Two simple harmonic motions are given by `y _(1) = 5 sin ( omegat = pi //3). y_(2) = 5 ( sin omegat+ sqrt(3) cos omegat)`. Ratio of their amplitudes is

Options

A. `1:2`
B. `1:4`
C. `1:5`
D. `1:8`

Answer

To find the ratio of the amplitudes of the two simple harmonic motions given by:

1. \( y_1 = 5 \sin(\omega t - \frac{\pi}{3}) \)
2. \( y_2 = 5 \left( \sin(\omega t) + \sqrt{3} \cos(\omega t) \right) \)

we will follow these steps:

### Step 1: Identify the amplitude of \( y_1 \)

The first equation is in the form \( A \sin(\omega t + \phi) \), where \( A \) is the amplitude. Here, the amplitude \( A_1 \) is simply the coefficient of the sine function.

\[ A_1 = 5 \]

### Step 2: Rewrite \( y_2 \) in standard form to find its amplitude

The second equation can be rewritten using the sine addition formula. We can express \( y_2 \) as:

\[ y_2 = 5 \left( \sin(\omega t) + \sqrt{3} \cos(\omega t) \right) \]

To find the amplitude, we can factor out the 5 and rewrite the sine and cosine terms in the form \( R \sin(\omega t + \phi) \).

### Step 3: Calculate the resultant amplitude \( R \)

Using the formula for the resultant amplitude \( R \) when combining sine and cosine:

\[ R = \sqrt{A^2 + B^2} \]

where \( A = 1 \) (the coefficient of \( \sin(\omega t) \)) and \( B = \sqrt{3} \) (the coefficient of \( \cos(\omega t) \)):

\[ R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

Thus, the amplitude \( A_2 \) of \( y_2 \) is:

\[ A_2 = 5R = 5 \times 2 = 10 \]

### Step 4: Find the ratio of the amplitudes

Now, we can find the ratio of the amplitudes \( A_1 \) and \( A_2 \):

\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{5}{10} = \frac{1}{2} \]

### Final Answer

The ratio of the amplitudes \( A_1 : A_2 \) is:

\[ \boxed{\frac{1}{2}} \]

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