Q:

Interpreting the Negative Sign in Simple Harmonic Motion

What I Know:
$$ \vec F = -k \vec x $$
where the negative sign indicates the Force acts in the opposite direction to the displacement.
If we were to take the integral so...
$$\int_{x_i}^{x_f} Fdx = -\Delta U$$
What would the negative sign in this instance represent? From my understanding, we cannot produce negative energy...or can we? I have attached the image below for the context of my confusion. Thank you.

A:

See first of all Potential Energy arises when work is done against a conservative force.This means that:
Change in Potential Energy = - Work done against conservative force
$$ΔU = - W$$
Thats where the negative sign comes from

What would the negative sign in this instance represent?

This means that work done against conservative force (or its  negative   value ) is equal to change in Potential Energy.

From my understanding, we cannot produce negative energy...or can we?

What you are missing here is that we are not producing 'negative energy' but negative change in energy.
The change in Energy may be positive or negative or zero.
Notice how I have stressed on change.

P.S. If you still have any doubt,comment below.

