Two particles are executing S.H.M. The equation of their motion are `y_(1) = 10 sin (omega t + (piT)/4),y_(2)=25 sin (omegat+(sqrt3piT/4))` . What is the ratio of their amplitude

Options

A. `1:1`
B. `2:5`
C. `1:2`
D. None of these

Answer

To find the ratio of the amplitudes of two particles executing Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Amplitude from the Equations:**
The equations of motion for the two particles are given as:
- \( y_1 = 10 \sin(\omega t + \frac{\pi}{4}) \)
- \( y_2 = 25 \sin(\omega t + \frac{\sqrt{3}\pi}{4}) \)

In the standard form of SHM, \( y = A \sin(\omega t + \phi) \), the amplitude \( A \) is the coefficient in front of the sine function.

2. **Extract the Amplitudes:**
From the equations:
- The amplitude \( A_1 \) for particle 1 is 10.
- The amplitude \( A_2 \) for particle 2 is 25.

3. **Calculate the Ratio of Amplitudes:**
The ratio of the amplitudes \( \frac{A_1}{A_2} \) is calculated as follows:
\[ \frac{A_1}{A_2} = \frac{10}{25} \]

4. **Simplify the Ratio:**
Simplifying the fraction:
\[ \frac{10}{25} = \frac{2}{5} \]

5. **Final Result:**
Thus, the ratio of the amplitudes of the two particles is \( \frac{2}{5} \).