A particle is subjected simultaneously to two SHMs, one along the x - axis and the other along the y - axis. The two vibrations are in phase and have unequal amplitudes. The particle will execute

Options

A. Straight line motion
B. Circular motion
C. Elliptic motion
D. Parabolic motion

Answer

To solve the problem, we need to analyze the motion of a particle subjected to two simple harmonic motions (SHMs) along the x-axis and y-axis. Let's break down the steps:

### Step-by-Step Solution

1. **Define the SHM along the y-axis**:
The equation for the particle's motion along the y-axis can be written as:
\[ y = A_1 \sin(\omega t) \]
where \( A_1 \) is the amplitude of the SHM along the y-axis.

2. **Define the SHM along the x-axis**:
The equation for the particle's motion along the x-axis can be expressed as:
\[ x = A_2 \sin(\omega t + \phi) \]
where \( A_2 \) is the amplitude of the SHM along the x-axis and \( \phi \) is the phase difference. Given that the vibrations are in phase, we can set \( \phi = 0 \). Thus, the equation becomes:
\[ x = A_2 \sin(\omega t) \]

3. **Substituting the phase angle**:
Since both SHMs are in phase, we can rewrite the x-axis equation as:
\[ x = A_2 \sin(\omega t) \]

4. **Combine the equations**:
We now have:
\[ x = A_2 \sin(\omega t) \]
\[ y = A_1 \sin(\omega t) \]

5. **Square and add the equations**:
To find the relationship between \( x \) and \( y \), we square both equations and add them:
\[ x^2 = A_2^2 \sin^2(\omega t) \]
\[ y^2 = A_1^2 \sin^2(\omega t) \]
Adding these gives:
\[ \frac{y^2}{A_1^2} + \frac{x^2}{A_2^2} = \sin^2(\omega t) + \sin^2(\omega t) \]
This simplifies to:
\[ \frac{y^2}{A_1^2} + \frac{x^2}{A_2^2} = \sin^2(\omega t) \]

6. **Recognize the resulting equation**:
The equation we derived is of the form:
\[ \frac{x^2}{A_2^2} + \frac{y^2}{A_1^2} = \sin^2(\omega t) \]
This represents an ellipse when plotted in the xy-plane.

### Conclusion
The particle will execute elliptical motion due to the combination of two SHMs along the x and y axes with different amplitudes.