## Oscillations and Waves

# Simple Harmonic Motion

- A body is said to be in SHM if it moves to and fro along a straight line about its mean position such that at any point its acceleration is directly proportional to its displacement in magnitude but opposite in direction and is always directed towards the mean position.
- Phase represents the state of vibration of the particle at any instant of time.
- The difference in the phase angles of two particles in SHM is known as the phase difference between them.
- The starting phase of oscillation is called epoch. If phase is zero y = A sin wt. If phase = \tt \frac{\pi}{2} y = A sin (wt + \tt \frac{\pi}{2})
- Velocity of SHM is given by \tt v = w \sqrt{A^{2} - y^{2}} where w is called angular velocity A = Amplitude and y = displacement of particle.

### View the Topic in this video From 0:21 To 05:23

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1. Simple Harmonic Motion (SHM) is that type of oscillatory motion in which the particle moves to and fro about a fixed point. Such that acceleration is proportional to negative of displacement.

The general equation of simple harmonic motion is given by *x*(*t*) = *A* cos (ω*t* + Φ)

2. The angular frequency ω is related to the period and frequency of the motion by

\omega = \frac{2 \pi}{T} = 2 \pi \nu

3. The particle velocity and acceleration during SHM as functions of time are given by

\tt v(t) = \frac{d}{dt}x(t) = - \omega A \sin (\omega t + \phi)

\tt a(t) = \frac{d}{dt}v(t) = - \omega^{2} A \cos (\omega t + \phi) = -\omega^{2} x(t)