Answer to Question #86498 in Physics for Varun kashyap

Question #86498
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.
1
Expert's answer
2019-03-21T10:39:42-0400

Find the amplitude applying the law of cosines:


A=82+1222812cos(100πt96πt)=208192cos(4πt).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 cos(4πt)\text{cos}(4\pi t):

Maximum will be at


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

Minimum will be at


4πt=π(1±2n), t=12(12±n), n=0,1,2,...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:


4πt=2πνt,4\pi t=2\pi\nu t,

ν=2 Hz.\nu=2\text{ Hz}.



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