The air 10 cm diameter cylinder, as shown below, is heated until the spring is compressed 50mm. Find the work done by the air on the frictionless piston. The spring as the beginning of this process has a displacement of 0.0m with a spring constant of K=10KN/m. (assume Patm=101.325Kpa)
To find the work done by the air on the frictionless piston, we need to calculate the change in volume and then use the ideal gas law to find the change in pressure.
First, let's consider the initial state of the air inside the cylinder. The initial volume can be calculated using the formula for the volume of a cylinder:
V_initial = π * (r^2) * h
where r is the radius of the cylinder (which is half of the diameter) and h is the height of the cylinder. Assuming the air cylinder is vertical and the height of the cylinder is negligible, we can ignore h in this case. The radius of the cylinder (r) can be calculated by dividing the diameter by 2:
r = diameter / 2 = 10 cm / 2 = 5 cm = 0.05 m
So, the initial volume (V_initial) is:
V_initial = π * (0.05^2) * h
Next, let's consider the final state of the air inside the cylinder. The final volume can be calculated by adding the change in volume to the initial volume:
V_final = V_initial + ΔV
The change in volume (ΔV) can be calculated based on the displacement of the spring. The displacement of the spring is given as 50 mm, and since we are assuming a cylindrical shape, we can use the formula for the change in volume of a cylinder:
ΔV = π * (r^2) * Δh
where Δh is the change in height. In this case, the change in height is given as 50 mm, so:
Δh = 50 mm = 0.05 m
Substituting the values into the equation, we can determine the change in volume:
ΔV = π * (0.05^2) * 0.05
Now, we can calculate the final volume:
V_final = V_initial + ΔV
Now, we have the initial and final volumes of the air inside the cylinder. To find the change in pressure, we can use the ideal gas law:
P_initial * V_initial = P_final * V_final
We are given the initial pressure (Patm = 101.325 Kpa) and we need to find the final pressure (P_final). Rearranging the equation, we can solve for P_final:
P_final = (P_initial * V_initial) / V_final
Substitute the values:
P_final = (101.325 Kpa * V_initial) / V_final
Finally, the work done by the air on the frictionless piston can be calculated using the formula:
Work = P_final * ΔV
Substitute the values:
Work = P_final * (π * (r^2) * Δh)
Calculate the values and you will have the work done by the air on the frictionless piston.