SHM: Equation relating acceleration and displacement

[andyrk]

A particle moves such that its acceleration is given by: a = -β(x-2). Here β is a positive constant and c is the distance from origin What is the time period of oscillation for the particle?

Solution: a = 0 at x = 2 (mean position)
a = -βX where X = x-2.
So, ω² = β ⇒ T = 2π/ω = 2π/√β

My question is, why do we need to substitute x-2 as X? Can't we solve the problem without doing this?

 

[BruceW]

you can do the whole problem without the substitution. I guess that using the substitution means that you can immediately recognise that the problem is SHM.

 

[andyrk]

So then, would we call x-2 as the distance from the mean position or x?

 

[andyrk]

There's not much depth involved in this. I am overthinking things.

 

[BruceW]

yeah :) x-2 is the displacement from the equilibrium position, which makes it a convenient choice to use as coordinate, instead of using x.