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
Gave rough beat frequency response below.
Plz provide ans
To calculate the maximum and minimum amplitudes and the frequency of amplitude modulation for the superposed harmonic oscillations x1 = 8sin(100πt) and x2 = 12sin(96πt), we need to find the resultant oscillation.
The resultant oscillation, x, is given by the sum of the two oscillations:
x = x1 + x2
x = 8sin(100πt) + 12sin(96πt)
To find the maximum and minimum amplitudes, we need to find the maximum and minimum values of x.
The maximum amplitude, A_max, occurs when both oscillations are at their maximum values. To find it, we need to find the maximum value of sin(100πt) and sin(96πt). The maximum value of sin function is 1, so:
A_max = 8 * 1 + 12 * 1
A_max = 20
Therefore, the maximum amplitude is 20.
The minimum amplitude, A_min, occurs when both oscillations are at their minimum values. To find it, we need to find the minimum value of sin(100πt) and sin(96πt). The minimum value of sin function is -1, so:
A_min = 8 * (-1) + 12 * (-1)
A_min = -20
Therefore, the minimum amplitude is -20.
To find the frequency of amplitude modulation, we need to find the difference between the frequencies of the two oscillations. The frequency of x1 is 100π and the frequency of x2 is 96π. Therefore, the frequency of amplitude modulation, f_am, is given by:
f_am = |100π - 96π|
f_am = 4π
Therefore, the frequency of amplitude modulation is 4π.
In summary:
- The maximum amplitude is 20.
- The minimum amplitude is -20.
- The frequency of amplitude modulation is 4π.