Centre of gravity of a quadrant of a circle For a plane quadrant of a circle of radius r, the centre of gravity (centroid) lies along the central radius at a distance from the circle's center equal to:

Introduction / Context:
Locating centroids of standard plane figures is essential for calculating bending stresses, locating neutral axes, and setting up composite-area problems in mechanics of materials.

Given Data / Assumptions:

• Figure: a quadrant (one-fourth) of a circle, radius r.
• Uniform area density (homogeneous lamina).
• Centroid lies along the central radius (the bisector of the right angle).

Concept / Approach:
The centroid of a circular arc, sector, and quadrant are standard results. For a quadrant of a circle (area), the distance of the centroid from the circle center along the bisector is x_c = 4r / (3π) ≈ 0.424 r. This is obtained by integrating the area’s first moments or from tabulated centroid formulae used in engineering handbooks.

Step-by-Step Solution:

Verification / Alternative check:
Compare with numerical value: 4/(3π) ≈ 0.424 < 0.5, hence centroid is inside the quadrant closer to the corner than to the midpoint of the radius, which matches intuition.

Why Other Options Are Wrong:

• r/2, 0.6 r, 3r/4, r/π: Do not match the standard derived centroid position; several are dimensionally fine but numerically incorrect.

Common Pitfalls:
Confusing centroid of a quadrant (area) with that of a circular arc; the arc centroid distance is 2r/π ≈ 0.637 r, not applicable here.

Final Answer:
4r / (3π)