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)

I don't see your figure, but why isn't the work done on the piston just equal to the final potential eneergy of the spring? You know what that is.

To find the work done by the air on the frictionless piston, we can first calculate the change in volume of the air when the spring is compressed.

1. The initial volume of the air can be calculated using the formula for the volume of a cylinder:
V_initial = π * (r_initial)^2 * h_initial
where r_initial = radius_initial and h_initial = height_initial.

2. Given that the diameter of the cylinder is 10 cm, the initial radius can be calculated as:
r_initial = diameter / 2 = 10 cm / 2 = 5 cm = 0.05 m.

3. The initial height of the air is not provided, so we cannot calculate the initial volume with the information given.

4. However, we can calculate the change in volume of the air when the spring is compressed.

5. The change in volume can be calculated using the formula:
ΔV = π * ((r_initial + Δr)^2 - r_initial^2) * Δh
where Δr = change in radius and Δh = change in height.

6. Given that the spring is compressed by 50 mm, the change in height can be calculated as:
Δh = 50 mm = 0.05 m.

7. The change in radius can be calculated using Hooke's Law:
F = -K * Δx,
where F = force, K = spring constant, and Δx = change in displacement.

8. The force exerted by the spring is equal to the force exerted by the air on the piston, which can be expressed as:
F = P * A,
where P = pressure and A = area of the piston.

9. The area of the piston can be calculated using the formula for the area of a circle:
A = π * (r_piston)^2

10. The pressure exerted by the air can be calculated as:
P = P_initial,
where P_initial = atmospheric pressure = 101.325 kPa.

11. Substituting these values into the equation for the force, we can solve for the change in radius:
-K * Δx = P_initial * π * (r_piston)^2,
Δx = (P_initial * π * (r_piston)^2) / -K

12. Substituting the given values, we have:
Δx = (101.325 kPa * π * (0.05 m)^2) / -10 kN/m
Δx = (101.325 * 1000 N/m^2 * 0.05 m^2 * π) / -10 N

13. Calculating the change in radius, we have:
Δr = Δx = (101.325 * 1000 N/m^2 * 0.05 m^2 * π) / -10 N
Δr ≈ -0.796 m

14. Now, we can substitute the values for the change in radius and change in height into the formula for the change in volume:
ΔV = π * ((0.05 m - 0.796 m)^2 - (0.05 m)^2) * 0.05 m
ΔV ≈ -0.032 m^3

15. The work done by the air on the frictionless piston can be calculated as:
W = -P_initial * ΔV
W = -101.325 kPa * -0.032 m^3
W ≈ 3.24 kJ

Therefore, the work done by the air on the frictionless piston is approximately 3.24 kJ.

To find the work done by the air on the frictionless piston, we need to calculate the change in the potential energy of the spring during the process.

The potential energy stored in a spring is given by the equation:

PE = (1/2)kx^2

Where:
PE = Potential Energy
k = Spring Constant
x = Displacement from the equilibrium position

In this case, the displacement of the spring is 50 mm, which is equal to 0.05 m. The spring constant is given as 10 kN/m, which can be converted to SI units by multiplying by 1000, giving us 10,000 N/m.

Now, we can calculate the potential energy of the spring at the beginning and end of the process:

PE_initial = (1/2)kx_initial^2
PE_final = (1/2)kx_final^2

Given that the initial displacement (x_initial) of the spring is 0 m, and the final displacement (x_final) is 0.05 m, we can calculate the potential energy at each position:

PE_initial = (1/2)(10,000 N/m)(0 m)^2 = 0 J
PE_final = (1/2)(10,000 N/m)(0.05 m)^2 = 2.5 J

The work done by the air on the frictionless piston is equal to the change in potential energy:

Work = PE_final - PE_initial
Work = 2.5 J - 0 J
Work = 2.5 J

Therefore, the work done by the air on the frictionless piston is 2.5 Joules.