Centres of gravity (C.G.) of common plane figures Identify the incorrect statement about the location of the centre of gravity (centroid) of the listed shapes.
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AThe C.G. of a circle is at its centre.
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BThe C.G. of a triangle is at the intersection of its medians.
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CThe C.G. of a rectangle is at the intersection of its diagonals.
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DThe C.G. of a semicircle is at a distance of r/2 from the centre.
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EThe C.G. of an ellipse is at the intersection of its axes.
Answer
Correct Answer: The C.G. of a semicircle is at a distance of r/2 from the centre.
Explanation
Introduction / Context:Remembering centroid locations of standard shapes is crucial for area moments, composite sections, and structural design. One of the statements below is intentionally wrong to test precise recall.
Given Data / Assumptions:
- Plane laminae of uniform thickness and density.
- Standard geometric shapes: circle, triangle, rectangle, semicircle, ellipse.
Concept / Approach:
For symmetric figures, the centroid lies at the intersection of symmetry axes. For semicircular area, the centroid lies along the axis of symmetry at a known distance from the circle’s centre, but it is not r/2.
Step-by-Step Solution:
(a) Circle: centroid at centre → correct.(b) Triangle: centroid at medians’ intersection → correct.(c) Rectangle: centroid at diagonal intersection → correct.(d) Semicircle: the correct distance from the circle centre to the area centroid is 4r/(3π) ≈ 0.424 r, not r/2 = 0.5 r → statement is incorrect.(e) Ellipse: centroid at intersection of major and minor axes → correct.Verification / Alternative check:
Using tabulated centroid formulae, the semicircle’s area centroid from the base is 4r/(3π) relative to the circle centre along the symmetry line, confirming (d) is wrong.
Why Other Options Are Wrong:
Here only (d) is wrong; the others match standard results.
Common Pitfalls:
Confusing the centroid of a semicircular arc (2r/π) with that of a semicircular area (4r/(3π)); mixing distances from base versus from centre.
Final Answer:
The C.G. of a semicircle is at a distance of r/2 from the centre.