twocollinear harmonic oscillationsx1=8sin(100pie(t)) and x2=12sin(96pie(t)) are superpose .calculate the

*maximum and minimum amplitudes

*the frequency of amplitude modulation

To tackle the problem of two collinear harmonic oscillations given by the equations \( x_1 = 8 \sin(100\pi t) \) and \( x_2 = 12 \sin(96\pi t) \), we need to analyze their superposition. This involves finding the resultant amplitude when these two waves combine and determining the frequency of amplitude modulation.

Understanding the Components of the Oscillations

Each oscillation can be characterized by its amplitude and angular frequency. The first oscillation, \( x_1 \), has an amplitude of 8 and an angular frequency of \( 100\pi \) rad/s. The second oscillation, \( x_2 \), has an amplitude of 12 and an angular frequency of \( 96\pi \) rad/s.

Finding the Resultant Amplitude

When two harmonic oscillations are superimposed, the resultant amplitude can be calculated using the principle of superposition. The general form for the superposition of two sine waves with different frequencies is:

Let \( A_1 = 8 \) and \( A_2 = 12 \). The resultant amplitude \( A \) can be found using the formula:

• Maximum Amplitude: \( A_{\text{max}} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\Delta \omega t)} \)
• Minimum Amplitude: \( A_{\text{min}} = |A_1 - A_2| \)

Here, \( \Delta \omega = \omega_1 - \omega_2 = 100\pi - 96\pi = 4\pi \) rad/s, which represents the difference in angular frequencies.

Calculating Maximum Amplitude

To find the maximum amplitude, we first calculate:

Maximum Amplitude:

\( A_{\text{max}} = \sqrt{8^2 + 12^2 + 2 \cdot 8 \cdot 12 \cdot \cos(4\pi t)} \)

Calculating the individual components:

• \( 8^2 = 64 \)
• \( 12^2 = 144 \)
• \( 2 \cdot 8 \cdot 12 = 192 \)

Thus, the maximum amplitude becomes:

\( A_{\text{max}} = \sqrt{64 + 144 + 192 \cos(4\pi t)} \)

When \( \cos(4\pi t) = 1 \) (maximum case), we have:

\( A_{\text{max}} = \sqrt{64 + 144 + 192} = \sqrt{400} = 20 \)

Calculating Minimum Amplitude

For the minimum amplitude, we use:

\( A_{\text{min}} = |8 - 12| = 4 \)

Frequency of Amplitude Modulation

The frequency of amplitude modulation can be derived from the difference in the angular frequencies of the two oscillations. The modulation frequency \( f_m \) is given by:

\( f_m = \frac{\Delta \omega}{2\pi} = \frac{4\pi}{2\pi} = 2 \) Hz

Summary of Results

In summary, for the superposition of the two harmonic oscillations:

• Maximum Amplitude: 20 units
• Minimum Amplitude: 4 units
• Frequency of Amplitude Modulation: 2 Hz

This analysis illustrates how two oscillations can interact, leading to varying amplitudes and a distinct modulation frequency. If you have any further questions or need clarification on any part of this process, feel free to ask!