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From the formulas I am seeing, these are two different physical concepts. Is oscillation the motion backwards and forwards motion, such as a spring moving backwards and forwards? And then, if a mass is attached to the spring, that would be simple harmonic motion? Am I correct in thinking this? If not, what is the difference between simple harmonic motion and oscillation.

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Simple harmonic oscillation is motion according to a sinusoidal wave.

$$ x(t) = A \sin (\omega t + B) $$

It arises from a Hooke's law-like restorative force.

$$ F(x) = - k x$$

This is because, given that force law, the acceleration can be written as so.

$$ a(x) = \frac{d^2x}{dt^2} = \frac{F(x)}{m} = - \frac{k}{m} x$$

And this has the general solution as given with $\omega = \sqrt{k/m}$.

Oscillation is any periodic motion, or any nearly period motion. For example, if you take a ruler, clamp one end to a table with the other end hanging over the table, and give it a flick. It will begin to vibrate and gradually the vibrations will decay. This is oscillation with a damping.

Or if you take a pendulum and push it very far from vertical and let it go, it won't be SHO because at the far ends of the oscillation the restoring force is not very close to Hooke's law.

Or if you take two pendulums and make some connection between them so they can exchange energy. The oscillations will be very complicated as the energy moves back and forth between them.

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From the formulas I am seeing, these are two different physical concepts. Is oscillation the motion backwards and forwards motion, such as a spring moving backwards and forwards?

Yes, that is a fine understanding of the meaning of the word "oscillation," which just has its ordinary English language meaning.

And then, if a mass is attached to the spring, that would be simple harmonic motion?

Basically, yes, with a minor caveat; the spring has to be "perfect"/"ideal" in the sense that the force must always be proportional to the stretch.

In more detail, "Simple harmonic motion" has a technical meaning. It means that Newton's equation of motion for the mass looks like: $$ -kx = m\ddot x\;, $$ which says that the force $F=-kx$ is proportional (with proportionality constant $-k$) to the "stretch" (denoted here by $x$, which indicates how much the spring is stretched away from its equilibrium position).

Am I correct in thinking this? If not, what is the difference between simple harmonic motion and oscillation.

Basically, yes. With the above caveat that the spring has to be ideal. Also, there are other way to have Simple Harmonic motion, since there are other ways that the force can be proportional to the stretch.

Furthermore, I should note that Simple Harmonic Motion is often an approximation to some other, more complex, form of motion. For example, the oscillation of a pendulum is approximately simple-harmonic for small-angle oscillations. But, for large-angle oscillations the motion fails to be simple harmonics.

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