Question

Circles Find parametrizations to model the motion of a particle that starts at \((a, 0)\) and traces the circle \(x^{2}+y^{2}=a^{2}, a>0,\) as indicated. (a) once clockwise }} & {\text { (b) once counterclockwise }} \\ {\text { (c) twice clockwise }} & {\text { (d) twice counterclockwise }}\end{array}$

Short answer

(a) Parametrization: \(x=a\cos t\), \(y=-a\sin t\); range for \(t\): [0, \(2\pi\)]. (b) Parametrization: \(x=a\cos t\), \(y=a\sin t\); range for \(t\): [0, \(2\pi\)]. (c) Parametrization: \(x=a\cos t\), \(y=-a\sin t\); range for \(t\): [0, \(4\pi\)]. (d) Parametrization: \(x=a\cos t\), \(y=a\sin t\); range for \(t\): [0, \(4\pi\)].

Step-by-step solution

Title

Parametrize the circle with direction. For clockwise motion, use \(x=a\cos t\), \(y=-a\sin t\), assuming \(t\) ranges from 0 to \(2\pi\). For counterclockwise motion, use \(x=a\cos t\), \(y=a\sin t\), with the same \(t\) range.

Title

Parametrize for once and twice movements. The range of \(t\) for once movement is [0, \(2\pi\)]. For twice movement, the range becomes [0, \(4\pi\)].

Title

Summarize the parametrizations. Based on step 1 and step 2, one can get the parametrization for each case. (a) \(x=a\cos t\), \(y=-a\sin t\) for \(t\in [0, 2\pi]\), (b) \(x=a\cos t\), \(y=a\sin t\) for \(t\in [0, 2\pi]\), (c) \(x=a\cos t\), \(y=-a\sin t\) for \(t\in [0, 4\pi]\), (d) \(x=a\cos t\), \(y=a\sin t\) for \(t\in [0, 4\pi]\).

Key concepts

Particle Motion in Calculus

Understanding particle motion in calculus can be quite fascinating. It involves the study of how an object, referred to as a 'particle', moves along a path over time. To describe this motion mathematically, we use parametric equations that specify the particle’s position in terms of a third variable, often time.

Consider a particle moving along a circular path. We need two equations to describe its position, one for the horizontal (x-coordinate) and one for the vertical (y-coordinate) components of motion. These equations are linked by a common parameter, typically designated as 't'. This parameter 't' not only indicates a point in time but also corresponds to an angle measure in radians when dealing with circular motion—making the connection between time and positional change explicit.

Trigonometric Parametrization

Basics of Trigonometric Parametrization

Parametrizing equations using trigonometric functions is a powerful way to describe circular motion. It's rooted in the fact that the functions sine and cosine inherently trace a circle on the unit plane. When a particle moves along the circumference of a circle, its coordinates can be expressed as functions of sine and cosine.

For a circle of radius 'a', the general parametric form is \(x = a\cos(t)\) and \(y = a\sin(t)\).These equations allow us to describe every point along the circle’s edge as 't'—our parameter, which often represents time—varies. Trigonometric parametrization makes calculating the position of a particle at any given time straightforward and is a fundamental aspect of analyzing periodic motion.

Parametrizations for Clockwise and Counterclockwise Motion

The direction of a particle's motion along a circle can be either clockwise or counterclockwise, and this is reflected in its parametrization. The choice of trigonometric functions and their signs are critical to represent the correct direction.

For a clockwise rotation, which is the opposite direction to how angles are typically measured, the parametric equations adjust. Movement to the right along the x-axis begins in the positive direction, but as the rotation is clockwise, the y-component should decrease initially. This leads to the equations \(x = a\cos(t)\) and \(y = -a\sin(t)\).In contrast, counterclockwise motion aligns with the standard angle measurement direction. Both the x and y components of the position increase initially, resulting in the equations \(x = a\cos(t)\) and \(y = a\sin(t)\).

• For clockwise motion, we use negative sine.
• For counterclockwise motion, we use positive sine.

Parametrizing Circular Motion in Calculus

Circular motion parametrization in calculus is not just restricted to defining direction but also the extent of the motion. Whether a particle completes one rotation or multiple rotations around the circle can be determined by the range of the parameter 't'.

Single vs. Multiple Rotations

For a single rotation, the range of 't' spans from 0 to \(2\pi\) radians, which equates to one full circle. When a particle traverses the path twice, the range of 't' extends to \(4\pi\) radians. It’s similar to the hands of a clock completing one full cycle (from 12 back to 12) or two full cycles.
Here’s how we summarize the parametrizations based on the number of rotations:

• For once clockwise: \(x=a\cos t\), \(y=-a\sin t\) for \(t\in [0, 2\pi]\)
• For once counterclockwise: \(x=a\cos t\), \(y=a\sin t\) for \(t\in [0, 2\pi]\)
• For twice clockwise: \(x=a\cos t\), \(y=-a\sin t\) for \(t\in [0, 4\pi]\)
• For twice counterclockwise: \(x=a\cos t\), \(y=a\sin t\) for \(t\in [0, 4\pi]\)

Understanding these parametrizations not only helps in visualizing particle motion but also lays the groundwork for solving more complex problems in calculus and physics.