# B Formula of S in simple harmonic oscillation

#### another_dude

In school we have numerous exercises that ask you to find the time when a body passes a certain point for the nth time in simple harmonic oscillation. But it is a bit mentally taxing to solve with the actual formula of x=Asin(ωt + φ), just because you have to sort out all the infinite solutions. It is even worse when you get to more challenging tasks like for example having to find various time intervals when the body gets from point A to point B. So I was wondering if it is possible to have a function f(t)=S where S is the distance travelled . This function must be 1-1 ( S is always increasing) and thus give only one solution. Then it must be relatively easy to get S from x and the opposite for any other possible calculations.

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#### sophiecentaur

Gold Member
I am not absolutely sure about what you mean. Are you trying to find the distance travelled (as opposed to displacement) after a time that could include several cycles of the oscillation? The distance travelled for a whole cycle is easy to find. So you can calculate the number of whole cycles and add that to the distance in the remaining fraction of a cycle.
From your post, I gather that you can deal with the questions you have already be set in school so it should be fairly straightforward for you to extend it to multiple cycles. (If indeed that's what you are trying to do.)

#### another_dude

You are correct. But I'm not trying to find travelled distance from time -in fact the opposite (time from distance travelled). The way it usually goes is you get displacement and in order to find time you solve the trig equation and filter out the solutions. But if you use distance travelled as a "middleman" you should get only one time and not have to sort out the solutions.

#### sophiecentaur

Gold Member
OK. A trig equation tends to have multiple solutions and you usually have to select the appropriate +2πn value. You still have to know the amplitude but why would it be any harder to do it that way round?

#### another_dude

Because you have to think which value to choose. Doesn't sound that big a deal, but when dealing with time intervals between two or maybe more points it can get a bit messy. That's when I think the way I proposed would be most useful.

"Formula of S in simple harmonic oscillation"

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