The equations of two SH M's are
`X_(1) = 4 sin (omega t + pi//2) . X_(2) = 3 sin(omega t + pi)`

Options

A. 6 units
B. 5 units
C. 1 units
D. 7 units

Explanation

To find the amplitude of the resultant wave from the two simple harmonic motions (SHM) given by the equations:

1. \( X_1 = 4 \sin(\omega t + \frac{\pi}{2}) \)
2. \( X_2 = 3 \sin(\omega t + \pi) \)

we can follow these steps:

### Step 1: Identify the Amplitudes and Phases
From the equations, we can identify:
- Amplitude \( A_1 = 4 \) (from \( X_1 \))
- Amplitude \( A_2 = 3 \) (from \( X_2 \))
- Phase of \( X_1 \) is \( \phi_1 = \frac{\pi}{2} \)
- Phase of \( X_2 \) is \( \phi_2 = \pi \)

### Step 2: Calculate the Phase Difference
The phase difference \( \Delta \phi \) between the two SHMs is given by:
\[ \Delta \phi = \phi_2 - \phi_1 = \pi - \frac{\pi}{2} = \frac{\pi}{2} \]

### Step 3: Use the Formula for Resultant Amplitude
The formula for the resultant amplitude \( A \) when two SHMs are combined is:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\Delta \phi)} \]
Substituting the values we have:
- \( A_1 = 4 \)
- \( A_2 = 3 \)
- \( \Delta \phi = \frac{\pi}{2} \)

### Step 4: Calculate \( \cos(\Delta \phi) \)
We know:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Step 5: Substitute and Simplify
Now substituting back into the formula:
\[ A = \sqrt{4^2 + 3^2 + 2 \cdot 4 \cdot 3 \cdot 0} \]
\[ A = \sqrt{16 + 9 + 0} \]
\[ A = \sqrt{25} \]
\[ A = 5 \]

### Conclusion
The amplitude of the resultant wave is \( 5 \).

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