---
title: "Piston with spring: final pressure when volume doubles"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 1
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:piston-spring-gas-k100-volume-doubles"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/piston-spring-gas-k100-volume-doubles
site: BCT Engineering
---

# Piston with spring: final pressure when volume doubles

Fundamental of Thermodynamics & Heat Transfer · Chapter 1: Introduction

- Asked in: 2068 Bhadra, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/piston-spring-gas-k100-volume-doubles
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-1

## Question

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.]

## 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.

## Answer

**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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With zero spring force initially, P1 is
   A. P_atm only
   B. P_atm + m g/A = 149.05 kPa
   C. P_atm - m g/A
   D. P_atm + k x/A
   Answer: B. Atmosphere plus piston weight pressure (49.05 kPa) balances the gas pressure. (Key point: Initial pressure)

2. With V going from 0.05 to 0.1 m^3 and A = 0.01 m^2, the piston travel is
   A. 0.5 m
   B. 50 m
   C. 5 m
   D. 0.05 m
   Answer: C. dx = (V2 - V1)/A = 0.05/0.01 = 5 m. (Key point: Piston travel)

3. Which expression gives the final pressure?
   A. P2 = P_atm + k dx
   B. P2 = P_atm + m g/A + k dx/A
   C. P2 = P1 - k dx/A
   D. P2 = k dx only
   Answer: B. Force balance at state 2 includes atmosphere, piston weight and spring force, each divided by A. (Key point: Final pressure formula)

4. The final pressure is about
   A. 149 kPa
   B. 500 kPa
   C. 5000 kPa
   D. 50 149 kPa
   Answer: D. Spring pressure is 100 000 x 5/0.01 = 5 x 10^7 Pa = 50 000 kPa; adding 149.05 gives about 50 149 kPa. (Key point: Final value)

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