# Oscillations of a vertical spring with non negligible mass

My physics professor recently covered the concept of simple harmonic motion and touched on vertical springs. However he mentioned that the spring (with a mass attached to its end) only undergoes simple harmonic motion if it is massless. However I am unsure why making the spring have non-negligible mass would affect its simple harmonic motion hence I am seeking clarification. Thank you!

• Effective mass (spring–mass system) Commented Jul 15, 2020 at 8:00
• It's a little unusual to ask questions in the answer, but in this case the answerer is waiting for some feedback from you, and they appear to be pretty flexible and willing to help.
– uhoh
Commented Nov 25, 2020 at 15:01

Do you know the equation $$\omega = 2\sqrt{\frac{k'}{m}}sin(\frac{ka}{2})$$ ?
If no, we can derive with your, i will postpone until your answer the derivation. If you know: $$\omega = 2\sqrt{\frac{k'}{m}}sin(\frac{ka}{2})$$ $$a \rightarrow 0$$ $$\omega = 2\sqrt{\frac{k'}{m}}(\frac{ka}{2})$$ but $$k'a = KL$$ $$\omega ^2 = K\frac{L}{\rho }k^2$$