1. What is a Cycloid Radius?

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. The radius of the generating circle can be calculated from the arc length of one complete arch of the cycloid.

2. How Does the Calculator Work?

The calculator uses the cycloid radius formula:

Where:

• \( r \) — Radius of the generating circle
• \( arc \) — Length of one complete arch of the cycloid

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

3. Importance of Cycloid Radius Calculation

Details: Calculating the radius is essential in mechanical engineering, physics, and mathematics when working with cycloidal motion or designing cycloidal gears and mechanisms.

4. Using the Calculator

Tips: Enter the arc length of one complete cycloid arch in any length units. The radius will be calculated in the same units.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between cycloid and circle?
A: A cycloid is generated by a point on a circle as it rolls along a straight line. The radius of that generating circle determines the cycloid's properties.

Q2: What are practical applications of cycloids?
A: Cycloids are used in gear design (cycloidal drives), pendulum clocks (cycloidal pendulum), and various mechanical systems requiring smooth periodic motion.

Q3: Why is the arc length exactly 8 times the radius?
A: This comes from the mathematical properties of the cycloid curve. One complete arch corresponds to one full rotation (2π radians) of the generating circle.

Q4: Can this calculator be used for partial arcs?
A: No, this formula specifically calculates the radius from one complete arch length (from cusp to cusp).

Q5: What units should I use?
A: Any consistent length units can be used (mm, cm, inches, etc.). The result will be in the same units as your input.