Let's say I have a pendulum hanging from a bar that's fixed to the wall of an elevator. Assume that there's no air or anything inside the elevator, that the string of the pendulum is very light and that the bob of the pendulum is more or less a heavy point mass. After setting the pendulum in motion, the elevator starts going down, increasing the period of the pendulum, until the cable holding the elevator runs out and brings the whole contraption into a free fall situation.
The formula for the period of a pendulum with length $L_0$ where the bob experiences a gravitational acceleration of $a_0$ is: $T = 2 \pi \sqrt{\frac{L_0}{a_0}}$. In free fall, $a_0 = 0$ so the pendulum wouldn't swing at all.
However, in my hypothetical situation, bob of the pendulum could've had a velocity right before going into free fall, so wouldn't the pendulum transition into a uniform circular motion which gives rise to a new period?
If so, shouldn't there be a better formula to describe the period of a pendulum that also correctly predicts the period depending on how the acceleration on the bob changes with respect to time?
Best Answer
As long as there is a net velocity on the pendulum bob at the moment the elevator goes into free fall, the pendulum will go into uniform circular motion.
The formula you have stated for the time period is only valid for a pendulum. Once the bob goes into circular motion, it is no longer a pendulum as there is no restoring force acting on the bob. The formula still makes logical sense as the bob will never reverse its direction and will hence take infinite time to come back to its starting path.