Definition — what is a centroid? In engineering mechanics of areas, the term “centroid” refers to which of the following?
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AThe geometric centre of area of a plane figure (point where first moments of area about any axis through it are zero).
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BExactly the same as the centre of gravity for every body in all conditions.
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CThe point of suspension of a rigid body.
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DThe point of application of the resultant of all forces tending to rotate a body about a certain axis.
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EThe instantaneous centre of rotation of a rigid link.
Answer
Correct Answer: The geometric centre of area of a plane figure (point where first moments of area about any axis through it are zero).
Explanation
Introduction / Context:The centroid is a geometric property of an area and is fundamental when computing bending stresses, shear flow, and deflections. It is often confused with the centre of gravity (C.G.).
Given Data / Assumptions:
- We consider plane areas (laminae) of uniform thickness and material, unless otherwise stated.
- Gravitational field is uniform when comparing centroid to C.G.
Concept / Approach:The centroid is the point where the first moments of area about any axis through that point are zero. For a thin homogeneous lamina in a uniform gravitational field, the centroid coincides with the centre of gravity. However, if density varies or the field is non-uniform, the C.G. may shift while the centroid, being purely geometric, does not.
Step-by-Step Solution:
Define centroid: location (x̄, ȳ) satisfying Σ(A_i x_i)/ΣA_i and Σ(A_i y_i)/ΣA_i.Relate to C.G.: C.G. coincides with centroid only for uniform density and gravity.Differentiate from “point of suspension” and “resultant force application,” which are unrelated concepts.Conclude the correct definition is the geometric centre of area.Verification / Alternative check:For symmetric shapes (e.g., rectangle), centroid lies at the intersection of symmetry axes, confirming the geometric definition.
Why Other Options Are Wrong:
- Same as C.G. always: Only true under uniform density and field; not universally true.
- Point of suspension / resultant of forces / instantaneous centre: These are dynamics/statics notions, not the area property.
Common Pitfalls:Using “centroid” and “centre of gravity” interchangeably without checking density and field assumptions.
Final Answer:The geometric centre of area of a plane figure (point where first moments of area about any axis through it are zero).