# Infinite period in Simple Harmonic Motion

I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period?

• When mass is 0.
• When mass is infinite.

With $\tau=2\pi/\sqrt{k/m}$.

Because if you have zero mass you can't use the Newtonian $F=m a$ to find force, you'll have a hard time making a physical argument for having zero mass. Likewise, you break all physical barriers when you go to an infinite mass. The best one can do is say that the mass approaches zero or approaches infinity.
As $m\to \infty$, $\frac{m}{k}\to \infty$, and so the period goes to infinity, which can be physically interpreted as the spring's $k$ constant is just not strong enough to accelerate the mass at all.
As $m\to 0$, $\frac{m}{k}\to 0$, and so the period goes to zero, which can be physically interpreted as very very fast vibrations.
The trick used here is the same one used in $\varepsilon$ - $\delta$ limit/derivative proofs in pure math. To avoid the problem of calculating $\infty-\infty$ or $\infty/\infty$, we just calculate behavior "arbitrarily large" or "arbitrarily small", but not actually infinite nor zero.