Two simple harmonic motion, are represented by the equations
`y _(1) = 10 sin ( 3 pi t + (pi)/(3))`
` y _(2) = 5 ( sin 3 pi t + sqrt3 cos 3 pi t )`
Ratio of amplitude of `y _(1)` to `y _(2) = x :1.` The value of x is __________

Answer

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

1. \( y_1 = 10 \sin(3\pi t + \frac{\pi}{3}) \)
2. \( y_2 = 5 \left( \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \right) \)

we will follow these steps:

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

From the equation \( y_1 = 10 \sin(3\pi t + \frac{\pi}{3}) \), we can see that the amplitude \( A_1 \) is simply the coefficient of the sine function.

\[ A_1 = 10 \]

### Step 2: Rewrite \( y_2 \) in standard form

The equation for \( y_2 \) is given as:

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

We can express this in the form \( R \sin(3\pi t + \phi) \) where \( R \) is the resultant amplitude. To find \( R \), we use the formula:

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

where \( A \) is the coefficient of \( \sin \) and \( B \) is the coefficient of \( \cos \).

In this case:

- \( A = 5 \)
- \( B = 5\sqrt{3} \)

Calculating \( R \):

\[ R = \sqrt{(5)^2 + (5\sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10 \]

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

\[ A_2 = 10 \]

### Step 3: Calculate the ratio of amplitudes

Now that we have both amplitudes:

- \( A_1 = 10 \)
- \( A_2 = 10 \)

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

\[ \frac{A_1}{A_2} = \frac{10}{10} = 1 \]

### Step 4: Express the ratio in the form \( x : 1 \)

From the ratio \( \frac{A_1}{A_2} = 1 \), we can express this as:

\[ x : 1 \quad \text{where } x = 1 \]

Thus, the final answer is:

\[ \text{The value of } x \text{ is } 1. \]

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