Particle Motion in the x-y Plane

The particle moves in the x-y plane according to the given rules:

x = a * sin(wt)
y = a * cos(wt * pi/2)

Let's analyze the motion of the particle in detail.

1. Sinusoidal Motion in the x-direction
The x-coordinate of the particle is given by x = a * sin(wt), where a represents the amplitude of the motion and w is the angular frequency. The sine function produces a periodic motion where the particle oscillates back and forth along the x-axis. The amplitude determines the maximum displacement of the particle from the equilibrium position, while the angular frequency controls the speed at which the particle oscillates.

2. Cosinusoidal Motion in the y-direction
The y-coordinate of the particle is given by y = a * cos(wt * pi/2), where a represents the amplitude of the motion and w is the angular frequency. The cosine function produces a periodic motion where the particle oscillates back and forth along the y-axis. The amplitude determines the maximum displacement of the particle from the equilibrium position, while the angular frequency controls the speed at which the particle oscillates. The presence of pi/2 in the argument of cosine function introduces a phase shift of pi/2 radians (or 90 degrees) compared to the x-motion.

3. Relationship between x and y
By substituting the given expressions for x and y into the equations, we can establish the relationship between them. We have:

x = a * sin(wt)
y = a * cos(wt * pi/2)

Comparing the equations, we can observe that the x-coordinate and y-coordinate have a phase difference of pi/2 radians (or 90 degrees). This means that as the particle moves along the x-axis, the corresponding y-coordinate lags behind by pi/2 radians.

4. Trajectory of the Particle
To understand the trajectory of the particle, we can plot its motion in the x-y plane. The particle moves in a circular path with a radius of magnitude a. The center of the circle is located at the origin (0,0) of the coordinate system. As time progresses, the particle completes one full revolution around the circle, maintaining a constant distance from the center.

Conclusion
In conclusion, the particle moves in the x-y plane according to the given rules, exhibiting sinusoidal motion along the x-axis and cosinusoidal motion along the y-axis. The x-coordinate and y-coordinate have a phase difference of pi/2 radians. The particle traces a circular trajectory with a radius of magnitude a.