17.1 Oscillations | Simple Harmonic Motion | Physics 12

What is Oscillatory Motion?

Oscillatory (vibratory) motion is the to-and-fro motion of a body about a fixed mean (equilibrium) position. The body repeatedly moves back and forth, passing through the mean position. Examples include:

A simple pendulum swinging
A mass attached to a spring
A vibrating guitar string
The prongs of a tuning fork

Key Terms

Displacement ( $x$ )

The instantaneous distance of the oscillating body from its mean (equilibrium) position. It is a vector quantity, measured in metres (m). Displacement can be positive or negative depending on the direction from the mean position.

Amplitude ( $x_{0}$ or $A$ )

The maximum displacement of the body from its mean position. It is always positive and measured in metres (m). The amplitude determines the energy of the oscillation.

Time Period ( $T$ )

The time taken for one complete oscillation (one full back-and-forth cycle). It is measured in seconds (s).

$T = \frac{1}{f}$

Frequency ( $f$ )

The number of complete oscillations per second. It is measured in Hertz (Hz), where 1 Hz = 1 oscillation per second.

$f = \frac{1}{T}$

Angular Frequency ( $ω$ )

Also called angular velocity, it relates frequency to the angle swept per unit time:

$ω = 2 π f = \frac{2 π}{T}$

Angular frequency is measured in radians per second (rad/s).

Phase ( $θ$ )

The phase of an oscillating body at any instant $t$ is the angle:

$θ = ω t$

Phase specifies both the displacement and the direction of motion of the oscillating body at that instant. It describes the complete state of the oscillator. Phase is measured in radians.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a special type of oscillatory motion in which the acceleration of the body is directly proportional to its displacement from the mean position and is always directed towards the mean position:

$a \propto - x ⟹ a = - ω^{2} x$

The negative sign indicates that acceleration always acts opposite to the direction of displacement — i.e., it is always a restoring acceleration directed towards the equilibrium position.

Condition for SHM

A body executes SHM if and only if:

Its acceleration is proportional to its displacement from the mean position.
Its acceleration is always directed towards the mean position.

Velocity and Acceleration in SHM

Position	Displacement	Velocity	Acceleration
Mean position	$x = 0$	Maximum: $v_{ma x} = ω x_{0}$	Zero
Extreme position	$x = \pm x_{0}$	Zero	Maximum: $a_{ma x} = ω^{2} x_{0}$

At the mean position, $a = - ω^{2} (0) = 0$ and velocity is maximum. At the extreme positions, velocity is zero and acceleration is maximum.

Time Period of SHM

Mass-Spring System

For a mass $m$ attached to a spring of spring constant $k$ , the restoring force is $F = - k x$ . Applying Newton's second law:

$ma = - k x \Rightarrow a = - \frac{k}{m} x$

Comparing with $a = - ω^{2} x$ , we get $ω^{2} = \frac{k}{m}$ , so:

$T = 2 π \frac{m}{k}$

Increasing mass $m$ increases the time period.
Increasing spring constant $k$ (stiffer spring) decreases the time period.
The time period is independent of amplitude (for ideal SHM).

What is Oscillatory Motion?

A simple pendulum swinging
A mass attached to a spring
A vibrating guitar string
The prongs of a tuning fork

Key Terms

Displacement ( $x$ )

Amplitude ( $x_{0}$ or $A$ )

The maximum displacement of the body from its mean position. It is always positive and measured in metres (m). The amplitude determines the energy of the oscillation.

Time Period ( $T$ )

The time taken for one complete oscillation (one full back-and-forth cycle). It is measured in seconds (s).

$T = \frac{1}{f}$

Frequency ( $f$ )

The number of complete oscillations per second. It is measured in Hertz (Hz), where 1 Hz = 1 oscillation per second.

$f = \frac{1}{T}$

Angular Frequency ( $ω$ )

Also called angular velocity, it relates frequency to the angle swept per unit time:

$ω = 2 π f = \frac{2 π}{T}$

Angular frequency is measured in radians per second (rad/s).

Phase ( $θ$ )

The phase of an oscillating body at any instant $t$ is the angle:

$θ = ω t$

Phase specifies both the displacement and the direction of motion of the oscillating body at that instant. It describes the complete state of the oscillator. Phase is measured in radians.

Simple Harmonic Motion (SHM)

$a \propto - x ⟹ a = - ω^{2} x$

The negative sign indicates that acceleration always acts opposite to the direction of displacement — i.e., it is always a restoring acceleration directed towards the equilibrium position.

Condition for SHM

A body executes SHM if and only if:

Its acceleration is proportional to its displacement from the mean position.
Its acceleration is always directed towards the mean position.

Velocity and Acceleration in SHM

Position	Displacement	Velocity	Acceleration
Mean position	$x = 0$	Maximum: $v_{ma x} = ω x_{0}$	Zero
Extreme position	$x = \pm x_{0}$	Zero	Maximum: $a_{ma x} = ω^{2} x_{0}$

At the mean position, $a = - ω^{2} (0) = 0$ and velocity is maximum. At the extreme positions, velocity is zero and acceleration is maximum.

Time Period of SHM

Mass-Spring System

For a mass $m$ attached to a spring of spring constant $k$ , the restoring force is $F = - k x$ . Applying Newton's second law:

$ma = - k x \Rightarrow a = - \frac{k}{m} x$

Comparing with $a = - ω^{2} x$ , we get $ω^{2} = \frac{k}{m}$ , so:

$T = 2 π \frac{m}{k}$

Increasing mass $m$ increases the time period.
Increasing spring constant $k$ (stiffer spring) decreases the time period.
The time period is independent of amplitude (for ideal SHM).