Mathematical Form of SHM

Understand the basic SHM equation and how sine and cosine functions describe oscillatory motion.

1. Why SHM Has a Mathematical Form

Simple Harmonic Motion is predictable and smooth, so it can be described beautifully using sine and cosine functions. These functions naturally repeat, just like oscillations do. Having a mathematical form makes it easy to find displacement, velocity, and acceleration at any instant.

I like to think of the SHM equations as a compact way of capturing the entire motion — every rise, fall, turning point, and return is encoded in a single expression.

2. General Equation of SHM

The displacement of an object performing SHM can be written using either a sine or cosine function. The most commonly used form is:

\( x(t) = A \sin(\omega t + \phi) \)

Here:

  • A is the amplitude (maximum displacement)
  • \( \omega \) is the angular frequency
  • t is the time
  • \( \phi \) is the phase constant, which adjusts the starting point of the motion

2.1. Why Sine and Cosine?

Sine and cosine naturally repeat and have smooth turning points, making them perfect for representing oscillations. Depending on where the object starts, the motion may be easier to describe using sine or cosine, but both represent the same type of motion.

2.2. Meaning of the Phase Constant

The phase constant \( \phi \) tells us the state of the motion at time \( t = 0 \). It adjusts the starting position. For example:

  • If the motion starts from the mean position and moves upward, \( \phi = 0 \).
  • If the motion begins at an extreme position, \( \phi = \dfrac{\pi}{2} \).

This makes the function flexible enough to represent any initial condition.

3. Angular Frequency and Its Meaning

Definition: Angular frequency \( \omega \) tells how rapidly the object oscillates and is related to the time period.

It is connected to the time period \(T\) and frequency \(f\) by:

\( \omega = 2\pi f = \dfrac{2\pi}{T} \)

A larger \( \omega \) means faster oscillations.

3.1. Visual Way to Understand \( \omega \)

I like to imagine \( \omega \) as the ‘speed of cycling’ around a circle. If the object cycles through the oscillation faster, the angular frequency is higher. That’s why \( \omega \) is linked to how many radians of oscillation take place per second.

4. Velocity as a Function of Time

Velocity in SHM is obtained by differentiating the displacement equation. If:

\( x(t) = A \sin(\omega t + \phi) \)

then velocity is:

\( v(t) = A\omega \cos(\omega t + \phi) \)

Velocity is highest at the mean position because the restoring force has accelerated the object the most there.

4.1. Phase Shift Between Displacement and Velocity

Velocity leads displacement by 90°. This means the velocity graph reaches its peak one-quarter cycle before the displacement graph does. This phase shift is a natural result of taking the derivative of a sine function.

5. Acceleration as a Function of Time

Acceleration comes from differentiating the velocity function:

\( a(t) = -A\omega^2 \sin(\omega t + \phi) \)

The negative sign shows that acceleration is always directed opposite to displacement — this is the very reason the motion stays oscillatory.

6. Graphical View of SHM Equations

Displacement, velocity, and acceleration each have their own waveform, and all three waveforms are sinusoidal but shifted relative to one another:

  • Displacement: sine wave
  • Velocity: cosine wave (90° ahead of displacement)
  • Acceleration: inverted sine wave (180° phase shift)

Looking at the graphs helps visualize when the object moves fastest, slows down, or changes direction.

7. Why the Mathematical Form is Useful

With one simple equation, we can describe every instant of the oscillation. The mathematical form makes it easy to calculate energy, understand resonance, study waves, and analyze electronic circuits later. Once the equation is clear, SHM becomes one of the most comfortable and predictable motions to work with.