Question

Two collinear harmonic oscillations x1 = 8 sin (100 πt) and x2 = 12 sin (96 πt) are superposed. Calculate the values of time when the amplitude of the resultant oscillation will be (i) maximum and (ii) minimum

Answer 1

Answer on Question #65634-Physics-Mechanics-Relativity

Two collinear harmonic oscillations $x1 = 8$ sin (100 nt) and $x2 = 12$ sin (96 nt) are superposed. Calculate the values of time when the amplitude of the resultant oscillation will be (i) maximum and (ii) minimum

Solution

The amplitude of the resultant oscillation will be

$$R = \sqrt{8^2 + 12^2 + 2(12)(5) \cos(100\ \pi t - 96\ \pi t)} = \sqrt{208 + 120\ \cos(4\ \pi t)}$$

(i) Maximum will be at

$$4\ \pi t = 2\pi n$$ $$t = \frac{n}{2},\ n = 0,1,2,\ \dots$$

(i) Minimum will be at

$$4\ \pi t = \pi \pm 2\pi n$$ $$t = \frac{1}{4} \pm \frac{n}{2},\ n = 0,1,2,\ \dots$$

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