Question

A simple harmonic oscillator has a time period of 10 seconds. Which equation relates its acceleration ‘a’ with displacement ‘x’?

Answer 1

Explanation:

The differential equation of linear S.H.M. is d2x/dt2 + (k/m)x = 0 where d2x/dt2 is the acceleration of the particle, x is the displacement of the particle, m is the mass of the particle and k is the force constant. We know that k/m = ω2 where ω is the angular frequency.

Answer 2

Equation relating Acceleration to Displacement in SHM:

We know that the basic criteria of an SHM is:

$F = - kx$

$\implies \: m \times a = - kx$

$\implies \: m \times \dfrac{dv}{dt} = - kx$

$\implies \: m \times \dfrac{ {d}^{2} x}{d {t}^{2} } = - kx$

$\boxed{ \implies \: \dfrac{ {d}^{2} x}{d {t}^{2} } + \dfrac{kx}{m} = 0}$

Or , we can also say :

$\boxed{a = - { \omega}^{2} x}$

Hence, both these equations can relate

Acceleration and Displacement.