two collinear harmonic oscillation x1=8sin(100pie(t)) and x2=12sin(96pie(t)) are superposed.calculate the
*maximum and minimum amplitudes
*the frequency of amplitude modulation
To calculate the maximum and minimum amplitudes of the superposed oscillation, we need to use the principle of superposition, where we add the two individual oscillations together.
The equation for the superposed oscillation is given by:
x = x1 + x2
Substituting the given values, we have:
x = 8sin(100πt) + 12sin(96πt)
Now, to find the maximum and minimum amplitudes, we can rewrite the expression in terms of a single sine function with an equivalent amplitude. Let's use trigonometric identities to simplify the equation:
x = 8sin(100πt) + 12sin(96πt)
x = 8sin(100πt) + 12(2sin(48πt)cos(48πt))
Using the double angle identity, where sin(2θ) = 2sin(θ)cos(θ), we can rewrite the equation:
x = 8sin(100πt) + 24sin(48πt)cos(48πt)
Now, to find the maximum and minimum amplitudes, we need to find the maximum and minimum values of the sinusoidal term, which appears as a coefficient of cos(48πt).
The maximum value occurs when sin(48πt) = 1, which yields:
x_max = 8 + 24 = 32
The minimum value occurs when sin(48πt) = -1, which yields:
x_min = 8 - 24 = -16
Therefore, the maximum amplitude of the superposed oscillation is 32, and the minimum amplitude is -16.
To find the frequency of amplitude modulation, we need to find the difference between the frequencies of the two individual oscillations. In this case, the frequency of oscillation for x1 is 100π and for x2 is 96π.
The frequency of amplitude modulation is given by the absolute difference between the two frequencies:
f_am = |100π - 96π|
= 4π
So, the frequency of amplitude modulation is 4π.