Question

What are the characteristics of the simple harmonic motion on a spring?

Answer 1

The acceleration is proportional to the displacement (x) of the mass and it acts toward the equilibrium position.

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The defining equation for any type of SHM is:
`x = – ω² a`
where `ω = 2πf`
The negative indicates that the acceleration always acts toward the equilibrium.

In the plane of the oscillation for a horizontal mass–spring system the only (significant) force is the tension in the spring.
From Hooke’s Law:
`T = kx`
Also the mass is not changing so we can apply the simplified form of Newton’s second law of motion to this example:
`T = ma`
Therefore, `ma = kx ⇒ a = (k/m)x`.

As you can see that satisfies the first condition of SHM as `k/m` is a constant, therefore `a prop x`. That equation only tells us about the magnitude of the acceleration. If we were to draw a diagram we would see in addition that the force and therefore the acceleration acts toward the equilibrium (the second condition of SHM).