Simple Harmonic Motion

INTRODUCTION

The motion of a particle moving along a straight line with an acceleration which is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point is called simple harmonic motion.

The best example of this phenomenon is a mass falling freely from spring.The force on the mass is

where k is spring constant. The ocsillations are sinusoidal type.The differential equations are

solved using differential calculus. For more information refer[1].

EXPLANATION OF DERIVATION

One of the common example is a mass on a spring.Magnitude of force on mass is

Using Newton’s second law , ma=-kx. Now the derivation starts from below.

or

Now let us come up with a trial solution to our differential equation. To see the detailed solution of these type of differential equations refer[2]. As a tentative solution,we write:

where a and b are constants. Differentiating this equation, we see that

and

Plugging this into our original differential equation ,we see that:

It is clear that ,if b^2 = k/m , then the equation is satisfied. Thus the equation governing simple m

harmonic oscillation is:

APPLICATIONS OF SIMPLE HARMONIC MOTION

The concept of Simple Harmnonic Motion is used in the Pendulum and swing.One type of pendulum is metronome, which registers the tempo or speed of music.Housed in a hollow box shaped like a pyramid,a metronome consists of a pendulum attached to a sliding weight,with a fixed weight attached to the bottom end of the pendulum.It includes a number scale indicat- ing the number of oscillations per minute , and by moving the upper weight, one can change the beatto be indicated.

Zhang Heng’s Seismoscope was invented using pendulum to develop the world’s first seismo- scope,an instrument for measuring motion on Earth’s surface as a result of earthquakes.Bain’s Fax Machine also used pendulum which works on principle ofsimple harmonic motion.The Foucalt’s Pendulum is also one of it’s application. SHM is also observed in bungee cords and rubber bands.

REFERENCES

1. Dr. PK Mittal and Shanti Narayana. Differential Calculus. S Chand, 1942.
2. HC Verma. Concepts of physics. Bharati Bhawan Publishers and Distributors, 1999.