Particle Moving in an x-y Plane

The motion of a particle in the x-y plane is given by the equations x = asin(Wt) and y = acos(Wt), where a is the amplitude, W is the angular frequency, and t is time.

(i) Elliptical Path:
To determine if the particle follows an elliptical path, we need to analyze the equations for x and y.
- The equation for x is x = asin(Wt), which represents a sinusoidal function. As t changes, the particle moves back and forth along the x-axis.
- The equation for y is y = acos(Wt), which also represents a sinusoidal function. As t changes, the particle moves back and forth along the y-axis.

Since the motion of the particle along both x and y directions is sinusoidal, the path traced by the particle is an ellipse. The major axis of the ellipse is along the x-axis, and the minor axis is along the y-axis.

(ii) Circular Path:
To determine if the particle follows a circular path, we need to check if the equations for x and y satisfy the equation of a circle.
- The equation for a circle is x^2 + y^2 = r^2, where r is the radius.
- Substituting the given equations x = asin(Wt) and y = acos(Wt) into the equation for a circle, we get (asin(Wt))^2 + (acos(Wt))^2 = r^2.
- Simplifying the equation, we have a^2(sin^2(Wt) + cos^2(Wt)) = r^2.
- Using the trigonometric identity sin^2θ + cos^2θ = 1, we get a^2 = r^2.

Therefore, the equation satisfies the equation of a circle, indicating that the particle follows a circular path with a radius equal to the amplitude a.

(iii) Parabolic Path:
To determine if the particle follows a parabolic path, we need to analyze the equations for x and y.
- The equation for x is x = asin(Wt), which represents a sinusoidal function. As t changes, the particle moves back and forth along the x-axis.
- The equation for y is y = acos(Wt), which also represents a sinusoidal function. As t changes, the particle moves back and forth along the y-axis.

Since the motion of the particle along both x and y directions is sinusoidal, the path traced by the particle is not a parabola. Therefore, the particle does not follow a parabolic path.

(iv) Straight Line Path Inclined Equally to x and y-axes:
To determine if the particle follows a straight line path inclined equally to the x and y-axes, we need to analyze the equations for x and y.
- The equation for x is x = asin(Wt), which represents a sinusoidal function. As t changes, the particle moves back and forth along the x-axis.
- The equation for y is y = acos(Wt), which also represents a sinusoidal function. As t changes, the particle moves back and forth along the y-axis.

Since the motion of the particle along both x and y directions is sinusoidal, the path traced by the particle is not a straight line inclined equally