1. What is a Cycloid?

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. It's a special case of a roulette curve.

2. How Does the Calculator Work?

The calculator uses the cycloid arc length formula:

Where:

• \( r \) — Radius of the generating circle

Explanation: The length of one complete arch of a cycloid is exactly 8 times the radius of the generating circle.

3. Importance of Cycloid Calculations

Details: Cycloids have important applications in physics and engineering, particularly in the study of pendulum motion and gear design.

4. Using the Calculator

Tips: Simply enter the radius of the generating circle. The radius must be a positive number.

5. Frequently Asked Questions (FAQ)

Q1: Why is the arc length 8 times the radius?
A: This is a fundamental property of cycloids derived through calculus. The integral of the curve's parametric equations leads to this simple relationship.

Q2: What are some real-world applications of cycloids?
A: Cycloidal gears, pendulum clocks (tautochrone property), and roller coasters often utilize cycloid properties.

Q3: Is this formula for a complete arch or partial cycloid?
A: This formula gives the length of one complete arch (from cusp to cusp). Partial lengths require more complex calculations.

Q4: Does this work for prolate or curtate cycloids?
A: No, this simple formula only applies to standard cycloids where the tracing point is exactly on the rim.

Q5: How was this formula historically derived?
A: The arc length of a cycloid was first calculated by Christopher Wren in 1658, before the development of calculus.