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Question
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
\[Y_1 , Y_2 \text { and } Y_3\] be the three vectors representing the motions, as shown in the figure given below.
According to the question:
\[\text { Angle between } Y_1 \text { and } Y_2 = 60 ^\circ\]
\[\text { Angle between } Y_2 \text{ and } Y_3 = 60 ^\circ\]
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°
\[= A + \frac{A}{2} + \frac{A}{2} = 2A\]
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