Oscillations - Simple Harmonic Motion (SHM)

Periodic Motion: A motion that repeats itself at regular intervals of time. Oscillatory motion is a specific type of periodic motion where the body moves to and fro about a fixed mean position.

Simple Harmonic Motion (SHM): A type of oscillation where the restoring force is directly proportional to the displacement $x$ from the mean position and acts in the opposite direction: $F = -kx$.

Displacement in SHM: The position of a particle at any time $t$ is given by $x(t) = A \cos(\omega t + \phi)$ or $x(t) = A \sin(\omega t + \phi)$, where $A$ is the amplitude and $\omega$ is the angular frequency.

Phase and Phase Constant: The term $(\omega t + \phi)$ is the phase, representing the state of motion. $\phi$ is the phase constant or initial phase at $t = 0$.

Velocity in SHM: Velocity is the rate of change of displacement, given by $v(t) = -A\omega \sin(\omega t + \phi)$. In terms of displacement: $v = \pm \omega \sqrt{A^2 - x^2}$. Velocity is maximum ($A\omega$) at the mean position and zero at the extremes.

Acceleration in SHM: Acceleration is $a(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x$. Acceleration is always directed towards the mean position and is maximum at the extreme positions.

Energy in SHM: The total mechanical energy is the sum of Kinetic Energy $K = \frac{1}{2} m v^2 = \frac{1}{2} k (A^2 - x^2)$ and Potential Energy $U = \frac{1}{2} k x^2$. The total energy $E = \frac{1}{2} k A^2$ remains constant.

Simple Pendulum: For small angular displacements, a simple pendulum executes SHM with a time period $T = 2\pi \sqrt{\frac{l}{g}}$, where $l$ is the length and $g$ is the acceleration due to gravity.