Sketch the path of a particle that moves in the xy plane according to x = x_m cos (ωt - π/2) and y = 2x_m cos ωt.

To sketch the path of a particle moving in the xy-plane with the given equations, we need to analyze the relationships between the x and y coordinates. The equations provided are:

• x = x_m cos(ωt - π/2)
• y = 2x_m cos(ωt)

First, let's simplify the equation for x. The term cos(ωt - π/2) can be rewritten using the trigonometric identity:

cos(θ - π/2) = sin(θ)

Thus, we can express x as:

x = x_m sin(ωt)

Now, substituting this expression for x into the equation for y gives us:

y = 2x_m cos(ωt)

Next, we can relate x and y directly by eliminating the time variable (t). Since we have:

• x = x_m sin(ωt)
• y = 2x_m cos(ωt)

We can express sin(ωt) and cos(ωt) in terms of x and y. From the first equation, we have:

sin(ωt) = x / x_m

From the second equation, we can express cos(ωt) as:

cos(ωt) = y / (2x_m)

Now, we can use the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

Substituting our expressions for sin(ωt) and cos(ωt) gives us:

(x / x_m)² + (y / (2x_m))² = 1

Multiplying through by (x_m)² to eliminate the denominators results in:

x² + (y² / 4) = x_m²

This equation represents an ellipse centered at the origin (0, 0) with a semi-major axis of length x_m along the x-axis and a semi-minor axis of length 2x_m along the y-axis. To visualize this, we can sketch the ellipse:

Sketching the Ellipse

1. **Identify the axes**: The semi-major axis is along the x-axis, extending from -x_m to x_m. The semi-minor axis is along the y-axis, extending from -2x_m to 2x_m.

2. **Plot the points**: Mark the points (x_m, 0), (-x_m, 0), (0, 2x_m), and (0, -2x_m) on the coordinate plane.

3. **Draw the ellipse**: Connect these points smoothly to form an oval shape, ensuring that it is wider along the x-axis and narrower along the y-axis.

Understanding the Motion

The particle's motion is periodic, oscillating between the maximum and minimum values defined by the ellipse. As time progresses, the particle moves along this elliptical path, with its position determined by the cosine and sine functions, which are inherently periodic. This means that the particle will repeat its path indefinitely, tracing the same ellipse over time.

In summary, the path of the particle is an ellipse defined by the equation x² + (y² / 4) = x_m², with the particle moving in a periodic manner as time varies. This relationship between x and y coordinates beautifully illustrates the connection between trigonometric functions and geometric shapes in motion.