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 all terms to sine form
First, we convert \( x_3 \) from cosine to sine. We know that:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Thus, we can rewrite \( x_3 \) as:
\[ x_3 = -15 \cos(\omega t) = -15 \sin\left(\omega t + \frac{\pi}{2}\right) \]

### Step 2: Express \( x_2 \) in terms of sine and cosine
Next, we express \( x_2 \) using the sine addition formula:
\[ x_2 = 5 \sin(\omega t + 37^\circ) = 5 \left( \sin(\omega t) \cos(37^\circ) + \cos(\omega t) \sin(37^\circ) \right) \]
Using the values \( \cos(37^\circ) = \frac{4}{5} \) and \( \sin(37^\circ) = \frac{3}{5} \), we have:
\[ x_2 = 5 \left( \sin(\omega t) \cdot \frac{4}{5} + \cos(\omega t) \cdot \frac{3}{5} \right) = 4 \sin(\omega t) + 3 \cos(\omega t) \]

### Step 3: Combine all terms
Now, we can combine \( x_1 \), \( x_2 \), and \( x_3 \):
\[ x = x_1 + x_2 + x_3 = 3 \sin(\omega t) + (4 \sin(\omega t) + 3 \cos(\omega t)) - 15 \sin\left(\omega t + \frac{\pi}{2}\right) \]
This simplifies to:
\[ x = (3 + 4) \sin(\omega t) + 3 \cos(\omega t) - 15 \sin\left(\omega t + \frac{\pi}{2}\right) \]
Since \( \sin\left(\omega t + \frac{\pi}{2}\right) = \cos(\omega t) \), we have:
\[ x = 7 \sin(\omega t) + 3 \cos(\omega t) - 15 \cos(\omega t) \]
This simplifies to:
\[ x = 7 \sin(\omega t) - 12 \cos(\omega t) \]

### Step 4: Calculate the resultant amplitude
The resultant amplitude \( A \) can be calculated using the formula:
\[ A = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2} \]
Where \( x_{\text{total}} = 7 \) and \( y_{\text{total}} = -12 \):
\[ A = \sqrt{7^2 + (-12)^2} = \sqrt{49 + 144} = \sqrt{193} \approx 13.89 \]

Thus, the resultant amplitude due to the superposition of the three simple harmonic motions is approximately \( 13.89 \) units.

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