Centre of gravity of a rectangle — location statement The centre of gravity (centroid) of a uniform rectangle lies at the intersection point of its two diagonals.
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AAgree
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BDisagree
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CAgree only if the rectangle is a square
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DAgree only if density varies linearly
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EDisagree unless sides are equal
Answer
Correct Answer: Agree
Explanation
Introduction / Context:The centroid (centre of gravity for uniform density) of standard shapes is fundamental in structural analysis, bending, and area-moment calculations. For rectangles, the centroid is famously at the intersection of diagonals.
Given Data / Assumptions:
- Uniform thickness and density.
- Planar, axis-aligned rectangle of width b and height h.
- No cut-outs or holes.
Concept / Approach:By symmetry, the centroid must lie midway between parallel sides. Therefore, it is located at coordinates (b/2, h/2) measured from a corner, which is exactly where the diagonals intersect.
Step-by-Step Solution:
Use symmetry about the rectangle’s midlines.Horizontal symmetry → x̄ = b/2.Vertical symmetry → ȳ = h/2.Intersection of diagonals also occurs at (b/2, h/2), confirming the centroid location.Verification / Alternative check:Apply the first-moment-of-area definition: x̄ = (∫ x dA)/A and ȳ = (∫ y dA)/A. For a uniform rectangle, both evaluate to mid-spans.
Why Other Options Are Wrong:
- Disagree / unless square: The result holds for any rectangle, not only squares.
- Density varies linearly: The statement assumes uniform density; non-uniformity would shift the centroid.
Common Pitfalls:Confusing “centre” with any geometric center point even when cut-outs exist; the centroid changes if holes or non-uniform density are present.
Final Answer:Agree