On which of the following does the energy of a simple harmonic motion depend

The correct option is a².

To understand why, let's look at the expression for the total energy of a particle executing Simple Harmonic Motion (SHM).

The total energy ($E$) of a system in simple harmonic motion is the sum of its kinetic energy ($K$) and potential energy ($U$) at any point in its path. At any displacement $x$ from the mean position, these are given by:

Potential Energy:
$U=\frac{1}{2}k{x}^2$

Kinetic Energy:
$K=\frac{1}{2}k(a^2-x^2)$
where $k$ is the spring constant (force constant) and $a$ is the amplitude of the motion.

The total mechanical energy ($E$) of the system is:
$E=K+U=\frac{1}{2}k(a^2-x^2)+\frac{1}{2}k{x}^2$

Simplifying the equation, we get:
$E=\frac{1}{2}k{a}^2$

Since the spring constant can also be written in terms of mass ($m$) and angular frequency ($\omega$) as $k=m{\omega }^2$, the total energy expression becomes:
$E=\frac{1}{2}m{\omega }^2{a}^2$

From this final relation, it is clear that the total energy of a particle in simple harmonic motion is directly proportional to the square of its amplitude ($a^2$).

Therefore, the energy of simple harmonic motion depends on a².