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Piston with spring: final pressure when volume doubles

Short answer

Final pressure is about 5.01×10⁴ kPa (50.1 MPa). Initial pressure from atmosphere and piston weight is 149.05 kPa; spring compression adds 50000 kPa, giving total 50149 kPa.

A piston cylinder device loaded with a linear spring with a spring constant of k = 100kN/m contains a gas initially at a pressure of Patm = 100 kPa and a volume of 0.05m³, as shown in figure below. The mass and cross sectional area of the piston are 50 kg and 0.01m² respectively. Heat is supplied to the system until its volume doubles, determine the final pressure. [Take g = 9.81m/s²] [Figure: vertical cylinder containing gas, closed by a piston held down by a linear spring above it.]

Given: k = 100 kN/m, P_atm = 100 kPa, V₁ = 0.05 m³, m = 50 kg, A = 0.01 m², V₂ = 2V₁ = 0.1 m³, g = 9.81 m/s².

Assumption: The spring is just touching the piston (zero force) at the initial state and is then compressed as the piston rises; spring force = k·Δx. (The data give a very long cylinder, since V/A = 5 m, but we use the numbers as given.)

Step 1: Piston weight pressure m g/A = (50 × 9.81)/0.01 = 49 050 Pa = 49.05 kPa

Step 2: Initial gas pressure (spring force zero) P₁ = P_atm + m g/A = 100 + 49.05 = 149.05 kPa

Step 3: Piston travel Δx = (V₂ − V₁)/A = (0.1 − 0.05)/0.01 = 5 m

Step 4: Spring force and pressure F_spring = k Δx = 100 000 × 5 = 500 000 N
F/A = 500 000/0.01 = 5.0 × 10⁷ Pa = 50 000 kPa

Step 5: Final pressure P₂ = P_atm + m g/A + k Δx/A = 100 + 49.05 + 50 000 ≈ 50 149 kPa

Answer: P₂ ≈ 5.01 × 10⁴ kPa (about 50 MPa). The result is large because the 0.01 m² area makes a 5 m stroke; the method is the same for any data: P₂ = P_atm + mg/A + k(V₂ − V₁)/A².

✓ Answer checked by Claude Opus 5.5 (high effort)

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