Overview: Oscillations

Any motion which repeats itself after a definite interval of time is called periodic motion.

A body performing periodic motion goes on repeating the same set of movements. The time taken for one such set of movements is called its period or periodic time.

Simple harmonic motion is an oscillatory motion in which the restoring force (or acceleration) is directly proportional to the displacement from the equilibrium position and is directed towards it.

Another type of periodic motion in which a particle repeatedly moves to and fro along the same path is the oscillatory or vibratory motion.

The smallest interval of time after which the to and fro motion is repeated is called its period (T).

The number of oscillations completed per unit time is called the frequency (n) of the periodic motion.

Linear S.HM. is defined as the linear periodic motion of a body, in which force (or acceleration) is always directed towards the mean position and its magnitude is proportional to the displacement from the mean position.

\[\frac{d^2x}{dt^2}+\omega^2x=0\] ⇒ x = ±A cos(ωt)

The maximum displacement of a particle performing S.H.M from its mean position is called the amplitude of S.H.M.

The time taken by the particle performing S.H.M. to complete one oscillation is called the period of S.H.M.

Mathematically,

T = 2π\[\sqrt{\frac{m}{k}}\]

The number of oscillations performed by a particle performing S.H.M. per unit time is called the frequency of S.H.M.

Mathematically,
n = \[\frac {1}{T}\] = \[\frac {ω}{2π}\] = \[\frac{1}{2\pi}\sqrt{\frac{k}{m}}\]

Phase in S.H.M. is the angular quantity θ = (ωt + ϕ) which specifies the state of oscillation of a particle at a given instant.

• In S.H.M., displacement, velocity, and acceleration change with time in sine or cosine form.
• Velocity is \[\frac {π}{2}\] ahead of displacement, and acceleration is opposite to displacement.
• Starting from the mean position → sine curve; starting from the extreme position → cosine curve.

• Total mechanical energy in S.H.M. is constant and equals
  E = \[\frac {1}{2}\]kA².
• At the mean position, K.E. is maximum, and P.E. is zero.
  At extreme position: P.E. is maximum, and K.E. is zero.
• During motion, energy continuously converts between kinetic and potential, but total energy remains conserved.

An ideal simple pendulum consists of a point mass suspended by a massless, inextensible, and perfectly flexible string from a fixed rigid support.

The distance between the point of suspension and the centre of gravity of the bob is called the length of the pendulum.

• For a small angular displacement, the restoring force of a simple pendulum is
  F = −\[\frac {mg}{L}\]x,
  hence the motion is simple harmonic.
• The time period of a simple pendulum is
  T = \[2\pi\sqrt{\frac{L}{g}},\]
  which is valid only for small oscillations.
• The period is directly proportional to \[\sqrt {L}\], inversely proportional to \[\sqrt{g}\], and is independent of mass and amplitude (for small angles).

A simple pendulum whose period is two seconds is called a second's pendulum. 

Angular S.H.M is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement. 

T = \[\frac {2π}{ω}\] = \[\frac {2\pi}{\sqrt{\text {angular acceleration per unit angular displacement}}}\]

T = \[2\pi\sqrt{\frac{I}{\mu B}}\]

Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations, and the oscillator is called a damped harmonic oscillator.