Oscillatory Motion

The periodic motion in which particle oscillates about mean position is known as oscillatory motion. It can repeat at fixed intervals or not.

For example the motion of a simple pendulum, motion of a loaded spring, etc.
Note: Every oscillatory motion is periodic motion but every periodic motion is not oscillatory motion.

Periodic Motion

A motion that repeats itself identically after a fixed interval of time is called periodic motion. e.g., the orbital motion of the earth around the sun, the motion of arms of a clock, motion of a simple pendulum, etc.

Harmonic Oscillation

The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function, is called harmonic oscillation.

Simple Harmonic Motion

Harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean Position is called Simple Harmonic Motion (SHM).

A simple harmonic oscillation can be expressed as
y = a sin ωt
or y = a cos ωt
Where a = amplitude of oscillation.

Characteristics of SHM are:

If a body is executing SHM, then
i) Its acceleration is directly proportional to displacement i.e. a ∝∝ y
ii) Its acceleration is always directed towards mean position i.e. a = -ky
iii) It is a periodic motion

Conditions for SHM:

Equation

The expression for displacement, velocity, acceleration and time period of simple harmonic motion

Equation

Displacement of the body undergoing simple harmonic motion is given by:

Equation

where,
• X is the displacement of the body at any instant from the equilibrium position.
• A is the maximum displacement of the body from the equilibrium position. It is also known as Amplitude of Simple Harmonic Motion.
• ω is the Angular frequency of the body executing the simple harmonic motion.
• ϕ is the phase constant of the oscillation.
Velocity is the first-order time derivative of the displacement of the body.
Therefore, velocity is given by:

Equation

Now, acceleration (a) is the second-order time derivative of the displacement of the body.
Therefore,

Equation

Here, Negative sign indicates that the direction of acceleration is opposite to the direction in which displacement increases, i.e., towards mean position.
At mean position y = 0 and acceleration is also zero.
At extreme position y = a and acceleration is maximum
Amax = – aω2

The time period in SHM is given by
T = 2π √Displacement / Acceleration

Graphical Representation:

(i) Displacement – Time Graph

Picture

(ii) Velocity – Time Graph

Picture

(iii) Acceleration – Time Graph

Picture