Obtain the force law for $SHM$ from the displacement of $SHM$ particle.

(D) Displacement of $SHM$ particle at time $t$ is given by:
$x(t) = A \cos (\omega t + \phi)$
where $A$ is the amplitude,$\omega$ is the angular frequency,and $\phi$ is the initial phase.
Differentiating the displacement equation with respect to time $t$ to find velocity:
$v(t) = \frac{d}{dt} [A \cos (\omega t + \phi)] = -A\omega \sin (\omega t + \phi)$
Differentiating the velocity equation with respect to time $t$ to find acceleration:
$a(t) = \frac{d}{dt} [-A\omega \sin (\omega t + \phi)] = -A\omega^2 \cos (\omega t + \phi)$
Since $x(t) = A \cos (\omega t + \phi)$,we can substitute this into the acceleration equation:
$a(t) = -\omega^2 x(t)$
According to Newton's second law,$F = ma$. Multiplying both sides by mass $m$:
$F = m a(t) = -m\omega^2 x(t)$
Defining the force constant $k = m\omega^2$,we get:
$F = -kx(t)$
This shows that the restoring force is directly proportional to the negative displacement,which is the force law for $SHM$.