1. What is Centroid?

The centroid is the geometric center of a plane figure or solid. For a shape of uniform density, it coincides with the center of mass. The x-coordinate of the centroid (x̄) is calculated by dividing the first moment of area about the y-axis by the total area.

2. How Does the Calculator Work?

The calculator uses the centroid formula:

Where:

• \( \bar{x} \) — x-coordinate of centroid (length units)
• \( A \) — Total area (square length units)
• \( \int x \, dA \) — First moment of area about y-axis (cubic length units)

Explanation: The formula calculates the average x-position weighted by the area distribution.

3. Importance of Centroid Calculation

Details: Centroid calculations are essential in engineering and physics for determining balance points, stress distribution, and structural stability.

4. Using the Calculator

Tips: Enter the total area and the value of the integral ∫x dA. Both values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between centroid and center of mass?
A: They coincide for uniform density objects. For non-uniform density, center of mass accounts for density variations.

Q2: How is the integral ∫x dA calculated?
A: For simple shapes, use standard formulas. For complex shapes, divide into simple parts or use numerical integration.

Q3: Can this calculator find ȳ (y-coordinate centroid)?
A: No, this calculates only x̄. For ȳ, use ∫y dA/A with the same approach.

Q4: What units should I use?
A: Use consistent units (e.g., all in meters or all in inches). The result will be in the same length unit.

Q5: How accurate is this calculation?
A: Accuracy depends on the precision of your area and moment of area inputs.