2.4. Reduced mass
Here, we derive the reduced mass for a system of two point-like masses connected by a force that obeys Hooke’s law.
We start out with the two point-like masses \(m_1\) and \(m_2\) at positions \(x_1\) and \(x_2\), assuming \(x_2>x_1\). The bond length is the difference \(r = x_2 - x_1\). The equilibrium bond length is \(r_\mathrm{e}\).
If the two masses are connected by a spring that follows Hooke’s law with force constant \(k\), the forces acting on the two are
Dividing the first equation by \(m_1\) and the second equation by \(m_2\), and then subtracting the first from the second equation gives
The expression in parentheses can be abbreviated by defining the reduced mass \(\mu\)
This gives finally
This equation is Hooke’s law for a single effective particle of mass \(\mu\) on a spring with force constant \(k\) and equilibrium length \(r_\mathrm{e}\).
This shows that the two spring-connected masses can be modeled as a single mass connected to a spring of the same force constant.