Question

What is the center of gravity of a semi-circular disc of radius (R)?

• A. \(\frac{2R}{\Pi}\)
• B. \(\frac{4R}{3\Pi}\)
• C. \(\frac{R}{2}\)
• D. \(\frac{3R}{8}\)

Answer 1

The Correct Option is B

The center of gravity (CG) of a semi-circular disc is a point where the entire mass of the disc can be considered to be concentrated. For a semi-circular disc of radius \( R \), let us determine the position of its center of gravity.

Conceptual Explanation

The center of gravity of a semi-circle lies along its axis of symmetry. For a semi-circular disc, this will be along the diameter line that forms the base of the semi-circle, at a certain distance from the center.

Mathematical Derivation

The center of gravity \( G \) of a semi-circular lamina occurs at a distance from the center of the semi-circle (along the y-axis in cartesian coordinates) given by the formula:

\(y = \frac{4R}{3\pi}\)

This distance \( y \) is from the flat diameter line (the base) towards the rounded edge, and it is measured along the axis of symmetry of the semi-circle.

Solution Justification

Given the options, we compare them against the derived position:

• \(\frac{2R}{\pi}\) - This option does not match the derived formula for the center of gravity of the semi-circle.
• \(\frac{4R}{3\pi}\) - This matches exactly with our derived formula, hence is the correct answer.
• \(\frac{R}{2}\) - This option is incorrect as it does not reflect any known formula for a semi-circular disc's center of gravity.
• \(\frac{3R}{8}\) - This option also does not correspond to the correct center of gravity for a semi-circle as derived.

Thus, the correct answer is \(\frac{4R}{3\pi}\), which indicates that the center of gravity is located at this distance from the diameter line of the semi-circle along the central axis.