Problem 33 - Calculus Early Transcendentals

Question

Find parametric equations for the path of a particle that moves along the circle \(x^{2}+(y-1)^{2}=4\) in the manner described. (a) Once around clockwise, starting at \((2,1)\) (b) Three times around counterclockwise, starting at \((2,1)\) (c) Halfway around counterclockwise, starting at \((0,3)\)

Step-by-Step Solution

Answer

(a) Use parametric: \(x = 2\cos(t), y = 1 - 2\sin(t), t \in [0, 2\pi]\). (b) Use \(x = 2\cos(t), y = 1 + 2\sin(t), t \in [0, 6\pi]\). (c) Use \(x = 2\cos(t), y = 1 + 2\sin(t)\), starting from \((0,3)\) and adjust \(t\).

Step 1: Identify the Circle Equation

The given circle equation is \(x^2 + (y-1)^2 = 4\). This represents a circle centered at \((0, 1)\) with a radius of \(2\).

Step 2: Parametric Equations for a Full Circle

The standard parametric equations for a particle moving counterclockwise along a circle centered at \((h, k)\) with radius \(r\) are \(x = h + r\cos(t)\) and \(y = k + r\sin(t)\). Using the center \((0, 1)\) and radius \(2\), we have \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\).

Step 3: Part (a) - Clockwise Direction

To move clockwise, reverse the direction of \(t\). Thus, parametric equations become \(x(t) = 2\cos(-t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(-t) = 1 - 2\sin(t)\). Since it goes once around, \(t\) varies from \(0\) to \(2\pi\).

Step 4: Part (b) - Three Times Counterclockwise

To move counterclockwise, use the standard direction. The parametric equations remain \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\). For three complete rotations, \(t\) ranges from \(0\) to \(6\pi\).

Step 5: Confirm Starting Points

Check starting points for each part: \((2, 1)\) for \(t = 0\). Substitute \(t = 0\) to confirm: both equations yield \((2, 1)\), confirming the starting point scenarios.

Step 6: Part (c) - Halfway Counterclockwise

Use the parametric equations \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\). Halfway around is \(\pi\), so \(t\) ranges from \(0\) to \(\pi\). Starting from \((0, 3)\), find an appropriate \(t\): at \(t = \frac{\pi}{2}\), \(x = 0\), and \(y = 3\). Modify \(t\) range accordingly.

Key Concepts

Circle Equation

The equation of a circle is a fundamental concept in mathematics, frequently appearing in geometry and algebra. A circle's equation, such as \(x^2 + (y-1)^2 = 4\), provides essential information about the circle's geometric properties. Here, the equation indicates a circle centered at the point \((0, 1)\) with a radius of \(2\). The general formula for a circle in the Cartesian plane is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. This particular circle equation reveals that the circle is shifted 1 unit upwards from the origin in the \(y\)-direction.

Trigonometric Functions

Trigonometric functions like cosine and sine are crucial in expressing circular motion, making them indispensable in parametric equations. These functions oscillate and provide a way to model periodic phenomena. When describing a particle's position on a circle, the function \(x(t) = r\cos(t)\) and \(y(t) = r\sin(t)\) allows us to calculate the particle's coordinates dynamically as it moves along the circle's path. The angles in trigonometric functions, denoted by \(t\), are measured in radians, which is a natural fit for circular motion due to their relation to the unit circle.

Particle Motion

Describing a particle's motion involves understanding how its position changes over time. In this context, parametric equations offer a powerful way to articulate this concept mathematically. With the circle equation previously identified, you can model the movement of a particle moving along the circle's edge. For example, parametric equations like \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\) define the path of the particle as it progresses around the circle over time. The variable \(t\) typically represents time, offering insight into the dynamical aspect of the particle's trajectory.

Parametrization

Parametrization involves defining a set of equations to express a curve or shape in terms of another parameter, such as time. This technique is particularly helpful for understanding complex motions or shapes that cannot be readily captured by simple equations. In the circle example, the parameter \(t\) is used to describe the particle's path around the circle. By adjusting \(t\), the equations \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\) can reflect different positions on the circle as time progresses, thus effectively mapping the circular path of the particle in a comprehensible manner.

Counterclockwise and Clockwise Directions

Directionality is a critical component when discussing particle motion around a circle. Moving counterclockwise or clockwise can be seen as analogous to forward and reverse in our understanding of rotational movement. To describe counterclockwise movement, which is the standard direction, the equations \(x(t) = 2\cos(t)\) and \(y(t) = 1 + 2\sin(t)\) suffice. However, to express motion in the clockwise direction, we must modify the equations slightly. This alteration involves replacing \(t\) with \(-t\) in the parametric equations, changing \(y(t) = 1 + 2\sin(t)\) to \(y(t) = 1 - 2\sin(t)\). Thus, by adapting these equations, one can clearly specify the intended direction of travel along the circle.