Two simple harmonic motions are represented by `y_(1)=4sin(4pit+pi//2)` and `y_(2)=3cos(4pit)`. The resultant amplitude is

Options

A. 7
B. 1
C. 5
D. `(2+sqrt(3))`

Explanation

To find the resultant amplitude of the two simple harmonic motions given by the equations \( y_1 = 4 \sin(4\pi t + \frac{\pi}{2}) \) and \( y_2 = 3 \cos(4\pi t) \), we can follow these steps:

### Step 1: Identify the amplitudes and phase difference
From the equations:
- For \( y_1 = 4 \sin(4\pi t + \frac{\pi}{2}) \), the amplitude \( a_1 = 4 \).
- For \( y_2 = 3 \cos(4\pi t) \), the amplitude \( a_2 = 3 \).

### Step 2: Convert \( y_1 \) to cosine form
We can rewrite \( y_1 \) in terms of cosine:
\[ y_1 = 4 \sin(4\pi t + \frac{\pi}{2}) = 4 \cos(4\pi t) \]
This shows that \( y_1 \) can be expressed as a cosine function with the same amplitude but a phase shift.

### Step 3: Determine the phase difference
Since both functions are now in terms of cosine:
- \( y_1 = 4 \cos(4\pi t) \)
- \( y_2 = 3 \cos(4\pi t) \)

The phase difference \( \phi \) between \( y_1 \) and \( y_2 \) is \( 0 \) because they are both in cosine form and have the same frequency.

### Step 4: Apply the formula for resultant amplitude
The formula for the resultant amplitude \( A \) of two simple harmonic motions is given by:
\[ A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2 \cos(\phi)} \]
Substituting the values:
- \( a_1 = 4 \)
- \( a_2 = 3 \)
- \( \phi = 0 \) (thus \( \cos(0) = 1 \))

We get:
\[ A = \sqrt{4^2 + 3^2 + 2 \cdot 4 \cdot 3 \cdot 1} \]
Calculating each term:
- \( 4^2 = 16 \)
- \( 3^2 = 9 \)
- \( 2 \cdot 4 \cdot 3 = 24 \)

Now, substituting these values:
\[ A = \sqrt{16 + 9 + 24} = \sqrt{49} = 7 \]

### Final Result
The resultant amplitude is \( A = 7 \).

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