The resultant amplitude due to superposition of three simple harmonic motions `x_(1) = 3sin omega t`,
`x_(2) = 5sin (omega t + 37^(@))` and `x_(3) = - 15cos omega t` is

Options

A. `18`
B. `10`
C. `12`
D. None of these

Explanation

To find the resultant amplitude due to the superposition of the three simple harmonic motions given by:

1. \( x_1 = 3 \sin(\omega t) \)
2. \( x_2 = 5 \sin(\omega t + 37^\circ) \)
3. \( x_3 = -15 \cos(\omega t) \)

we will follow these steps:

### Step 1: Convert \( x_3 \) to sine form
The equation \( x_3 = -15 \cos(\omega t) \) can be converted to sine form using the identity \( \cos(\theta) = \sin(\theta + \frac{\pi}{2}) \):

\[ x_3 = -15 \cos(\omega t) = 15 \sin\left(\omega t - \frac{\pi}{2}\right) \]

### Step 2: Rewrite the equations
Now we have:
- \( x_1 = 3 \sin(\omega t) \)
- \( x_2 = 5 \sin(\omega t + 37^\circ) \)
- \( x_3 = 15 \sin\left(\omega t - \frac{\pi}{2}\right) \)

### Step 3: Break down \( x_2 \) into components
Using the angle addition formula, we can express \( x_2 \):

\[ x_2 = 5 \left( \sin(\omega t) \cos(37^\circ) + \cos(\omega t) \sin(37^\circ) \right) \]

Calculating \( \cos(37^\circ) \) and \( \sin(37^\circ) \):
- \( \cos(37^\circ) \approx \frac{4}{5} \)
- \( \sin(37^\circ) \approx \frac{3}{5} \)

Thus, we can write:

\[ x_2 = 5 \left( \sin(\omega t) \cdot \frac{4}{5} + \cos(\omega t) \cdot \frac{3}{5} \right) = 4 \sin(\omega t) + 3 \cos(\omega t) \]

### Step 4: Combine all components
Now we can express \( x_1 \), \( x_2 \), and \( x_3 \) in terms of sine and cosine:

\[ x_1 + x_2 + x_3 = (3 + 4) \sin(\omega t) + (3 - 15) \cos(\omega t) \]
\[ = 7 \sin(\omega t) - 12 \cos(\omega t) \]

### Step 5: Find the resultant amplitude
To find the resultant amplitude \( A \), we can use the formula:

\[ A = \sqrt{X^2 + Y^2} \]

where \( X = 7 \) (coefficient of \( \sin(\omega t) \)) and \( Y = -12 \) (coefficient of \( \cos(\omega t) \)):

\[ A = \sqrt{7^2 + (-12)^2} = \sqrt{49 + 144} = \sqrt{193} \approx 13.89 \]

### Final Result
The resultant amplitude is approximately \( 13.89 \, \text{cm} \).

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