Simple Harmonic Motion
9.0 Basic Differential Equation of SHM
9.0 Basic Differential Equation of SHM
The force required to produce SHM is of nature, $$F=-kx$$
From the equation of motion $F\ =\ ma$ and remembering that in rectilinear motion $$a = \frac{{{d^2}x}}{{d{t^2}}}$$ we may write the above equation as,
$$m\frac{{{d^2}x}}{{d{t^2}}} = - kx\ \ \ or\ \ \ m\frac{{{d^2}x}}{{d{t^2}}} + kx = 0$$ Setting, $$\begin{equation} \begin{aligned} {\omega ^2} = \frac{k}{m},{\text{ we have}} \\ \frac{{{d^2}x}}{{d{t^2}}} + {\omega ^2}x = 0 \\\end{aligned} \end{equation} $$
This is an equation which relates displacement with acceleration. The solution of the equation is cosine or sine functions $\omega t$. For instance, substituting $x = Acos\left( {\omega t + \phi } \right)$ and its second time derivative in the above equation, the equation is satisfied, regardless of the values of $A$ and $\phi$. Thus, $$x = Acos\left( {\omega t + \phi } \right)$$ is a general solution of differential equation of SHM. Similarly, by the same process, we can see that $$x = Acos\left( {\omega t + \phi } \right)\ \ and\ \ x = Asin\omega t + Bcos\omega t$$ are also general solution of differential equation of SHM.
