IGNOU BPHCT 131 2026 - Two collinear harmonic oscillations, each of frequency , have amp

Question

Two collinear harmonic oscillations, each of frequency $\omega_0$ , have amplitudes, a₁ and a₂ and initial phases, $\phi_1 = \frac{\pi}{3}$ and $\phi_2 = \frac{4\pi}{3}$ , respectively. Show that, when these oscillations are superposed, the amplitude of the resultant motion is equal to (a₁ - a₂). What will be the value of the resultant amplitude when $\phi_1 = \frac{\pi}{3}$ and $\phi_2 = \frac{7\pi}{3}$ .

19 Jan 2026

Answer :

Word Count : 245

The two collinear harmonic oscillations can be represented as: [ x_1 = a_1 \cos(\omega_0 t + \phi_1), \quad x_2 = a_2 \cos(\omega_0 t + \phi_2) ] The resultant displacement is: [ x = x_1 + x_2 = a_1 \cos(\omega_0 t + \phi_1) + a_2 \cos(\omega_0 t + \phi_2) ] The amplitude of the resultant motion is given by the formula for the sum of two harmonic oscillations of the same frequency: [ R = \sqrt{a_1^2 + a_2^2 + 2 a_1 a_2 \cos(\phi_1 - \phi_2)} ] Case 1: (\phi_1

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