Problem 27 - Thomas Calculus

Question

Find parametric equations and a parameter interval for the motion of a particle starting at the point \((2,0)\) and tracing the top half of the circle \(x^{2}+y^{2}=4\) four times.

Step-by-Step Solution

Answer

Parametric equations: \(x(t) = 2\cos(t), y(t) = 2\sin(t)\) with \(0 \leq t \leq 4\pi\).

Step 1: Understand the Problem

We need to find the parametric equations for a particle that starts at the point \((2,0)\) and traces the top half of the circle \(x^2 + y^2 = 4\). The circle is centered at \((0,0)\) with a radius of \(2\). The particle traces this path four times.

Step 2: Set up the Parametric Equations

A circle with radius \(r\) and centered at \((0,0)\) can be parametrically defined as:\[x(t) = r \cos(t), \y(t) = r \sin(t).\]Since our radius \(r = 2\), the equations become:\[x(t) = 2 \cos(t), \y(t) = 2 \sin(t).\]To ensure the top half is traced, we consider \(0 \leq t \leq \pi\).

Step 3: Adjust for Starting Point

The starting point in our setup is \((2,0)\) which corresponds to \(t = 0\), aligning with the parametric definition since \(x(0) = 2\cos(0) = 2\).

Step 4: Determine Parameter Interval for Four Circulations

To trace the top half of the circle four times, the parameter \(t\) will cover the interval \([0, 4\pi]\). Each traversal of \([0, \pi]\) covers the top half of the circle once.

Step 5: Final Parametric Equations and Interval

The parametric equations are:\[x(t) = 2\cos(t), \y(t) = 2\sin(t)\quad \text{for} \quad 0 \leq t \leq 4\pi.\] This setup ensures the particle traces the top half of the circle four times.

Key Concepts

Circle Equation

The equation of a circle is a fundamental concept in geometry. For a circle centered at the origin \((0,0)\), the equation can be expressed as \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In our problem, the circle has a radius of 2, so the equation becomes \(x^2 + y^2 = 4\).- **Components**: - \(x^2\) and \(y^2\): These represent the square of the respective coordinates. - \(r^2\): Represents the square of the radius of the circle.Understanding this allows us to explore how different positions \((x, y)\) fit within the curve defined by the circle's equation to make sure we map all points correctly.

Trigonometric Functions

Trigonometric functions are essential for parametric equations of circles. In this case, we use \(\cos(t)\) and \(\sin(t)\) to describe the circular motion parametrically. These trigonometric functions relate angles to the coordinates on a circle.- **Cosine Function \(\cos(t)\)**: - Determines the horizontal position \(x(t) = r\cos(t)\).- **Sine Function \(\sin(t)\)**: - Determines the vertical position \(y(t) = r\sin(t)\).This relationship helps us express the position of a particle moving around a circle smoothly by changing the parameter \(t\).

Parameter Interval

The parameter \(t\) is a crucial part of parametric equations. Its interval defines how far along the path the particle will move. By varying \(t\), we control the length and portions of the circle covered.For this exercise, the parameter interval is initially \(0 \leq t \leq \pi\) for tracing the top half ofthe circle once. Since the top half is covered with each complete cycle of the interval, repeating this interval four times results in \(0 \leq t \leq 4\pi\). Hence, the complete path is traced multiple times by the particle.

Particle Motion

Particle motion describes how a particle progresses along a path over time. In our scenario, the particle starts at \((2,0)\) and moves along the top half of the circle.- **Start Point**: - The particle commences at \((2,0)\) when \(t = 0\).- **Path**: - It follows the path defined by the parametric equations \(x(t) = 2\cos(t)\) and \(y(t) = 2\sin(t)\).Understanding particle motion helps us appreciate how a point's coordinates change continuously over intervals of time or parameter \(t\). This concept is especially useful in physics and computer graphics.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. In this problem,our circle has a radius of 2, which determines the extent and size of the circle but does not affectits parametric definitions aside from being the scaler.- **Role in Parametric Equations**: - It scales the trigonometric function outputs so the circle has the correct size: - \(x(t) = 2\cos(t)\) - \(y(t) = 2\sin(t)\)Being aware of the radius allows for accurate modeling of the circle and adjusting parameters to fit within the set boundaries of our problem.

Top Half of a Circle

Tracing only the top half of a circle involves careful manipulation of the parameter \(t\) to ensureonly the desired part of the circle is covered. The top half corresponds to the interval where the sinefunction \(\sin(t)\) is non-negative.- **Interval**: - Encountering \(0 \leq t \leq \pi\) covers the top half of the circle once as both the x and y coordinates keep their signs positive reflecting the upper semicircle path.By selecting \(t\) carefully, you can create and define specific sections of a circle and repeat them as needed, as shown in this problem that repeats the top half path four times by extending the parameter interval to \(0 \leq t \leq 4\pi\).