Two SHW are represented by the equations `x_1 = 20 sin [5pit +pi/4] and x_2 = 10 (sin5pit+sqrt(3) cos 5 pit]` . The ratio of the amplitudes of the two motions is

Options

A. 0.5
B. 1
C. 0.25
D. `sqrt3/2`

Answer

To find the ratio of the amplitudes of the two simple harmonic waves represented by the equations \( x_1 = 20 \sin(5\pi t + \frac{\pi}{4}) \) and \( x_2 = 10 \left( \sin(5\pi t) + \sqrt{3} \cos(5\pi t) \right) \), we will follow these steps:

### Step 1: Identify the amplitude of the first wave
The first wave is given by the equation:
\[ x_1 = 20 \sin(5\pi t + \frac{\pi}{4}) \]
Here, the amplitude \( A_1 \) is simply the coefficient of the sine function, which is:
\[ A_1 = 20 \]

### Step 2: Rewrite the second wave in standard form
The second wave is given by:
\[ x_2 = 10 \left( \sin(5\pi t) + \sqrt{3} \cos(5\pi t) \right) \]
To find the amplitude of this wave, we need to express it in the form \( R \sin(5\pi t + \phi) \), where \( R \) is the amplitude.

### Step 3: Calculate the resultant amplitude for the second wave
Using the formula for the resultant amplitude \( R \) when combining sine and cosine:
\[ R = \sqrt{A^2 + B^2} \]
where \( A \) is the coefficient of \( \sin(5\pi t) \) (which is 1) and \( B \) is the coefficient of \( \cos(5\pi t) \) (which is \( \sqrt{3} \)). Thus, we have:
\[ R = \sqrt{(1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
Now, since the wave \( x_2 \) has a coefficient of 10, the amplitude \( A_2 \) is:
\[ A_2 = 10 \times R = 10 \times 2 = 20 \]

### Step 4: Calculate the ratio of the amplitudes
Now that we have both amplitudes:
\[ A_1 = 20 \quad \text{and} \quad A_2 = 20 \]
The ratio of the amplitudes \( \frac{A_1}{A_2} \) is:
\[ \frac{A_1}{A_2} = \frac{20}{20} = 1 \]

### Final Answer
The ratio of the amplitudes of the two motions is \( 1 \).

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