Define simple harmonic motion. Obtain an expression for the velocity and acceleration of SHM.

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Accepted Answer

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> [!TLDR]
> In SHM, $v=\omega\sqrt{A^2-x^2}$ and $a=-\omega^2x$ — acceleration is always proportional to displacement, directed toward the mean position.
>
> **Definition:** Simple harmonic motion (SHM) is a type of periodic motion in which the restoring force (and hence the acceleration) acting on the particle is directly proportional to its displacement from a fixed mean (equilibrium) position, and is always directed towards that mean position:
> $$F = -kx \quad	ext{or equivalently}\quad a = -\omega^2 x$$
> where $\omega$ is the angular frequency, related to $k$ and mass $m$ by $\omega=\sqrt{k/m}$.
>
> **Displacement:** The general solution for the displacement of a particle in SHM is
> $$x(t) = A\sin(\omega t + \phi)$$
> where $A$ is the amplitude and $\phi$ is the initial phase.
>
> **Velocity:** Differentiating $x(t)$ with respect to time,
> $$v = \frac{dx}{dt} = A\omega\cos(\omega t+\phi)$$
> Using $\cos^2	heta = 1-\sin^2	heta$ and $\sin(\omega t+\phi) = x/A$:
> $$v = A\omega\sqrt{1-\sin^2(\omega t+\phi)} = \omega\sqrt{A^2 - x^2}$$ …

Q.Define simple harmonic motion. Obtain an expression for the velocity and acceleration of SHM.

Concept understanding — Simple Harmonic Motion

Simple Harmonic Motion: The Natural Rhythm of Things

The Intuition: A Restoring Force That Fights Displacement

The Precise Statement

What Does This Motion Look Like?

The Mathematical Description (Derived from $F = -kx$ )

Q.Define simple harmonic motion. Obtain an expression for the velocity and acceleration of SHM.

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