Question

What is the equation of a simple harmonic motion?

Answer 1

Simple harmonic motion:

1. Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position.
2. The direction of this restoring force is always towards the mean position.
3. The general expression for the wave is given as,

$y=A\sin \left(\omega t+\theta \right)$, where $A$ is the amplitude of the wave, $\omega$ is the angular velocity, $t$ is the time period, and $\theta$ is the phase angle.

Velocity of the particle in SHM:

1. The velocity of a particle executing simple harmonic motion at any instant is defined as the rate of change of its displacement at that instant.
2. Mathematically, it can be expressed as,

$$\begin{gathered} v=\frac{dy}{dt} \\ v=\frac{d\left(A\sin \left(\omega t+\theta \right)\right)}{dt} \\ v=A\omega \cos \left(\omega t+\theta \right) \end{gathered}$$

Acceleration of the particle in SHM:

1. The acceleration of a particle executing simple harmonic motion at any instant is defined as the rate of change of its velocity at that instant.
2. Mathematically, it can be expressed as,

$$\begin{gathered} a=\frac{dv}{dt} \\ a=\frac{d\left(A\omega \cos \left(\omega t+\theta \right)\right)}{dt} \\ a=-A{\omega }^2\sin \left(\omega t+\theta \right) \end{gathered}$$