---
title: "Piston with linear spring, air supply (5 kg)"
subject: "Fundamental of Thermodynamics & Heat Transfer"
chapter: 1
key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:piston-spring-air-pressure-100mm"
source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/piston-spring-air-pressure-100mm
site: BCT Engineering
---

# Piston with linear spring, air supply (5 kg)

Fundamental of Thermodynamics & Heat Transfer · Chapter 1: Introduction

- Asked in: 2075 Baisakh, 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-air-pressure-100mm
- Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-1

## Question

A 5 kg piston in a cylinder with diameter of 100 mm is loaded with a linear spring and the outside atmospheric pressure of 100 kPa. The spring exerts no force on the piston when it is at the bottom of the cylinder and for the state shown in figure below, the pressure is 400 kPa with volume of 0.4 L. The valve is opened to let some air in, causing the piston to rise 2 cm. Find the new pressure.
[Figure: vertical cylinder containing air, closed by a piston held down by a linear spring above it; an air supply line with a valve connects to the cylinder.]

## Short answer

Force balance includes piston weight, atmosphere, and linear spring with zero force at bottom. Area = 0.007854 m², weight pressure = 6.245 kPa. Initial spring pressure = 293.75 kPa, height = 0.05093 m. For 2 cm rise, extra spring pressure = 115.4 kPa. New pressure = 515.4 kPa.

## Answer

**Given:** m = 5 kg, D = 100 mm, P_atm = 100 kPa, P₁ = 400 kPa, V₁ = 0.4 L = 0.0004 m³, piston rises Δx = 2 cm. Take g = 9.81 m/s².

**Idea:** The spring force is zero when the piston is at the bottom, so the spring force is F = kx, where x = V/A is the height of the piston above the bottom. Force balance on the piston:
P·A = P_atm·A + m g + k x

**Step 1: Area**
A = (π/4)(0.1)² = 0.007854 m²

**Step 2: Piston weight pressure**
m g / A = (5 × 9.81)/0.007854 = 6245 Pa = 6.245 kPa

**Step 3: Spring contribution at state 1**
k x₁ / A = P₁ − P_atm − m g/A = 400 − 100 − 6.245 = 293.75 kPa

**Step 4: Initial height**
x₁ = V₁/A = 0.0004/0.007854 = 0.05093 m

**Step 5: Spring constant**
k = (293.75 × 1000 × 0.007854)/0.05093 = 4.53 × 10⁴ N/m ≈ 45.3 kN/m

**Step 6: New state** (x₂ = x₁ + 0.02 = 0.07093 m)
Extra spring pressure = k Δx / A = (45 300 × 0.02)/0.007854 = 115.4 kPa

P₂ = 400 + 115.4 = **515.4 kPa**

**Answer: New pressure ≈ 515 kPa**

## Check yourself

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

1. Which force balance on the piston is used?
   A. P A = P_atm A + m g + k x
   B. P A = P_atm A - m g - k x
   C. P A = m g + k x only
   D. P A = P_atm A + k x only
   Answer: A. Gas force upward balances atmosphere, piston weight and spring force acting downward. (Key point: Force balance)

2. Spring pressure at state 1 is 400 - 100 - 6.245 = 293.75 kPa. Using x1 = V1/A = 0.0509 m and A = 0.007854 m^2, the spring constant k is about
   A. 4.53 x 10^3 N/m
   B. 4.53 x 10^5 N/m
   C. 45.3 N/m
   D. 4.53 x 10^4 N/m
   Answer: D. k = (293.75 x 1000 x 0.007854)/0.05093 = 4.53 x 10^4 N/m. (Key point: Spring constant)

3. How does the new pressure follow when the piston rises 2 cm?
   A. P2 = P1 - k dx/A, since the gas expands
   B. P2 stays at 400 kPa
   C. P2 = P1 + k dx/A, since the spring force increases
   D. P2 = P_atm + k dx/A
   Answer: C. Extra spring compression adds k dx/A = 115.4 kPa to the pressure, giving P2 = 515.4 kPa. (Key point: Pressure rise)

4. What is the new pressure?
   A. About 400 kPa
   B. About 285 kPa
   C. About 115 kPa
   D. About 515 kPa
   Answer: D. P2 = 400 + 115.4 = 515.4 kPa. (Key point: Final result)

## Close variants set in other exams

- [Piston with linear spring, air supply (15 kg)](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Apiston-spring-air-15kg) — Fundamental of Thermodynamics & Heat Transfer, chapter 1, asked 2×

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