Question

The first moment of area of a semicircular area about its diameter ‘d’ is given by:

• A. \( \frac{d^3}{36} \)
• B. \( \frac{d^3}{12} \)
• C. \( \frac{d^3}{24} \)
• D. \( \frac{d^3}{6} \)

Hint:

- The first moment of area is useful for determining centroids. - For a semicircle, its centroid lies at \( \frac{4r}{3\pi} \) from the diameter. - The moment of area formula helps in shear stress calculations.

Answer 1

The Correct Option is C

Step 1: Definition of First Moment of Area. The first moment of area of a semicircle about its diameter is given by: \[ Q = \frac{A \cdot \bar{y}}{2} \] For a semicircle of diameter \( d \), the centroid is located at \( \bar{y} = \frac{4r}{3\pi} \), where \( r = \frac{d}{2} \).
Step 2: Calculation. \[ Q = \frac{\pi r^2 \times \frac{4r}{3\pi}}{2} \] \[ = \frac{\frac{\pi d^2}{4} \times \frac{4(d/2)}{3\pi}}{2} \] \[ = \frac{d^3}{24} \] Thus, the correct answer is (c) \( \frac{d^3}{24} \).