Two particle are executing SHMs .The equations of their motions are
`y_(1)=10"sin"(omegat+(pi)/(4)) " and "y_(2)=5 "sin"(omegat+(sqrt(3)pi)/(4))`
What is the ratio of their amplitudes.

Options

A. `1:1`
B. `2:1`
C. `1:2`
D. None of these

Explanation

To find the ratio of the amplitudes of the two particles executing simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the equations of motion**:
The equations given are:
\[ y_1 = 10 \sin\left(\omega t + \frac{\pi}{4}\right) \]
\[ y_2 = 5 \sin\left(\omega t + \frac{\sqrt{3}\pi}{4}\right) \]

2. **Recognize the standard form of SHM**:
The standard equation of SHM is:
\[ y = A \sin(\omega t + \phi) \]
where \(A\) is the amplitude.

3. **Extract the amplitudes from the equations**:
- From the first equation \(y_1\), the amplitude \(A_1\) is:
\[ A_1 = 10 \]
- From the second equation \(y_2\), the amplitude \(A_2\) is:
\[ A_2 = 5 \]

4. **Calculate the ratio of the amplitudes**:
The ratio of the amplitudes \(A_1\) to \(A_2\) is given by:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{10}{5} = 2 \]

5. **Express the ratio in the form of \(a : b\)**:
Thus, the ratio of the amplitudes can be expressed as:
\[ A_1 : A_2 = 2 : 1 \]

6. **Conclusion**:
Therefore, the ratio of the amplitudes of the two particles is \(2 : 1\).

### Final Answer:
The ratio of their amplitudes is \(2 : 1\).

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