# What is the meaning of a pendulum with a period of 0 seconds?

This question has been bugging me for a while, so i hope that some people could answer my question.

My question is, what is the exact meaning behind a pendulum which has a period 0s,What will its motion look like?

I thought of this problem when reading about a pendulum in a free falling elevator where its period tends to infinity thus the pendulum will be in a static state. So what is the motion of the pendulum when it has a period of 0 seconds .

Futhermore, can this be analysed through a pendulum in an elevator going upwards with infinite acceleration? If yes/no can you also explain at that state?

Im a high school student, so it will be helpful if can explain it in a way that i could understand.

A mathematical pendulum has the period of small oscillations given by: $$T = 2\pi\sqrt{\frac{l}{g}},$$ where $$l$$ is the pendulum length and $$g$$ is the free fall acceleration. The situation mentioned in the question - pendulum in a falling elevator - corresponds to zero free fall acceleration in the elevator reference frame, i.e. we need to take $$g\rightarrow 0$$ in the above equation, which means that $$T\rightarrow +\infty$$. Note that we may also have a different situation - of an extremely long pendulum, $$l\rightarrow +\infty$$ with the same result.

The elevator moving with constant high acceleration will have g that is almost equal to this acceleration, so your reasoning is correct. One could also associate this with the situation $$l=0$$, but in this case we have no pendulum.

• Note that you also have no pendulum in the case of infinite acceleration because infinite acceleration is not possible. Either way, $T=0$ is not physical, and physics has nothing to say about situations that are unphysical. Commented Dec 18, 2020 at 12:31
• @garyp $T=0$ and $T\rightarrow$ are limiting cases, which stand behind many approximations used in physics. Commented Dec 18, 2020 at 12:38
• Way before you reach T near zero the classical approximation fails and you need to use relativistic mechanics.
– nasu
Commented Dec 18, 2020 at 13:15
• @nasu any value in physics can be big or small only in comparison to something else. Whether we can take $T$ to be zero or not depends on the specific problem. Commented Dec 18, 2020 at 13:29
• The relativistic regime depends on the speed compared to the speed of light. As the period decreases the maximum speed increases and will approach the speed of light. The classic formula for the period won't be valid anymore.
– nasu
Commented Dec 18, 2020 at 15:12