Explain and draw the graphs of kinetic energy, potential energy, and mechanical energy as a function of time for a particle in Simple Harmonic Motion $(SHM)$.

(N/A) For a particle in $SHM$, the displacement is given by $x(t) = A \sin(\omega t + \phi)$.
The kinetic energy $K(t)$ is given by:
$K(t) = \frac{1}{2} m v^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \phi)$
The potential energy $U(t)$ is given by:
$U(t) = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \sin^2(\omega t + \phi)$
The total mechanical energy $E$ is:
$E = K(t) + U(t) = \frac{1}{2} k A^2 (\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)) = \frac{1}{2} k A^2$
Since $\sin^2 \theta + \cos^2 \theta = 1$, the total mechanical energy $E$ remains constant over time.

Time $(t)$	Kinetic Energy $(K)$	Potential Energy $(U)$	Total Energy $(E)$
$0$	$\frac{1}{2} k A^2$	$0$	$\frac{1}{2} k A^2$
$T/4$	$0$	$\frac{1}{2} k A^2$	$\frac{1}{2} k A^2$
$T/2$	$\frac{1}{2} k A^2$	$0$	$\frac{1}{2} k A^2$
$3T/4$	$0$	$\frac{1}{2} k A^2$	$\frac{1}{2} k A^2$
$T$	$\frac{1}{2} k A^2$	$0$	$\frac{1}{2} k A^2$