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Amplitude is a fundamental concept in simple harmonic motion (SHM) that represents the maximum displacement from the equilibrium position. Think of it as how far the object moves away from its center point. When multiple harmonic motions interact, as in the given problem, their amplitudes contribute to the overall magnitude of the resultant motion.
In this case, we have three equal amplitude harmonic motions, denoted by \( A \). Each has the same time period, indicating they oscillate at the same rate but get displaced by different initial phase angles. We aim to find the amplitude of their combined effect, the resultant motion.
To combine these motions, each with amplitude \( A \), we employ trigonometric identities to determine how they reinforce or cancel each other out. The process shows the real power of the amplitude concept in describing and predicting motion patterns easily.

Understanding phase difference is key in analyzing waves and SHM. It indicates the relative timing of two periodic functions, often measured in degrees or radians. In our problem, the phase difference is given explicitly: the second motion is \( 60^{\circ} \) ahead of the first, and the third is another \( 60^{\circ} \) ahead of the second.
When combining waves or motions, phase differences determine how these displacements add up at any point in time. They influence whether the waves constructively add to produce a larger amplitude, or destructively interfere, possibly reducing the overall amplitude.

• If the phase difference is \( 0^{\circ} \), waves are in phase and add constructively.
• A \( 180^{\circ} \) phase difference causes destructive interference, cancelling the amplitudes.
• Any other angle, like \( 60^{\circ} \), leads to partial interference where magnitudes vary based on the specific configuration and timing.

Recognizing these patterns helps predict the behavior of the resultant wave as reflected in the problem solution.

Trigonometric identities are crucial tools in simplifying the equations of harmonic motions. They help express complex wave interactions in more manageable terms, often revealing underlying patterns and simplifying calculations.
In our solution, trigonometric identities are employed to handle phase differences easily. For example, the second motion in the problem is expressed using the identity:

• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)

This results in simplifications such as:

• \( x_2 = A[\cos(\omega t)\cos(60^{\circ}) - \sin(\omega t)\sin(60^{\circ})] \)
• \( x_3 = A[\cos(\omega t)\cos(120^{\circ}) - \sin(\omega t)\sin(120^{\circ})] \)

These transformations allow us to understand how each phase-shifted motion contributes to the resultant motion. Using identities correctly aligns our calculations and reveals the clear path to the final amplitude solution, showing that the resultant amplitude is indeed \( 2A \).