituting the values we have:
75 cm + ρgh₂ = 75 cm
Since the atmospheric pressure and the pressure from the air in the cylinder are equal, we can simplify this to:
ρgh₂ = 0
This means that the maximum height of the mercury column that can be placed on the piston is determined by the point at which the pressure exerted by the mercury equals the atmospheric pressure. To find this height, we can rearrange the equation:
h₂ = P₀ / (ρg)
Substituting Values
Now, let’s substitute the values into the equation:
First, convert atmospheric pressure from cm of mercury to pascals (Pa) for consistency:
P₀ = 75 cm × 1000 kg/m³ × 9.81 m/s² = 73575 Pa
Now, substituting into the height equation:
h₂ = 73575 Pa / (13600 kg/m³ × 9.81 m/s²)
h₂ ≈ 0.55 m or 55 cm
Final Thoughts
The maximum height of the mercury column that can be placed on the piston is approximately 55 cm. This calculation illustrates how atmospheric pressure and the pressure exerted by a fluid column interact in a closed system. Understanding these principles will help you solve similar problems in fluid mechanics, where pressure, height, and density are key factors. Keep practicing, and soon these concepts will become second nature!
