Problem 38 - Thomas Calculus

Question

Trochoids \(A\) wheel of radius \(a\) rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point \(P\) on a spoke of the wheel \(b\) units from its center. As parameter, use the angle \(\theta\) through which the wheel turns. The curve is called a trochoid, which is a cycloid when \(b=a\) .

Step-by-Step Solution

Answer

The parametric equations are \( x(\theta) = a\theta - b\sin(\theta) \) and \( y(\theta) = a - b\cos(\theta) \).

Step 1: Understand the Wheel's Movement

The wheel rolls along a horizontal line without slipping. Each time the wheel rotates through an angle \( \theta \), it travels a linear distance along the horizontal equal to the circumference arc length corresponding to \( \theta \), which is \( a \theta \).

Step 2: Define the Fixed Point on the Wheel

The point \( P \) is located \( b \) units from the wheel's center. Think of \( P \) as being on a spoke that rotates around the wheel's center. When the wheel rotates, this point moves in a circular path around the center of the wheel, besides translating horizontally.

Step 3: Parametric Equation for Horizontal Displacement

As the wheel rotates and translates, the horizontal displacement of the point \( P \) is the sum of the wheel's movement along the line and the relative horizontal position of \( P \) on the wheel. This gives the equation \( x(\theta) = a\theta - b\sin(\theta) \).

Step 4: Parametric Equation for Vertical Displacement

The vertical position of point \( P \) is determined by its position relative to the wheel's center, which results in the motion of the point up and down by \( b \) units. Hence, the vertical displacement is given by \( y(\theta) = a - b\cos(\theta) \).

Step 5: Combine into Parametric Equations

The curve traced out by the point \( P \) on the wheel is fully described by the parametric equations: \( x(\theta) = a\theta - b\sin(\theta) \) and \( y(\theta) = a - b\cos(\theta) \).

Key Concepts

Trochoid

A trochoid is the path traced by a point on a spoke of a wheel as it rolls along a straight line. This concept can be quite fascinating when you dive into its formation. When a wheel rolls without slipping, every bit of rotation translates into forward movement. Now, imagine a point on a spoke of the wheel which is not at the edge but somewhere along its radius.

This point will move in both a circular path and a linear path as the wheel rolls. The combination of these two movements creates the trochoid. The general form of the parametric equations for a trochoid are:

• Horizontal position: \( x(\theta) = a\theta - b\sin(\theta) \)
• Vertical position: \( y(\theta) = a - b\cos(\theta) \)

Here, \( a \) is the radius of the wheel, \( b \) is the distance from the wheel's center to the point on the spoke, and \( \theta \) is the angle by which the wheel has turned. This shows how unique curves can be created just by varying the position of point \( P \).

Cycloid

A cycloid is a special type of trochoid where the point you are following is exactly on the rim of the wheel, not somewhere along a spoke away from the rim. In other words, when the distance \( b \) is equal to the radius \( a \) of the wheel, the point traces a cycloid. This results in an elegant, periodic curve that has captivated mathematicians and physicists for centuries.

When you trace a cycloid, the point \( P \) starts at the bottom of the wheel and sweeps upwards in a smooth arch before descending back down—completing this pattern as a loop with each full rotation. The parametric equations simplify to:

• Horizontal position: \( x(\theta) = a(\theta - \sin(\theta)) \)
• Vertical position: \( y(\theta) = a(1 - \cos(\theta)) \)

This simplification reflects that the motion is purely determined by the wheel's movement without an added offset from the center.

Curve Tracing

Tracing curves like trochoids and cycloids involves finding the precise mathematical descriptions that capture the path a point follows due to a dynamic motion. With parametric equations, we can describe more complex paths than traditional Cartesian equations. Here’s how it works:

• Role of Parameters: Parameters like \( \theta \) are used to systematically cover every point on a curve by varying a single variable.
• Intuitive Understanding: Instead of thinking of curves as \( y \) being the dependent variable of \( x \) or vice versa, in parametric equations, both \(x\) and \(y\) are functions of \( \theta \).
• Dynamic Exploration: By adjusting the parameter \( \theta \), we see how the curve unfolds as a point moves along its path, helping to visualize the motion behind abstract equations.

Studying curve tracing in this way not only simplifies the math but also enriches our geometric intuition, revealing how small changes in parameters can dramatically alter the paths we observe.