Answer to Question #86498 in Physics for Varun kashyap

Question

Two collinear harmonic oscillations
8sin (100 ) 1
x = πt
and 12sin (96 ) 2
x = πt
are superposed. Calculate the (i) maximum and minimum amplitudes, and (ii) the
frequency of amplitude modulation.

Answer 1

Find the amplitude applying the law of cosines:

$$A=\sqrt{8^2+12^2-2\cdot 8\cdot 12\cdot \text{cos}(100\pi t-96\pi t)}=\sqrt{208-192\text{cos}(4\pi t)}.$$

Solve this as an equation with maximum and minimum value of $\text{cos}(4\pi t)$:

Maximum will be at

$$4\pi t=2\pi n, \space t=2n, \space n=0,1,2,...$$

Minimum will be at

$$4\pi t=\pi (1\pm 2n), \space t=\frac{1}{2}(\frac{1}{2}\pm n), \space n=0,1,2,...$$

The frequency of amplitude modulation:

$$4\pi t=2\pi\nu t,$$

$$\nu=2\text{ Hz}.$$