Three simple harmonic motions of equal amplitude A and equal time periods in the same direction combine. The phase of the second motion is 60° ahead of the first and the phase of the third motion is 60° ahead of the second. Find the amplitude of the resultant motion.

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Solution

It is given that three S.H.M.s of equal amplitudes A and equal time periods are combined in the same direction.

Let $Y_{1}, Y_{2} and Y_{3}$ be the three vectors representing the motions, as shown in the figure given below.

According to the question:
$Angle between Y_{1} and Y_{2} = 60 ° Angle between Y_{2} and Y_{3} = 60 °$ .

By using the vector method, we can find the resultant vector.
Resultant amplitude = Vector sum of the three vectors
= A + A cos 60° + A cos 60°
$= A + \frac{A}{2} + \frac{A}{2} = 2 A$

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