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. 1
B. 2
C. `0.5`
D. `sqrt2`

Answer

To solve the problem, we need to find the ratio of the amplitudes of the two simple harmonic motions (SHM) 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 1: Identify the amplitude of \( y_1 \)

The amplitude of \( y_1 \) is straightforward since it is given directly in the equation:
- Amplitude of \( y_1 = A \)

### Step 2: Simplify \( y_2 \)

We need to express \( y_2 \) in a standard form to identify its amplitude. The equation is:
\[ y_2 = \frac{A}{2} \sin(\omega t) + \frac{A}{2} \cos(\omega t) \]

### Step 3: Factor out the common term

We can factor out \( \frac{A}{2} \):
\[ y_2 = \frac{A}{2} \left( \sin(\omega t) + \cos(\omega t) \right) \]

### Step 4: Use the trigonometric identity

To combine \( \sin(\omega t) + \cos(\omega t) \), we can use the identity:
\[ \sin(\theta) + \cos(\theta) = \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]
This means:
\[ \sin(\omega t) + \cos(\omega t) = \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]

### Step 5: Substitute back into \( y_2 \)

Now substituting this back into the equation for \( y_2 \):
\[ y_2 = \frac{A}{2} \cdot \sqrt{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]
This simplifies to:
\[ y_2 = \frac{A \sqrt{2}}{2} \sin\left(\omega t + \frac{\pi}{4}\right) \]

### Step 6: Identify the amplitude of \( y_2 \)

From the equation, we can see that the amplitude of \( y_2 \) is:
- Amplitude of \( y_2 = \frac{A \sqrt{2}}{2} \)

### Step 7: Calculate the ratio of the amplitudes

Now we can find the ratio of the amplitudes of \( y_1 \) and \( y_2 \):
\[ \text{Amplitude ratio} = \frac{\text{Amplitude of } y_1}{\text{Amplitude of } y_2} = \frac{A}{\frac{A \sqrt{2}}{2}} = \frac{A \cdot 2}{A \sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer

The ratio of the amplitudes of the two motions is:
\[ \sqrt{2} \]

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