# Periodic Motion

A motion that repeats itself at regular intervals of time is called periodic motion.

**Note** that both the curved parts in Fig (c) are sections of a parabola given by Newton’s equation of motion with different values of *u *in each case. These are examples of periodic motion.

*Fig. Examples of periodic motion. The period T is shown in each case.*

## Oscillations

If the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, giving rise to oscillations or vibrations.

**For example: –** A ball placed in a bowl will be in equilibrium at the bottom. If displaced a little from the point, it will perform oscillations in the bowl. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Circular motion is a periodic motion, but it is not oscillatory.

There is **no significant difference between oscillations and vibrations**.

- When the frequency is
**small**, we call it**oscillation**(like the oscillation of a branch of a tree) - When the frequency is
**high**, we call it**vibration**(like the vibration of a string of a musical instrument).

## Period and frequency

### Period:

A motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval of time after which the motion is repeated is called its **period.**

Let us denote the period by the symbol **T**.

Its **SI unit is second.**

The period of vibrations of a quartz crystal is expressed in units of microseconds (10^{-6} s) abbreviated as μs.

### Frequency:

The reciprocal of *T *gives the number of repetitions that occur per unit time. This quantity is called the **frequency of the periodic motion**. It is represented by the symbol ν . The relation between *v *and *T *is

The **unit of ν is thus** **s ^{-1}**. It is called hertz (abbreviated as Hz). Thus,

1 hertz = 1Hz = 1 oscillation per second = 1s^{-1}

### Displacement

An oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable [see Fig.(b)]. in the context of position only.

There can be many other kinds of displacement variables. The voltage across a capacitor, changing with time in an a.c. circuit is also a displacement variable.

The displacement can be represented by a **mathematical function of time**. In case of periodic motion, this function is periodic in time.

- One of the simplest periodic functions is given by

- If the argument of this function, ωt, is increased by an integral multiple of 2π radians, the value of the function remains the same.
- The function
*f*(*t*) is the periodic and its period, T, is given by

- Thus, the function f(t) is periodic with period T.

- The same result is obviously correct if we consider a sine function,

- Further, a linear combination of sine and cosine functions like

is also a periodic function with the same period T. Taking,

Eq.(1) can be written as,

- Here D and ϕ are constant given by

**Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients**.