Question

What is simple harmonic motion equation?

Answer 1

Step1: Assumptions for the simple harmonic motion equation

Consider a particle of mass $\left(\mathrm{m}\right)$ in SHM moving along a path as shown in the figure, with the mean position at $\mathrm{O}$. When the particle is at position $\mathrm{P}$, let its speed be ${\mathrm{V}}_0$ (at a distance ${\mathrm{X}}_0$ from $\mathrm{O}$).

The particle at $\mathrm{P}$ at $\mathrm{t}=0$ (moving towards the right).

At $\mathrm{t}=\mathrm{t}$.the particle is at $\mathrm{Q}$ with velocity $\mathrm{V}$.

Step2: Finding the equation of SHM,

The restoring force at $\mathrm{Q}$ is given by,

$\Rightarrow$$\overrightarrow{F}=-k\overrightarrow{x}$,

where,$\overrightarrow{F}$ is the force, $k$ is a positive constant.

The force at $\mathrm{Q}$ is also equal to,

$\Rightarrow$$\overrightarrow{F}=m\overrightarrow{a}$,
where,$m$ is the mass of the body, $\overrightarrow{a}$ is the acceleration at $\mathrm{Q}$.

We can equate the two equations to get,

$\Rightarrow$$m\overrightarrow{a}=-k\overrightarrow{x}$

$\Rightarrow$$\overrightarrow{a}=-\frac{k}{m}\overrightarrow{x}$

Now assume $\frac{k}{m}={\omega }^2$,

$\Rightarrow$$\overrightarrow{a}=-{\omega }^2\overrightarrow{x}$, we know that acceleration is the second derivative of displacement $\overrightarrow{a}=\frac{d^2\overrightarrow{x}}{d{t}^2}$.

$\Rightarrow$$\frac{d^2\overrightarrow{x}}{d{t}^2}=-{\omega }^2\overrightarrow{x}$

Final Answer

$\frac{d^2\overrightarrow{x}}{d{t}^2}+{\omega }^2\overrightarrow{x}=0$, is the differential equation for linear simple harmonic motion.