Equations ` y_(1) =A sinomegat and y_(2) = A/2 sin omegat + A/2 cos omega t `represent S.H.M. The ratio of the amplitudes of the two motions is

Options

A. 0.5
B. `sqrt(2)`
C. 1
D. 2

Explanation

To solve the problem, we need to find the ratio of the amplitudes of the two simple harmonic motions represented by the equations:

1. \( y_1 = A \sin(\omega t) \)
2. \( y_2 = \frac{A}{2} \sin(\omega t) + \frac{A}{2} \cos(\omega t) \)

### Step-by-Step Solution:

**Step 1: Identify the amplitude of the first equation.**
- The first equation \( y_1 = A \sin(\omega t) \) has an amplitude \( A_1 = A \).

**Step 2: Analyze the second equation.**
- The second equation can be rewritten in a more useful form. We can express \( y_2 \) as a single sine function using the sine addition formula:
\[ y_2 = \frac{A}{2} \sin(\omega t) + \frac{A}{2} \cos(\omega t) \]

**Step 3: Use the sine addition formula.**
- The sine addition formula states that:
\[ R \sin(\omega t + \phi) = R \sin(\omega t) \cos(\phi) + R \cos(\omega t) \sin(\phi) \]
- Here, we can identify \( R \) as the resultant amplitude and \( \phi \) as the phase angle.

**Step 4: Calculate the resultant amplitude.**
- We can find the resultant amplitude \( A_2 \) using the coefficients of sine and cosine:
\[ A_2 = \sqrt{\left(\frac{A}{2}\right)^2 + \left(\frac{A}{2}\right)^2} \]
- This simplifies to:
\[ A_2 = \sqrt{\frac{A^2}{4} + \frac{A^2}{4}} = \sqrt{\frac{A^2}{2}} = \frac{A}{\sqrt{2}} \]

**Step 5: Find the ratio of the amplitudes.**
- The ratio of the amplitudes \( \frac{A_1}{A_2} \) is:
\[ \frac{A}{\frac{A}{\sqrt{2}}} = \sqrt{2} \]

### Final Result:
- The ratio of the amplitudes of the two motions is \( \sqrt{2} \).