Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

. A point \(P\) on the rim of a wheel of radius \(a\) is initially at the origin. As the wheel rolls to the right along the \(x\)-axis, \(P\) traces out a curve called a cycloid (see Figure 18). Derive parametric equations for the cycloid as follows. The parameter is \(\theta\). (a) Show that \(\overline{O T}=a \theta\). (b) Convince yourself that \(\overline{P Q}=a \sin \theta, \overline{Q C}=a \cos \theta\), \(0 \leq \theta \leq \pi / 2\).

Step-by-Step Solution

Answer

\( OT = a \theta \); \( PQ = a \sin \theta \); \( QC = a \cos \theta \).

Step 1: Understanding the Geometry

The problem revolves around a circle (the wheel) rolling along the x-axis, creating a cycloid. As the wheel rolls, the point \( P \) moves. The wheel makes \( \theta \) radians of rotation, which will help us calculate various segment lengths in the subsequent steps.

Step 2: Calculate OT as a Linear Distance

When the wheel rolls, the arc length corresponding to \( \theta \) is equal to the linear distance traveled by any point on the wheel. Since the radius is \( a \), the arc length or distance \( \overline{OT} \) is given by \( a \theta \). This is because arc length \( s = r \theta \), where \( r \) is the radius.

Step 3: Analyzing the Motion of Point P

Point \( P \), initially at the origin, moves as the wheel rotates. However, due to the rolling nature, the vertical and horizontal components of its motion can be described using right-angle trigonometry related to the angle \( \theta \):

Step 4: Show \( \overline{PQ} = a \sin \theta \)

The wheel also rotates upwards, lifting point \( P \) vertically. Using trigonometry, the vertical distance from \( P \) to the x-axis (line \( OT \)) is given by \( \overline{PQ} = a \sin \theta \). This is derived from considering \( P \) as the vertex of an angle \( \theta \) on a radius (of the wheel) forming a right triangle, where the opposite side is \( a \sin \theta \).

Step 5: Show \( \overline{QC} = a \cos \theta \)

Similarly, the horizontal distance (backward from \( C \) to \( Q \) on the wheel's direct path) due to rotation is \( \overline{QC} = a \cos \theta \). Again, this arises from the circle's geometry, where the adjacent side of the triangle mentioned earlier corresponds to this expression.

Key Concepts

Parametric Equations

When a wheel rolls along a straight line without slipping, it creates a path called a cycloid, traced by a fixed point on the wheel's rim. To describe this path, we use parametric equations. These equations use a parameter, often an angle, instead of a traditional coordinate system like Cartesian coordinates.

• Parameter: In this context, the parameter is \(\theta\), which represents the angle of rotation of the wheel.
• Equation Components: The movement along the x-axis and the movement in other directions are separately expressed using the angle \(\theta\).

Here, each position of the point on the wheel is calculated in terms of \(\theta\). This allows us to find coordinates of different points on the cycloid by varying \(\theta\). It comes down to understanding how the wheel's rotation translates into the motion of the point, which is a blend of linear and circular movement.

Trigonometry

Trigonometry helps in calculating the positions of various points, crucial for understanding cycloids. In this scenario, it helps break movement into horizontal and vertical components.

• Vertical Movement: As the wheel rolls, the point's vertical position, \((\overline{PQ})\), is determined using \(a \sin \theta\), where \(a\) is the wheel's radius.
• Horizontal Movement: The horizontal position relation, \((\overline{QC})\), is determined using \(a \cos \theta\).

These are derived from a concept we often encounter in trigonometry: right triangle relationships, where the sides of the triangle are expressed as \(\sin\) and \(\cos\) functions of \(\theta\).
This understanding links back to how these functions work within a circle, using the arch of the circle's edge as a guide.

Arc Length

The concept of arc length is essential for understanding the distance traveled by the point on the wheel's rim. The arc length \(\overline{OT}\) for a circle is the linear distance a point travels during rotation. For a circle of radius \(a\), this is calculated by the expression \(a \theta\), based on rotational radians.

• Formula: \(s = r \theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the angle in radians.
• Understanding this in Motion: As the wheel rolls, \(\theta\) increases, and \(\overline{OT}\) provides the linear distance along the x-axis.

This principle explains how a point on the wheel translates rotational motion into linear motion. This seamless conversion is why the arc length directly contributes to understanding cycloid paths, tying together the rotational movement of the wheel with the linear path of the point.

Geometry of Circles

The geometry of circles gives foundational insights needed to understand the wheel's rolling and the creation of a cycloid. Here’s how:

• Circle Dynamics: A circle rotates by spinning around its center. When its rim touches a surface, its point follows both the circular path and the linear path on the ground.
• Radius Usage: Given the radius \(a\), the circle’s movement ensures that every \(\theta\) unit of rotation corresponds to a direct distance relation, owing to consistent circumferential properties.

By using the circle's radius and basic geometric principles, we divide the motion into understandable parts: linear technics of the circle’s path and cyclic techniques of its edges. These ideas make described phenomena, like the cycloid, evident and relatable.