Three simple harmonic motions with equal amplitudes A and identical time periods in the same direction are combined. The second motion has a phase 60^ ahead of the first, while the third motion has a phase 60^ ahead of the second. Determine the amplitude of the resultant motion.

Step-by-step solution

The three simple harmonic motions can be represented as phasors of length A with angles 0^ , 60^ , and 120^ respectively. The resultant amplitude R can be found by resolving the phasors into horizontal and vertical components. R_x = A (0^ ) + A (60^ ) + A (120^ ) R_x = A (1 + 1 2 - 1 2 ) = A R_y = A (0^ ) + A (60^ ) + A (120^ ) R_y = A (0 + 3 2 + 3 2 ) = 3 A The resultant amplitude is given by R = R_x^2 + R_y^2 R = A^2 + ( 3 A)^2 = A^2 + 3A^2 = 4A^2 = 2A Alternatively, the resultant of the first and third phasors (