Locate the centroid, measured from the base AB, of a rectangular section 200 mm wide and 300 mm high from which a circle of diameter 150 mm, centred 100 mm below the top edge, has been removed.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 04/12/2023 (Morning)
Detailed Explanation
For a section with a hole, treat the hole as a negative area. Take moments of area about the base AB: the solid rectangle counts positive, the hole negative. The section is symmetrical about the vertical axis, so only ȳ is needed.
Rectangle: \(A_1 = 200 \times 300 = 60000\ \text{mm}^2\), \(y_1 = 150\ \text{mm}\)
Hole: \(A_2 = \frac{\pi}{4}(150)^2 = 17671\ \text{mm}^2\), \(y_2 = 300 - 100 = 200\ \text{mm}\)
\[ \bar y = \frac{A_1y_1 - A_2y_2}{A_1 - A_2} = \frac{9\,000\,000 - 3\,534\,292}{60000 - 17671} = \frac{5\,465\,708}{42\,329} \approx 129.1\ \text{mm} \]
Why “129.1 mm” Is Correct
Removing material from the upper part moves the centroid down from 150 mm. The calculation gives 129.1 mm above AB — option (a).
Why the Other Options Are Wrong
- B. 133.68 mm: 133.68 mm does not come from the correct areas and distances.
- C. 105.49 mm: 105.49 mm moves the centroid down too far.
- D. 112.44 mm: 112.44 mm also moves it down too far.
Key Point
Composite area with a hole: ȳ = (ΣAy for solids − ΣAy for holes)/(net area).
Key Concepts for Students
- Removing area above the centroid lowers it; removing area below raises it.
- Choose the reference axis (here AB) before taking moments.
- The same method gives the centroid of T, I and L sections.
Practice more: this is question 2 in Moment MCQs – Part 1. Try the whole set and check every answer instantly.