Askiitians Tutor Team
Last Activity: 4 Months ago
To tackle this problem, we need to understand how pressure works in a closed system, particularly in the context of a vertical cylinder with a piston. The key here is to apply the principles of fluid mechanics and the concept of pressure equilibrium. Let's break it down step by step.
Understanding the System
We have a vertical cylinder that is closed at the top by a piston. The air inside the cylinder is at a constant temperature, and the atmospheric pressure acting on the piston is equivalent to 75 cm of mercury (Hg). When mercury is poured onto the piston, it exerts additional pressure on the air inside the cylinder.
Pressure Basics
Pressure is defined as force per unit area. In this scenario, we have two types of pressure to consider:
- Atmospheric Pressure (Pₐ): This is the pressure exerted by the weight of the air above us, which in this case is 75 cm of mercury.
- Hydrostatic Pressure (Pₕ): This is the pressure exerted by the mercury column that we will add on top of the piston.
Calculating the Maximum Height of Mercury
To find the maximum height of the mercury column that can be placed on the piston, we need to ensure that the total pressure inside the cylinder does not exceed the atmospheric pressure outside. The pressure inside the cylinder can be expressed as:
Pₐ + Pₕ = Pₐ (atmospheric pressure)
Where:
- Pₕ = ρgh
- ρ = density of mercury (approximately 13,600 kg/m³)
- g = acceleration due to gravity (approximately 9.81 m/s²)
- h = height of the mercury column in meters
Given that the atmospheric pressure is equivalent to 75 cm of mercury, we can convert this to meters:
75 cm = 0.75 m
Setting Up the Equation
Now, we can set up the equation for pressure equilibrium:
0.75 m + ρgh = 0.75 m
Since we want to find the maximum height of the mercury column (h), we can rearrange the equation:
ρgh = 0.75 m
Substituting Values
Now, substituting the known values:
h = (0.75 m) / (ρg)
Substituting the values for ρ and g:
h = (0.75 m) / (13,600 kg/m³ * 9.81 m/s²)
Calculating this gives:
h = 0.75 m / (133,322.68 N/m²) = 0.00000562 m
However, this is not the correct approach. Instead, we should realize that the maximum height of the mercury column that can be added is directly related to the atmospheric pressure. The pressure exerted by the mercury column must equal the atmospheric pressure. Thus:
h = Pₐ / (ρg)
Substituting the values:
h = (75 cm * 1000 kg/m³) / (13,600 kg/m³ * 9.81 m/s²)
Calculating this gives us the maximum height of the mercury column that can be placed on the piston.
Final Thoughts
In summary, the maximum height of the mercury column that can be placed on the piston is determined by balancing the pressures. The atmospheric pressure acts as a limit, and by understanding the relationship between pressure, height, and density, you can solve similar problems effectively. If you practice these concepts, you'll find that they become much clearer over time!
