Problem 73 - Calculus Early Transcendentals

Question

A string is wound around a circle and then unwound while being held taut. The curve traced by the point \(P\) at the end of the string is called the involute of the circle. If the circle has radius \(r\) and center \(O\) and the initial position of \(P\) is \((r, 0),\) and if the parameter \(\theta\) is chosen as in the figure, show that parametric equations of the involute are $$x=r(\cos \theta+\theta \sin \theta) \quad y=r(\sin \theta-\theta \cos \theta)$$

Step-by-Step Solution

Answer

The parametric equations of the involute are given by \[ x = r(\cos \theta + \theta \sin \theta), \; y = r(\sin \theta - \theta \cos \theta) \].

Step 1: Understand the Problem

We need to find the parametric equations of the involute of a circle starting from the point \((r, 0)\). The involute is the path traced by the end of a taut string unwound from the circle, parameterized by \(\theta\).

Step 2: Define Relevant Geometry

Consider a circle of radius \(r\) centered at the origin \(O\). The point \(P\) starts at \((r, 0)\) and follows the circle as the string unwinds. The length of the string unwound at angle \(\theta\) is equal to the arc on the circle, which is \(r\theta\).

Step 3: Determine Coordinates using Geometry

To find the coordinates of \(P\) as the string extends, think about the new position of the point \(P\). The angle \(\theta\) shifts the point from \((r, 0)\) along the circle to initial position due to the unwound arc, which rotates the coordinates clockwise with respect to the circle's tangent lines. Our task is to find this position geometrically in terms of \(\theta\).

Step 4: Set Up Parametric Equations

To obtain the new position \((x, y)\), we start with the circle's parametric equations \((r\cos \theta, r\sin \theta)\) for angle \(\theta\). The tangent to this position is perpendicular to the radius vector. As the string unwinds, the tracing point \(P\) has moved \(r\theta\) further along the line tangent to the circle.

Step 5: Derive x-coordinate

The x-coordinate is adjusted from the original \(r\cos \theta\) by adding the component of the tangent vector extended by \(r\theta\) on the x-axis direction. This involves component-wise addition with tangent length: \[ x = r\cos(\theta) + r\theta \sin(\theta) \]

Step 6: Derive y-coordinate

Similarly, to find the y-coordinate, we start from \(r\sin \theta\) and adjust using the tangent vector component on the y-axis: \[ y = r\sin(\theta) - r\theta \cos(\theta) \]

Step 7: Formulate Complete Parametric Equations

After adjusting both coordinates using the movement along the tangent, we can now write the full parametric equations for \(P\): \[ x = r(\cos \theta + \theta \sin \theta) \] \[ y = r(\sin \theta - \theta \cos \theta) \]

Step 8: Verify Through Consistency

Check if these equations satisfy the problem's scenario visually or by plugging in known values to confirm the mathematical tracing of the string's path as intended.

Key Concepts

Parametric Equations

Parametric equations are a powerful way to represent curves. Instead of expressing variables such as x and y independently, parametric equations use a parameter, typically denoted as \( \theta \), to define both x and y in terms of this parameter.
This is especially useful in problems involving motion along a path, like the involute of a circle. By using parametric equations, we can effectively describe the position of point \( P \) as it traces the curve as the string unwinds.
For the involute of a circle, the parametric equations are derived based on the geometric properties of the circle and the linear motion of the taut string's end. These equations are given by:

• \( x = r(\cos \theta + \theta \sin \theta) \)
• \( y = r(\sin \theta - \theta \cos \theta) \)

By understanding parametric equations, one can graph or model complex curves accurately, showing how crucial they are in calculus and geometry.

Geometry

Geometry plays a foundational role in understanding the involute of a circle. The circle’s radius \( r \) is a key element from which the string is initially wound. When the string unwinds, it creates a path on a plane, tracing the involute.
Visualizing this, the string initially tangential to the circle at point \( P = (r, 0) \) unfurls. The point \( P \), therefore, moves not in a linear direction, but in a path dependent on the circle's curvature.
Key geometric considerations include:

• The initial position of the point \( P \) on the circle, rotated as the string unwinds.
• The arc length equivalent to the string length unwound at any angle \( \theta \), given as \( r\theta \).
• The radial line from the circle's center to any point on its edge, which forms a right angle with the tangent line for that point.

This geometric understanding helps transform the problem from abstract geometry into tangible mathematics using parametric equations.

Calculus Concepts

Calculus introduces concepts like arc length and differentiation, which are invaluable in deriving the involute's equations. When dealing with involutes, calculus helps in understanding how a string unwinding along a circle translates into continuous motion.
The unwinding string can be likened to evaluating how a space curve evolves. With calculus concepts, we can:

• Calculate changes in position on the x and y coordinates as functions of \( \theta \).
• Use derivatives to analyze slope or rate of change at any point of the path.

The employment of derivatives when adjusting the traditional parametric form \((r \cos \theta, r \sin \theta)\) helps include the motion along the tangent, effectively modifying the original parametric form to account for the effect of the unwound tangent line over the circle’s circumference. This is a clear example of how calculus extends and modifies geometric interpretations into functional equations.

Trigonometry

Understanding trigonometry is crucial for analyzing the involute's parametric equations. These equations involve trigonometric functions \( \sin \theta \) and \( \cos \theta \), which describe the circular motion of point \( P \) as it follows the circle’s circumference.
Trigonometry enables us to:

• Express the position of \( P \) using \( \cos \theta \) and \( \sin \theta \) in terms of angle \( \theta \).
• Understand the rotational components that shift \( P \) from its tangential starting point to a new position.
• Decompose the unwinding string's movement into x and y components, attributing changes to trigonometric functions multiplied by the angle \( \theta \).

Using these trigonometric identities and their properties, we can clearly define how the string’s end moves away from its initial tangency along a detailed path, making trigonometry a powerful tool in solving such rotational motion problems.