Simple Harmonic Motion: Simple Harmonic Motion (SHM) is a periodic motion the body moves to and fro about its mean position. The time period of such a motion is given by T = 2 ∏ (displacement/acceleration)^1/2

Simple pendulum:

A simple pendulum is defined as an object that has a small mass (pendulum bob), which is suspended from a wire or string having negligible mass.     

• When the pendulum bob is displaced it oscillates on a plane about the vertical line through the support.
• Simple pendulum can be set into oscillatory motion by pulling it to one side of equilibrium position and then releasing it
• The time period in simple pendulum is given by T = 2 ∏ (l/g)^1/2 where l is the length of the pendulum and g is the acceleration due to gravity

Why is oscillation in a simple pendulum simple harmonic?

• In a pendulum, the particle moves to and fro about the mean position.
• In simple harmonic motion, the time period is directly proportional to the displacement from the mean position and inversely proportional to acceleration. While in case of a pendulum, the time period is proportional to the length which is indirectly proportional to the displacement and inversely proportional to the acceleration.

The effect on energy when amplitude of SHM is doubled:

Let the amplitude of the SHM be A. Hence, the corresponding energy is given by ½ mA²

E₁ = ½ mA²

Now as amplitude is doubled, A becomes 2A. Hence, E₂ = ½ m(2A)² = 4 [ ½ m A² ] = 4 E₁

Hence, when amplitude is doubled, the energy becomes 4 times the original.