Problem 19 - Thomas Calculus

Question

The involute of a circle If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end \(P\) traces an involute of the circle \(x^{2}+y^{2}=1\) and the tracing point circle in question is the circle \(x^{2}+y^{2}=1\) and the tracing point starts at \((1,0) .\) The unwound portion of the string is tangent to the circle at \(Q,\) and \(t\) is the radian measure of the angle from the positive \(x\) -axis to segment \(O Q .\) Derive the parametric equations $$ x=\cos t+t \sin t, \quad y=\sin t-t \cos t, \quad t>0 $$ of the point \(P(x, y)\) for the involute. Graph cannot copy

Step-by-Step Solution

Answer

Parametric equations are \(x=\cos t+t \sin t\), \(y=\sin t-t \cos t\).

Step 1: Understanding the Problem

We begin by understanding that an involute is created by unwinding a string from a circle, maintaining the string's tautness, and tracing its end. The circle has the equation \(x^2 + y^2 = 1\) with the starting point \((1, 0)\). We need to find parametric equations for the trajectory of the string's end.

Step 2: Analyzing the Geometry

As the string unwinds, its length from point \(Q\) (tangent point) to \(P\) (end-point) is equal to the arc length. The point \(Q\) is determined by the angle \(t\), with coordinates \((\cos t, \sin t)\). The angle \(t\) represents the arc from the positive \(x\)-axis to the tangent point \(Q\) on the unit circle.

Step 3: Deriving the Tangent Vector

The unit tangent vector at point \(Q\) is perpendicular to the radius \((\cos t, \sin t)\), meaning it has components \((-\sin t, \cos t)\). The length of this vector from \(Q\) to \(P\) is equal to the arc length \(t\) since it's being unwound from the circle.

Step 4: Constructing the Parametric Equations

To derive the point \(P(x, y)\), we start from \(Q(\cos t, \sin t)\) and move \(t\) units along the tangent vector \((-\sin t, \cos t)\). Therefore, we calculate:\[x = \cos t + t(-\sin t) = \cos t + t \sin t\]\[y = \sin t + t(\cos t) = \sin t - t \cos t\]

Step 5: Verifying the Equations

Finally, verify our parametric equations by considering the special case when \(t = 0\), which gives the starting point \((1,0)\), matching the initial condition. As \(t\) increases, \(P\) moves along the involute path, confirming the equations.

Key Concepts

Parametric Equations

When dealing with problems of geometric shapes and curves, parametric equations play a crucial role. They define a set of equations that express coordinates as functions of a parameter, usually denoted as \(t\). This parameter often represents time or an angle, allowing us to describe complex paths and patterns in a straightforward way.

For the involute of a circle, we work with parametric equations to represent the path traced by the end of a string as it unwinds from the circle. The traced path is characterized by the angle \(t\) and the motion around the unit circle, starting at point \((1, 0)\).

• The starting coordinates are \(x = \,\cos t + t \sin t\).
• Meanwhile, \(y = \,\sin t - t \cos t\).

These equations account for both the rotation caused by the circle's geometry and the linear extension as the string unwraps, ensuring the path of the end-point \(P(x, y)\) is accurately traced. It combines rotational motion (\(\cos t\) and \(\sin t\)) with an extension that scales with \(t\). This dual nature is why parametric equations are vital in accurately describing the motion involved.

Arc Length

The concept of arc length is pivotal when dealing with curves like the involute of a circle. Arc length refers to the distance along a curved path, in this case, the path that the unwound string travels as it unwraps from the circle.

For a unit circle, the arc length from the positive x-axis to any point \(Q\) on the circumference is measured by the angle \(t\), expressed in radians. This is because each radian represents a length equivalent to the radius, which for a unit circle is 1. Therefore, as \(t\) varies, it directly corresponds to the amount of string that has been unwound.

• The more the string unwinds (the larger \(t\)), the longer the arc.
• This transition translates into the distance point \(P\) moves away from its starting point \((1, 0)\).

The crucial takeaway is how the angle \(t\), as it increases, denotes both the distance travelled along the circle and the length of string that has been released, intimately linking with parametric expressions of the path.

Unit Circle Geometry

Unit circle geometry underpins the entire problem of the involute of a circle. A unit circle is a circle with a radius of 1, centered at the origin \((0,0)\) in the coordinate plane, described by the equation \(x^2 + y^2 = 1\). In this problem, it serves as the foundational geometry from which the string is unwound.

The key aspects of unit circle geometry are:

• The coordinates of any point on the circle can be represented as \( (\cos t, \sin t) \), where \(t\) is the angle formed by the positive x-axis and the line segment from the origin to the point.
• The circle's symmetry ensures routes and angles are easily predictable, which is crucial in defining the correct path of the involute.

Understanding this geometry allows us to work seamlessly with the trigonometric identities essential in deriving the path of \(P(x, y)\). The tangential and normal properties of the circle enable clear resolution of the problem, by allowing the determination of tangent vectors necessary for expressing the direction and path traveled by point \(P\). Ultimately, unit circle geometry provides both a simple yet powerful framework for analyzing the mechanics of the involute trace.