A particle is subjected to two simple harmonic motions along x and y directions according to `x=3sin100pit`, `y=4sin100pit`.

Options

A. Motion of particle will be an ellipse traversing it in clockwise direction.
B. Motion of particle will be one a straight line with slope 4/3.
C. Motion will be a simple harmonic motion along x - axis with amplitude 5.
D. Phase difference between two motions is `pi//2`

Answer

To solve the problem, we need to analyze the given equations of motion for the particle in the x and y directions:

1. **Equations of Motion**:
- \( x = 3 \sin(100 \pi t) \)
- \( y = 4 \sin(100 \pi t) \)

2. **Finding the Relationship Between x and y**:
To find the relationship between \( x \) and \( y \), we can take the ratio of the two equations:
\[ \frac{y}{x} = \frac{4 \sin(100 \pi t)}{3 \sin(100 \pi t)} \]
This simplifies to:
\[ \frac{y}{x} = \frac{4}{3} \]
Rearranging gives us:
\[ y = \frac{4}{3} x \]
This equation represents a straight line with a slope of \( \frac{4}{3} \).

3. **Identifying the Motion**:
Since the relationship between \( y \) and \( x \) is linear, the motion of the particle is along a straight line, confirming that option B is correct.

4. **Finding the Resultant Amplitude**:
The amplitudes of the simple harmonic motions in the x and y directions are:
- Amplitude in x direction, \( A_x = 3 \)
- Amplitude in y direction, \( A_y = 4 \)

The resultant amplitude \( A \) can be calculated using the Pythagorean theorem, as the two motions are perpendicular to each other:
\[ A = \sqrt{A_x^2 + A_y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the resultant amplitude is 5 units, confirming that option C is correct.

5. **Finding the Phase Difference**:
The phase of both motions is the same:
- Phase of \( x \): \( 100 \pi t \)
- Phase of \( y \): \( 100 \pi t \)

The phase difference \( \Delta \phi \) is:
\[ \Delta \phi = 100 \pi t - 100 \pi t = 0 \]
Since the phase difference is 0, option D is incorrect.

### Summary of Correct Options:
- Option B: The motion of the particle will be a straight line with slope \( \frac{4}{3} \) (Correct).
- Option C: The amplitude of the resultant motion is 5 units (Correct).
- Option A: Incorrect (the motion is not an ellipse).
- Option D: Incorrect (the phase difference is not \( \frac{\pi}{2} \)).