Simple Harmonic Motion: 

The type of oscillatory motion, in which acceleration of the body at any instant is directly proportional to the displacement from the mean position and directed towards the mean position, is called simple harmonic motion (SHM).

                a ∝ - x

Simple Pendulum:

A simple pendulum consists of a small heavy mass attached to a light and inextensible string suspended with frictionless support.




Explanation:

Consider a bob of mass m attached to the end of a lightweight string of length l. When the simple pendulum is displaced from its mean position through a small angle θ and released then it starts to oscillate about mean position O.

Components of weight:

When the simple pendulum is displaced from its mean position through a small angle Î¸, then by resolving the weight w into two rectangular components, we get w cosθ (x-component) and w sinθ (y-component),

Where the x-component w cosθ is balanced by the tension force in the string 

Tension in string = w cosθ

The y-component of weight w sinθ is responsible for the motion of the simple pendulum, which brings the bob back towards its mean position and acts as the restoring force for the bob.

Restoring force= F = -w sinθ
where w = mg, thus

        F= -mg sinθ --------------------- (1)

The negative sign shows that restoring force is directed towards the mean position and the direction of restoring force is opposite to displacement,

Also, we know that

        F = ma ------------------- (2)

Comparing above equations (1) and (2) we get

        ma = -mg sinθ

        a = -g sinθ

For the small value of angle Î¸,   sinθ ≈ Î¸

So,     a = -g θ ----------------(3)

From figure sinθ ≈ Î¸ = `\frac {arc bar {OA}}{l}` = `\frac {x}{l}` [as Î¸ is very very small so arc `\bar {OA}` = x]

Putting `\frac {x}{l}` in equation (3)

        a = -g `\frac {x}{l}`  

or        
        a = - (`\frac {g}{l}`)x ----------------(4)

This is an expression for the acceleration of a simple pendulum,

[as the value of g and l are constant ] so,

         a = - (constant) x  [as the value of g and l are constant ]

or

        a ∝ - x  ----------------(5)

This proves that the motion of the simple pendulum is SHM.


Time Period of Simple Pendulum:

Definition: The time required to complete one vibration by the bob of the simple Pendulum is called the time period. The time period for SHM can be expressed as,

        T = `\frac {2ã…ˆ}{w}`   ----------------(6)      

We also know that simple harmonic motion 

        a = -à´§² x  ----------------(7)

So by comparing the equation (7) and (4)

        à´§² = `\frac {g}{l}`

or (by taking square root on both sides)

        à´§ = `\sqrt frac {g}{l}` ----------------(8)


Putting the value from equation (6) we get
          
        T = `\frac {2ã…ˆ}{sqrt frac {g}{l}}`

or 

        T = 2ã…ˆ`\sqrt frac {g}{l}`

This is the equation for the time period of the simple pendulum which shows that the time period of a simple pendulum depends upon the length of the pendulum and acceleration due to gravity.

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