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# Sinusoidal oscillations

## Sinusoidal oscillations

The varying quantity in most oscillations follows a sinusoidal pattern with respect to time. This means that the physical quantity can be described by a sinusoidal function or graph.

The physical quantity increases from equilibrium at a high rate but then begins to level off at a maximum value. It then begins to decrease with a gradually increasing rate back to equilibrium.

The process is then repeated in a reversed manner, with the quantity decreasing to a minimum value and then returning to equilibrium.

## Characteristics of sinusoidal oscillations

Graph of a sinusoidal oscillation

The amplitude $x_{0}$ (or $A$) of an oscillation is the magnitude of the maximum displacement (from the equilibrium point) of the physical quantity.

The amplitude of a mass oscillating vertically on a spring is the distance from its highest point to its midpoint (or its lowest point to the midpoint) of oscillation. The midpoint is the equilibrium point.

The period $(T)$ is the time taken to complete one oscillation (i.e. one complete "back-and-forth" movement).

The frequency $(f)$ is the number of oscillations completed per unit of time. The SI unit of frequency is hertz $(\text{Hz})$. The frequency is related to the period by:$$f=\frac{1}{T}$$

This means that the period and the frequency are just two different ways of expressing the same thing. The period is the time for one oscillation and the frequency is the number of oscillations per unit time.

## Angular frequency

The angular frequency $(\omega)$ is the rate of change of the angle $(\theta)$ (the argument) of the sinusoidal function which describes an oscillation (recall that most oscillations can be described by sinusoidal functions).

$\omega$ is related to $\theta$ and the time $t$ by: $\theta=\omega t$. It is a scalar quantity.

The angle $\theta$ and the angular frequency $\omega$ have no physical significance (i.e. they cannot be physically measured). They are simply mathematical entities that are inserted into the sinusoidal function to describe how the physical quantity varies.

The change in the argument of the sinusoidal function after one complete oscillation is equivalent to $2\pi\text{ rad}$. The angular frequency is simply $2\pi$ divided by the time taken for one oscillation (i.e. the period).

The unit of angular frequency is radians per second $(\text{rad s}^{-1})$. It is given by:$$\omega=2\pi f=\frac{2\pi}{T}$$

The angular frequency is the the rate of change of the angle traced out by the circular plate.

## Relationship between angular frequency and angular velocity

The angular frequency is similar to the angular velocity in circular motion.

Continuous circular motion can be thought of as a type of oscillation because the object always returns to the starting point after each complete circle.

The magnitude of angular velocity and angular frequency are numerically equivalent in the case of circular motion (they even have the same symbol $\omega$).