Write the force law for $SHM$ and obtain the formula for the period of an $SHM$ particle.

(N/A) The acceleration of an $SHM$ particle is given by:
$a(t) = -\omega^{2} x(t)$
where $x(t)$ is the displacement at time $t$.
According to Newton's second law of motion,the force $F$ exerted on the particle is:
$F = m a(t)$
Substituting the expression for acceleration:
$F = -m \omega^{2} x(t) \quad (1)$
In $SHM$,the restoring force is directly proportional to the displacement and directed towards the mean position:
$F = -k x(t) \quad (2)$
where $k$ is the force constant.
Comparing equations $(1)$ and $(2)$:
$k = m \omega^{2}$
$\omega = \sqrt{\frac{k}{m}}$
Since the angular frequency $\omega = \frac{2 \pi}{T}$,where $T$ is the time period:
$\frac{2 \pi}{T} = \sqrt{\frac{k}{m}}$
$T = 2 \pi \sqrt{\frac{m}{k}}$