Application of Mathematics

Generally, we consider mathematics as a different subject from science. But in fact, mathematics is an intimate part of basic science and daily life activities.

Mathematics can be used to find the rate of disintegration of radioactive isotopes and to find its half life.

It is used for periodic function and periodic motion such as:

• (i) simple harmonic motion,
• (ii)progressive sound waves and
• (iii) A.C. current.

Using logarithmic function it can be used to find the concentration of P.H. value of acidic, basic and neutral solution in chemistry.

The concept of increasing and decreasing function in mathematics can be used for the diagrammatic representation of Boyle's Law, Charles Law and Ohm's Law in chemistry.

Using the anti-derivative of mathematics we can find the gravitational potential and electrostatic potential as well.

The concept of anti-derivative can also be used in laws of thermos-dynamics and so on.

Here we will discuss how the periodic function of mathematics is used in Simple Harmonic Motion.

What is periodic function?

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Thus a function y=f(x) is said to be a periodic function of period P if P is the least positive real number such that f(x+P)=f(x).The most important example of periodic function are the trigonometric functions, whose values repeat over certain interval.

For example: sin(2π+x)=sinx, cos(2π+x)=cosx ,so sine and cosine are periodic functions of period 2π.

Also, tan(π+x)=tanx , so it is a periodic function of period π and so on.

Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity.

Example:

A graph of the sine function showing two complete periods

For example, the sine function is periodic with period 2π, since for all values of x, this function repeats on intervals of length 2π. (see the graph above).

Periodic functions frequently arise in various fields of science and engineering, business and in some daily life activities such as:

• i. Brain waves and heart beating are periodic.
• ii. Household voltage and electric current are periodic.
• iii. The cash flows in seasonal business are periodic.
• iv. The seasons and weather are periodic.
• v. The motion of wrist watch and pendulum clock are periodic.
• vi. The phases of the moon are periodic.

What is Simple Harmonic Motion?

In mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.

Equation of simple Harmonic Motion

The mathematical representation of simple harmonic motion can be expressed by the relation y=A sint ; where y=displacement from mean position ,

A= Amplitude , = angular velocity with = 2π/T; T = time period & t = time.

This simple harmonic motion can be represented by the motion of simple pendulum.

Fig: sinusoidal Wave representation of simple Pendulum.

By the equation of Simple Harmonic Motion, i.e. y=A sint,

when t=0, then y=0

t=T/4, then y=A

t=T/2, then y=0

t=3T/4, then y=-A

t=T, then y=0

So from above, its clear that position of the bob of the pendulum repeats in every T/2 interval. Hence the motion of simple pendulum is periodic.

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