Displacement

The Meaning of Displacement in Oscillatory Motion

When we talk about displacement in the context of oscillations, we are not just talking about how far something has moved from where it started. In oscillatory motion, displacement has a very specific meaning: it is the distance of the oscillating particle from its mean position (or equilibrium position) at any given instant.

Think of a simple pendulum. The mean position is the point where the bob hangs vertically downwards, at rest. If you pull the bob to one side and release it, the displacement at any moment is the distance of the bob from that central, vertical line. This displacement can be positive (to the right of the mean position) or negative (to the left), depending on which side the bob is on.

The key idea is that displacement in an oscillation is always measured from the equilibrium point, not from the starting point of the motion. This is a crucial distinction from the displacement you might calculate in kinematics problems.

The Mathematical Description of Displacement

For the simplest kind of oscillation — simple harmonic motion (SHM) — the displacement $x(t)$ varies sinusoidally with time $t$. The most general way to write this is:

Here, each symbol has a precise physical meaning:

• $x(t)$ is the displacement of the particle from its mean position at time $t$.
• $A$ is the amplitude of the motion. It is the maximum magnitude of displacement. The particle oscillates between $x = +A$ and $x = -A$.
• $\omega$ is the angular frequency of the motion. It tells you how fast the oscillation is happening in terms of radians per second. It is related to the time period $T$ and frequency $f$ by $\omega = 2\pi f = \frac{2\pi}{T}$.
• $\phi$ is the initial phase (or phase constant). It determines the displacement of the particle at time $t = 0$. If $\phi = 0$, then $x(0) = A$, meaning the particle starts at its maximum positive displacement.

This single equation is the foundation for describing the position of any particle executing SHM.

Why Use a Cosine Function?

You might wonder why we use a cosine and not a sine. The answer is that both are valid, and the choice simply affects the value of the phase constant $\phi$. A sine function is just a cosine function shifted in time:

So, if you prefer to write $x(t) = A \sin(\omega t + \phi')$, you can, and it will describe the same physical motion, just with a different phase constant. The textbook uses the cosine form as its standard representation, and we will stick with that for consistency.

The Four Key Quantities of SHM

From the displacement equation $x(t) = A \cos(\omega t + \phi)$, we can derive everything else about the motion. The textbook lists four fundamental quantities that completely define a simple harmonic motion:

1. Amplitude ($A$): The maximum displacement from the mean position. It is always a positive constant.
2. Angular Frequency ($\omega$): A measure of how rapidly the oscillations occur. It is a positive constant for a given system.
3. Phase ($\omega t + \phi$): The argument of the cosine function. It tells you the current "state" of the oscillation — where the particle is in its cycle.
4. Initial Phase ($\phi$): The phase at time $t = 0$. It sets the initial conditions of the motion.

Properties of the Displacement Function

The textbook then examines the mathematical properties of the cosine function to understand the physical behaviour of the oscillating particle. Let's go through them one by one.

›Proof

Property (I): The motion is periodic.

A function $f(t)$ is periodic with period $T$ if $f(t+T) = f(t)$ for all $t$. For our displacement function $x(t) = A \cos(\omega t + \phi)$, we need to find a $T$ such that:

$A \cos(\omega (t+T) + \phi) = A \cos(\omega t + \phi)$

We know that the cosine function repeats its value when its argument increases by $2\pi$:

$\cos(\theta + 2\pi) = \cos(\theta)$

Therefore, we require:

$\omega (t+T) + \phi = \omega t + \phi + 2\pi$

Simplifying:

$\omega t + \omega T + \phi = \omega t + \phi + 2\pi$

Cancelling $\omega t$ and $\phi$ from both sides gives:

$\omega T = 2\pi$

So the time period $T$ is:

$T = \frac{2\pi}{\omega}$

This proves that $x(t)$ is a periodic function with a period $T = 2\pi/\omega$. The particle repeats its motion after every interval $T$.

›Proof

Property (II): The motion is oscillatory about the mean position.

An oscillatory motion is one in which the particle moves back and forth about a fixed point (the mean position). The displacement function $x(t) = A \cos(\omega t + \phi)$ clearly shows this. The cosine function oscillates between $-1$ and $+1$. Therefore, $x(t)$ oscillates between $-A$ and $+A$. The value $x=0$ (the mean position) is crossed twice in every cycle. This back-and-forth motion about $x=0$ is the very definition of oscillation.

›Proof

Property (III): The amplitude $A$ is the maximum displacement.

The maximum value of $\cos(\omega t + \phi)$ is $+1$. Therefore, the maximum value of $x(t)$ is:

$x_{\text{max}} = A \times (+1) = A$

The minimum value of $\cos(\omega t + \phi)$ is $-1$. Therefore, the minimum value of $x(t)$ is:

$x_{\text{min}} = A \times (-1) = -A$

The magnitude of the maximum displacement from the mean position is $|A|$. Since amplitude is defined as this maximum magnitude, it is $A$ (taking $A$ to be positive).

›Proof

Property (IV): The phase $(\omega t + \phi)$ determines the state of motion.

…