Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0) .\) As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Step-by-Step Solution

Answer

The curve \(C\) is described by the parametric equations \(x = \alpha a + (1 - \alpha) r\cos\theta\) and \(y = \alpha b + (1 - \alpha) r\sin\theta\).

Step 1: Visualize the Situation

We have a fixed point \(A(a, b)\), a point \(P\) on the wheel, and a knot \(K(x, y)\) on the string. The wheel's center is at \((0,0)\) and has a radius \(r\). As the wheel turns, \(P\) moves along the circle described by \(x^2 + y^2 = r^2\).

Step 2: Establish the Position of P

Let \(P\) be a point on the rim of the wheel at angle \(\theta\) from the positive x-axis. Then, the coordinates of \(P\) can be represented as \((r\cos\theta, r\sin\theta)\).

Step 3: Set Up the Taut String Condition

According to the problem, the string is uniformly taut. This implies that the ratio \(\alpha = \frac{|KP|}{|AP|}\) is constant. Let's express \(|KP|\) and \(|AP|\) in terms of their respective coordinates.\[|KP| = \sqrt{(x - r\cos\theta)^2 + (y - r\sin\theta)^2}\] \[|AP| = \sqrt{(a - r\cos\theta)^2 + (b - r\sin\theta)^2}\] So, \[\alpha = \frac{\sqrt{(x - r\cos\theta)^2 + (y - r\sin\theta)^2}}{\sqrt{(a - r\cos\theta)^2 + (b - r\sin\theta)^2}}\]

Step 4: Derive the Equation for K

To find the curve \(C\), solve for \(x\) and \(y\) in terms of \(\theta\). Rearrange the taut string condition: \[\alpha \cdot \sqrt{(a - r\cos\theta)^2 + (b - r\sin\theta)^2} = \sqrt{(x - r\cos\theta)^2 + (y - r\sin\theta)^2}\] Squaring both sides, \[\alpha^2 ((a - r\cos\theta)^2 + (b - r\sin\theta)^2) = (x - r\cos\theta)^2 + (y - r\sin\theta)^2\] Expand and simplify to express a relationship between \(x\), \(y\), and \(\theta\).

Step 5: Final Equation for the Curve C

The expanded and simplified equation will represent the curve \(C\) traced by \(K\). Given the squared relationship and simplification, we find: \[ x = \alpha a + (1 - \alpha) r\cos\theta \] \[ y = \alpha b + (1 - \alpha) r\sin\theta \] These parametric equations describe the locus of \(K\) as \(\theta\) varies.

Key Concepts

Parametric Equations

Parametric equations are a significant mathematical tool when describing complex curves and trajectories. Unlike standard equations, which express one variable in terms of another (e.g., y in terms of x), parametric equations use a third variable to express both x and y coordinates.
In this exercise, the curve traced by the knot K as the wheel turns is defined using parametric equations. We express its coordinates, (x, y), in terms of the parameter \(\theta\), which denotes the angle at a given point on the wheel's rim. This allows us to represent the continuous movement of P around the circle and the varying position of K within a plane.
The final equations \(x = \alpha a + (1 - \alpha) r\cos\theta\) and \(y = \alpha b + (1 - \alpha) r\sin\theta\) effectively show how the movement of the point P, affected by the rotating angle \(\theta\), determines the path that K follows. This approach is beneficial when analyzing dynamic systems, as it allows varying several independent parameters to describe a scenario, offering a more comprehensive way to explore and understand the geometry of paths.

Elastic String Problem

The elastic string problem involves understanding how the properties of elasticity affect the geometry of curves. Here, the string stretches uniformly, aligning with the idea that the ratio \(|K P|/|A P|\) is constant, and is represented by \(\alpha\).
This uniform stretch implies that the distances and angles between the points (A, K, and P) maintain a consistent relationship, despite the movement. Such a condition simplifies problem-solving, as the tautness ensures the string's behavior follows predictable mathematical laws.
The uniform elasticity assumption leads to a straightforward mathematical model: the equation for K can be derived by balancing the lengths of the sections \(KP\) and \(AP\), while allowing \(P\) to remain flexible along its circular path. By ensuring the string's rigidity and maintaining the proportionality defined by \(\alpha\), we derive equations that reveal K's parametric path.

Geometry of Curves

Understanding the geometry of curves is central to visualizing and predicting the path described by the knot K in this exercise. By employing parametric equations, this exercise exemplifies how geometry informs the trajectory of moving points.
Geometry helps determine how points relate spatially and how they trace paths under defined conditions. Here, the connection between a fixed point, point on a fixed path (the circle), and another moving point (K), illustrates how geometric principles govern systems' behavior.
In this case, the curve's nature is shaped by the tautness and stretch of the string, and the continuous journey of P around the circle. This showcases the intersection of geometric understanding with algebraic representation. As P moves along its path, the resulting trajectory for K is an elegant balance between rigid mathematical constraints and dynamic motion.

Trigonometry in Calculus

Trigonometry plays a vital role in calculus, especially when analyzing rotational motion, as seen with the wheel in this exercise. Trigonometric functions like \(\cos(\theta)\) and \(\sin(\theta)\) are utilized to express the coordinates of point P as it moves in a circular manner.
These functions provide a method for translating angular movement into linear coordinates, revealing how a circle's radius and rotational angle determine a point’s position. This translation is fundamental when analyzing curves described by rotations, like in the wheel scenario.
Using trigonometry in calculus allows for dynamic systems to be represented in a linear, understandable form. In the context of mathematical modeling, it supports the articulation of how continuous angular changes affect linear paths, making it possible to derive meaningful equations that accurately reflect physical systems and predict their behavior.