Periodic motion

Periodic phenomena or event repeats after certain interval like seasons, motion of planets, swings, pendulum etc. In this module, our focus, however, is the physical movement of particle or body, which shows a pattern recurring after certain fixed time interval.

Representation of periodic motion has a basic pattern, which is repeated at regular intervals. What it means that if we know the basic form (building block), then we can describe the motion by following the pattern again and again.

Periodic attributes

A periodic motion can be described with respect to different quantities. A given periodic motion can have a host of attributes which may undergo periodic variations. Consider, for example, the case of a pendulum. We can choose any of the attributes like angle (θ), horizontal displacement (x), vertical displacement (y), kinetic energy (K), potential energy (U) etc. The values of these quantities undergo periodic alteration with respect to time. These attributes constitute periodic attributes of the periodic motion.

There are, however, other attributes, which may remain constant during periodic motion. If we consider pendulum executing simple harmonic motion (it is a particular periodic motion), then total mechanical energy of the system is constant and as such is independent of time. Hence, we need to pick appropriate attribute(s) to describe a periodic motion in accordance with problem situation in hand.

Description of periodic motion

We need a mathematical model to describe periodic motion. For this, we employ certain mathematical functions. The important feature of a periodic function is that its value is repeated after certain interval. We call this interval as “period”. In case, this period refers to time, then the same is called “time period”. Mathematically, a periodic function is defined as :

Periodic motion

A function is said to be periodic if there exists a positive real number “T”, which is independent of “t”, such that $f\left(t+T\right)=f\left(t\right)$ .

The least positive real number “T” (T>0) is known as the fundamental period or simply the period of the function. The “T” is not a unique positive number. All integral multiple of “T” is also the period of the function.

In the context of periodic function, an “aperiodic” function is one, which in not periodic. On the other hand, a function is said to be anti-periodic if :

$$f\left(t+T\right)=-f\left(t\right)$$

Example

Problem 1: Prove that " $\sin t$ " is a periodic function. Also find its period.

Solution :

We know that :

$$\sin \left(n\pi +t\right)={\left(-1\right)}^n\sin t,\text{when “n” is an integer.}$$

$$\sin \left(n\pi +t\right)=\sin t,\text{when “n” is an even integer.}$$

Thus, there exists T>0 such that f(t+T) = f(t). Further “nπ” is independent of “t”. Hence, “ $\sin t$ ” is a periodic function. Its period is the least value, when n = 2 (first even positive integer),

$$\Rightarrow T=2\pi$$

The period of "sine" function is collaborated from the figure also. Note that we can not build a sine curve with a half cycle of period “π” (upper figure). We require a full cycle of “2π” to build a sine curve (lower figure).