State the formula for period of sham in terms of acceleration and displacement ​

Finding time period of SHM from equation of displacement

Explanation:

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Say for example I've got the equation of a SHM as:

x=Acos(ωt+ϕ)

where A is the amplitude.

How do I find the time period of this motion?

I tried by finding the second order differential of the given equation.

a=d2xdt2=−Aω2cos(ωt+ϕ)

Comparing it with the general equation for acceleration a=−ω2x, we can find ω from here.

But that is where the problem is coming. It makes no sense if I write ω=ωA−−√.

What is the correct method to find the time period of the SHM? What am I missing?

There is a very simple mistake in your math. Notice A is part of x, it is factored so you'll get to ω=ω again. If you want to find a meaning to ωT=2π, consider the fact that cos (or sin) are periodic functions with period 2π. Hence, every time you have a time difference such that ω(t1−t2)=2π you are back at the same point. Hence the period is given by ωT=2π.