A 50 kg piston with a radius of 50 cm floats on a cylinder that is filled with air. The air in the cylinder fills the entire volume, and begins at a height of 1 m. The cylinder is then heated, so that the column rises to a height of 2 m. What is the work done by the air as it expands in the cylinder? Assume that the internal dimension of the cylinder is the same as that of the piston.
To find the work done by the air as it expands in the cylinder, we can use the formula:
Work = Force x Distance
First, let's calculate the force exerted by the air on the piston. The force is equal to the weight of the piston. We can calculate the weight using the formula:
Weight = mass x gravity
Given that the mass of the piston is 50 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the piston is:
Weight = 50 kg x 9.8 m/s^2
Weight = 490 N
Since the piston is floating, the force exerted by the air is equal to the weight of the piston.
Therefore, the force exerted by the air on the piston is 490 N.
Next, let's calculate the distance through which the air expands. The distance is equal to the change in height of the air column, which is given as 2 m - 1 m = 1 m.
Therefore, the distance through which the air expands is 1 m.
Now, we can calculate the work done by the air as it expands in the cylinder:
Work = Force x Distance
Work = 490 N x 1 m
Work = 490 Nm (or Joules)
Therefore, the work done by the air as it expands in the cylinder is 490 Joules.
To calculate the work done by the air as it expands in the cylinder, we need to determine the change in volume of the air and the pressure at which the expansion occurs.
Let's start by calculating the change in volume of the air. The initial volume (Vi) of the air in the cylinder can be calculated using the formula for the volume of a cylinder:
Vi = π * r^2 * hi
where r is the radius of the cylinder/piston (50 cm or 0.5 m) and hi is the initial height of the air column (1 m).
Vi = π * (0.5 m)^2 * 1 m
Vi = π/4 m^3
Similarly, we can calculate the final volume (Vf) of the air when the column rises to a height of 2 m:
Vf = π * r^2 * hf
where hf is the final height of the air column (2 m).
Vf = π * (0.5 m)^2 * 2 m
Vf = π m^3
The change in volume (ΔV) can be calculated as:
ΔV = Vf - Vi
ΔV = π m^3 - π/4 m^3
ΔV = 3π/4 m^3
Now, we need to determine the pressure at which the expansion occurs. To do this, we can use Pascal's law, which states that the pressure is transmitted undiminished in an enclosed fluid. Since the piston is floating, the pressure exerted by the air column pushing on the piston is equal to the atmospheric pressure, which we can assume to be constant.
The work done by the expanding air can then be calculated using the formula:
Work = Pressure * ΔV
Considering that the pressure is constant, we can use the atmospheric pressure as the value for Pressure. Let's assume the atmospheric pressure is 1 atmosphere (atm), which is approximately 101,325 pascals (Pa).
Work = 101,325 Pa * (3π/4 m^3)
Work ≈ 238,732.4 J
Therefore, the work done by the air as it expands in the cylinder is approximately 238,732.4 Joules.