# SHM Phase and Phase Difference

An oscillating pendulum, a cork bobbing up and down in water or the periodic motion of the mass attached to a spring have one thing in common. They are all executing simple harmonic motion (SHM). Simple harmonic motion is a special type of oscillation. The concept of phase and phase difference associated with SHM is generally difficult to comprehend. In this demonstration this concept of phase and phase difference can be understood easily.

## Procedure

You need a stand, two pendulums made by attaching two similar plastic balls with two threads.

1. Hang the two threads attached to the plastic balls to the stand to make the two pendulums.
2. Make the length of the threads same.
3. Take both the balls to one extreme and release them together. Both the balls oscillate in a similar manner i.e., they have same time period, reach the other extreme in the same time and are at same position at any instant of time. We say that both are in the same phase and their phase difference is zero.
4. Now take one ball to one extreme and the other ball to the other extreme and release them simultaneously. See that the balls move opposite to one another at all times i.e., if one moves right the other moves left. Also if one is at the right extreme, the other is at the left extreme. We say that both are in opposite phase and their phase difference is $$\pi$$.
5. Now make the length of one of the pendulums slightly less than the other. Again release both of them simultaneously from the extreme position. This time the two do not move together, one with the smaller length reaches the other extreme earlier, so we say that they are not in phase. But the phase difference between them does not remain constant, first it increases with time and becomes equal to $$\pi$$, then the phase difference starts decreasing and becomes equal to zero and again it increases to $$\pi$$ and then goes to zero. This process repeats itself and this phenomenon is called beats.

## Discussion

SHM is an oscillation in which a particle moving in a straight line experiences a force which directs it towards its mean position and the magnitude of the force is proportional to the displacement from the mean position. It can be represented by a sine or cosine function. The argument of the sine or cosine function is called the phase of the particle executing SHM. Phase tells about the state of the particle at any instant.

In the above case the SHM of the pendulums which are released from their extreme position can be represented by the equations $$X_1=A_1\cos(\omega_1 t+\theta)$$ and $$X_2=A_2\cos(\omega_2 t+\phi)$$, where $$A_1$$ and $$A_2$$ are the amplitudes, $$\theta$$ and $$\phi$$ are the initial phases, $$\omega_1$$ and $$\omega_2$$ are the angular frequencies and the arguments ($$\omega_1t+\theta$$), ($$\omega_2 t+\phi$$) are the phases of the two pendulums at any instant.

Since the frequency of the pendulum is dependent on its length, when the length of the threads is kept same $$\omega_1=\omega_2$$. Now if the balls are released together from

1. Same extreme, initial phases $$\theta=\phi=0$$ and phase difference = 0
2. Opposite extremes, $$\theta= 0$$ and $$\phi = \pi$$ and phase difference $$=\phi-\theta=\pi$$. Also the amplitudes $$A_1 = A_2$$.

When the length of the threads is different $$\omega_1\neq \omega_2$$. Now if the balls are released together initial phases $$\theta=\phi=0$$ but the phase difference is $$(\omega_1 t-\omega_2 t)$$ which changes with time and oscillates between the value $$0$$ and $$\pi$$ causing beats.

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