---
title: "Two oscillating masses: motion, centre of mass, energy"
subject: "Applied Mechanics (Dynamics)"
chapter: 3
key: "semester_3/applied_mechanics_dynamics:two-masses-springs-equations-energy"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/two-masses-springs-equations-energy
site: BCT Engineering
---

# Two oscillating masses: motion, centre of mass, energy

Applied Mechanics (Dynamics) · Chapter 3: System of particles

- Asked in: 2074 Chaitra, 6 marks
- Video: [Double Spring Oscillation : Differential Equation](https://www.youtube.com/watch?v=7vzqKTk3Pnc) — Indra Lal Subedi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/two-masses-springs-equations-energy
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-3

## Question

Two masses shown in figure oscillate on the smooth plane in the x-direction. a) Write the differential equation of motion for each mass. b) Find the equation of motion for the center of the mass. c) Write the expression for kinetic and potential energy of the system of particles. [Figure: wall, spring $k_1$, mass $m_1$ (displacement $x_1$), spring $k_2$, mass $m_2$ (displacement $x_2$), with force $F$ acting on $m_2$.]

## Short answer

a) m1 x1'' = -k1 x1 + k2(x2 - x1), m2 x2'' = -k2(x2 - x1) + F. b) (m1+m2)xG'' = F - k1 x1. c) T = 1/2 m1 (x1')^2 + 1/2 m2 (x2')^2, V = 1/2 k1 x1^2 + 1/2 k2(x2 - x1)^2.

## Answer

Let $x_1$ and $x_2$ be the displacements of $m_1$ and $m_2$ from the positions where both springs are unstretched, measured to the right. The plane is smooth, so only the spring forces and $F$ act horizontally. Extension of spring $k_1$ is $x_1$; extension of spring $k_2$ is $(x_2 - x_1)$.

#### (a) Equations of motion
Spring $k_1$ pulls $m_1$ back with $k_1x_1$; spring $k_2$ pulls $m_1$ forward with $k_2(x_2-x_1)$:

$$
m_1\ddot x_1 = -k_1x_1 + k_2(x_2 - x_1)
$$

On $m_2$, spring $k_2$ pulls backward and $F$ pushes forward:

$$
m_2\ddot x_2 = -k_2(x_2 - x_1) + F
$$

#### (b) Motion of the centre of mass
$$
x_G = \frac{m_1x_1 + m_2x_2}{m_1+m_2}
$$

Adding the two equations, the internal spring force $k_2(x_2-x_1)$ cancels:

$$
(m_1+m_2)\,\ddot x_G = F - k_1x_1
$$

So the centre of mass moves as if the total mass were acted on by the external forces only: $F$ and the wall-spring force $k_1x_1$.

#### (c) Energy of the system
$$
T = \tfrac12 m_1\dot x_1^2 + \tfrac12 m_2\dot x_2^2
$$

$$
V = \tfrac12 k_1x_1^2 + \tfrac12 k_2(x_2 - x_1)^2
$$

Equivalently, with total mass $M = m_1+m_2$, $T = \tfrac12 M\dot x_G^2 + \tfrac12 m_1(\dot x_1-\dot x_G)^2 + \tfrac12 m_2(\dot x_2-\dot x_G)^2$ (kinetic energy of the centre of mass plus kinetic energy relative to it). The work done by $F$ equals the change in $T+V$.

## Check yourself

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

1. What is the extension of the middle spring k2 in terms of x1 and x2?
   A. x2
   B. x2 - x1
   C. x1 - x2 + F/k2
   D. x1
   Answer: B. Spring k2 joins the two masses, so its extension is the relative displacement x2 - x1. (Key point: Spring k2 extension)

2. Which is the equation of motion for m1?
   A. m1 x1'' = -k1 x1 - k2(x2 - x1)
   B. m1 x1'' = -k1 x1 + k2(x2 - x1)
   C. m1 x1'' = -k1 x1 + F
   D. m1 x1'' = k1 x1 + k2(x2 - x1)
   Answer: B. k1 pulls m1 back (-k1 x1), while stretched k2 pulls m1 forward with k2(x2 - x1). (Key point: Equation of motion of m1)

3. Adding the two equations of motion gives (m1 + m2) xG'' equal to what, and why?
   A. F - k1 x1, because the internal spring force k2(x2 - x1) cancels
   B. F + k2(x2 - x1), because spring forces add
   C. F - k1 x1 - k2(x2 - x1)
   D. 0, because springs are conservative
   Answer: A. The force of k2 on the two masses is equal and opposite (internal), so only the external forces F and the wall-spring force -k1 x1 remain. (Key point: Centre of mass motion)

4. Which expression is the potential energy of the system?
   A. (1/2)k1 x1^2 + (1/2)k2 x2^2
   B. (1/2)k1 x1^2 + (1/2)k2 (x2 - x1)^2
   C. (1/2)m1 x1'^2 + (1/2)m2 x2'^2
   D. (1/2)k1 x1^2 + (1/2)k2 (x2 + x1)^2
   Answer: B. Each spring stores (1/2)k times extension squared, with extensions x1 and x2 - x1. (Key point: Potential energy)

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