I just found a question in my textbook which asked how the period of the vertical oscillation will change if the spring and block system is moved to the moon, and the gravity due to acceleration is cut to one-sixth of the gravity on Earth.
The answer is that the period stays the same, because gravity does not affect the period of the spring-block oscillator. I know this from the equation used to find the period, but I’d like to know just why exactly does gravity not make a difference in the speed in which a cycle is completed. Is it because the block is slowed down while going upwards just as much as it is sped up when going downwards?
EDIT: Hey guys, I’m just a high school student who barely understands physics. I really, really appreciate you all taking the time to answer my question, but I don’t understand your explanations. If possible, a simpler explanation would be super helpful.
Best Answer
In the case of 1-D harmonic motion a constant force cannot change the the time period. The constant force simply shifts the equilibrium position of the harmonic motion. That is to say, it shifts the motion by a distance $\frac{mg}{k}$ (the distance the spring would stretch due to the mass if there was no brownian motion).
We can think of it in this way: If the string is stretched by a distance $\frac{mg}{k}$, a constant force equal to $mg$ is applied on the block by the spring (in a direction opposite to gravity). This force is cancelled out by the constant gravitational force. There is no difference in the motion if we stretch the spring a bit and apply a constant force -- just a displacement of the whole motion $$ma=-kx+mg $$
replace $x$ by $x_{eff}$ such that $x_{eff}=x- \frac{mg}{k}$. Therefore
$ a_{eff}=a \hspace{2cm} ma_{eff}=-kx_{eff}$
Therefore by adding a constant to $x$ we can get the standard harmonic motion expression
You can also explain vaguely based on the slowing down and fastening up without considering the change in the equilibrium position. If we see the motion in the displaced frame there is absolutely no change in motion. It is better to see that the constan force is completely nullified due to the stretch in the spring