What is the centroid of a plane lamina and how does it differ from centre of gravity?

In one line

The centroid is the geometric centre of an area, the point where the whole area can be imagined to be concentrated for taking first moments.

Explained simply

Cut a shape out of cardboard and try to balance it on a pencil tip. The balancing point is its centre of gravity, and for a thin, uniform sheet this is exactly the centroid of its shape. The centroid depends only on the shape, not on the material.

Technically, the centroid (\( \bar{x}, \bar{y} \)) of an area A is the point about which the first moment of area is zero, so \( A\bar{x} = \int x\,dA \) and \( A\bar{y} = \int y\,dA \).

How it works

1. Centre of gravity (CG): point through which the resultant of all the weights of the body's particles acts. Depends on how weight is distributed.
2. Centre of mass: point where the whole mass may be taken to be concentrated. Same as CG in a uniform gravity field.
3. Centroid: purely geometric centre of an area (2D), a line, or a volume. For a uniform, homogeneous body of constant thickness, CG = centre of mass = centroid.
4. Symmetry rule: if an area has an axis of symmetry, the centroid lies on it; with two axes, it lies at their intersection.
5. The centroid can lie outside the material, e.g. for a ring or a C-section.

Key formula

\( \bar{x} = \dfrac{\int x\,dA}{\int dA} \), \( \bar{y} = \dfrac{\int y\,dA}{\int dA} \); for composite areas \( \bar{x} = \dfrac{\Sigma A_i x_i}{\Sigma A_i} \), \( \bar{y} = \dfrac{\Sigma A_i y_i}{\Sigma A_i} \). Here \( A_i \) (mm\(^2\)) is the area of part i and \( x_i, y_i \) (mm) the coordinates of its own centroid; \( \Sigma A_i x_i \) is the first moment of area (mm\(^3\)).

Quick example

A rectangle 60 mm wide and 40 mm high with its corner at the origin has two axes of symmetry, so its centroid is at (30, 20) mm without any calculation. Beam designers use the centroid of the cross-section to locate the neutral axis, which is the reason this topic matters later in Strength of Materials.

Common mistakes

• Using 'centroid' for a body whose density varies; then the centre of mass differs from the geometric centroid.
• Assuming the centroid must lie inside the material.
• Mixing up the first moment of area (\( \Sigma A x \), mm\(^3\)) with the area itself.

How MU asks it

A 3-mark short part in Q.1: 'Define centroid and centre of gravity; differentiate between them' or 'State the significance of the axis of symmetry in locating the centroid'. Write crisp definitions, the formula, and one line on when they coincide. The main marks of this module come from the composite-lamina numerical.