Center of Gravity, Center of Mass & Centroid – Comprehensive Notes

Definition
• Point where the entire weight of a body is considered to act, regardless of the body’s orientation.
• Equivalent statement: the point about which the algebraic sum of gravitational‐force moments is zero: $\sum Wi xi = 0,\; \sum Wi yi = 0$ when referenced to that point.
Fundamental Properties
• Unique for any rigid body; does not shift when the object is reoriented.
• Depends on weight distribution; therefore varies with changes in gravitational field strength only through the change in weight itself.
Conceptual Significance
• Serves as the ideal location to apply a single resultant weight force in statics/dynamics problems.
• In engineering design (e.g., vehicles, structures, robots) stability is assessed with respect to whether the projected C.o.G. lies within the support base.

Definition
• Point at which the whole mass of a body may be assumed to be concentrated for translational motion analysis.
Key Attributes
• Represents the average location of mass distribution: $m = \sum mi, \qquad \vec r{cm} = \frac{\sum mi \vec ri}{\sum m_i}$ .
• Independent of gravitational field; remains fixed when $g$ varies or is zero (e.g., space-flight).
• For bodies of uniform density in a uniform gravitational field, C.o.G. coincides with C.o.M.
Practical Insight
• Crucial in dynamics (e.g., projectile motion, spacecraft attitude) where external forces act on the mass center.

Definition for Plane Areas
• The geometric center at which the entire area of a 2-D lamina is assumed to be concentrated.
• For laminae of uniform thickness/density in a uniform gravitational field: Centroid ≡ C.o.G. ≡ C.o.M.
Coordinate Formulas for a Plane Area
• $\bar x = \frac{\intA x \, dA}{\intA dA}, \qquad \bar y = \frac{\intA y \, dA}{\intA dA}$
• Discrete composite form (used in examples): $\bar x = \frac{\sum Ai xi}{\sum Ai}, \qquad \bar y = \frac{\sum Ai yi}{\sum Ai}$

C.o.G. → weight focus; C.o.M. → mass focus; Centroid → area (or volume) focus.
In uniformly dense, homogeneous bodies in a uniform gravitational field: all three points coincide.
Under variable density or non-uniform $g$ : C.o.M. remains unchanged while C.o.G. may shift; centroid remains purely geometric.

Decompose the body/area into basic shapes (segments): rectangles, triangles, circles, annuli, sectors, cylinders, hemispheres, etc.
For each segment identify its individual centroid coordinates $(xi, yi)$ relative to a chosen reference axis system.
Construct a summary table:
• Segment label
• Area (or mass/weight) $Ai$ • Coordinates $xi$ , $yi$ • Area–moment products $Ai xi$ and $Ai y_i$
Compute totals $\sum Ai$ , $\sum Ai xi$ , $\sum Ai y_i$ .
Apply composite formulas to locate the overall centroid:
$\bar x = \frac{\sum Ai xi}{\sum Ai}, \qquad \bar y = \frac{\sum Ai yi}{\sum Ai}$ .
For bodies with cut-outs (negative areas) treat removed segments with negative $A_i$ .

Rectangle (width $b$ , height $h$ )
• $\bar x = b/2$ , $\bar y = h/2$
Right Triangle (base $b$ along x-axis, height $h$ along y-axis, right angle at origin)
• $\bar x = b/3$ , $\bar y = h/3$
Circle (radius $R$ )
• Centroid at center (0,0)
Semicircle (radius $R$ , flat side on x-axis)
• $\bar x = 0$ , $\bar y = 4R/(3\pi)$ from the diameter line
Quarter-circle (radius $R$ , along positive x & y)
• $\bar x = \bar y = 4R/(3\pi)$ from the corner
Circular Sector (radius $R$ , angle $\theta$ in radians, apex at origin)
• $\bar r = \frac{2R \sin(\theta/2)}{3\theta/2}$ along the bisector
• For thin sector plates the polar form or rectangular projections may be used.

Example 5.1 – Trapezoid
• Shape divided into simpler figures (e.g., rectangle + triangle) or treated via trapezoid formula.
• Apply composite method to find $(\bar x, \bar y)$ .
Example 5.2 – Complex Lamina (millimetres)
• Possibly involves holes or stepped geometry; negative areas accounted for.
• Reinforces systematic tabulation.
Example 5.3 – Thin Plate with Dimension $a = 30\,\text{cm}$
• Emphasises scalability: centroid location scales proportionally with $a$ .
• Encourages expressing final coordinates as multiples of $a$ (dimensionless form).
• All examples use the tabular form: $Ai, xi, yi, Ai xi, Ai y_i$ .

Tutorial 6.2 (Fig. 5.9)
• Thin plate with parameters $a = 1\,\text{cm},\; b = 6\,\text{cm},\; c = d = 3\,\text{cm}$ .
• Likely composite of rectangles/triangles; aims to solidify student practice.
Tutorial 6.4 (Fig. 5.11)
• Another irregular plate; students to apply identical procedure.
• Reinforces attention to sign convention and coordinate reference selection.

Structural design: Knowing centroid/C.o.G. essential for stress distribution, bending analysis, and stability.
Robotics & Aerospace: Control algorithms use C.o.M. to predict motion and orientation.
Ergonomics & Safety: Position of C.o.G. influences tipping, gait stability, and crash performance.
Philosophical note: Although C.o.G./C.o.M. are abstract points, they encapsulate the aggregate behaviour of distributed matter—demonstrating the power of mathematical abstraction in engineering.

Composite centroid of area:
$\bar x = \frac{\sum Ai xi}{\sum Ai}, \qquad \bar y = \frac{\sum Ai yi}{\sum Ai}$
Mass analogue:
$\vec r{cm} = \frac{\sum mi \vec ri}{M}, \qquad M = \sum mi$
Moment-zero definition of C.o.G. with respect to any axis through C.o.G.:
$\sum Wi (xi - \bar x) = 0, \qquad \sum Wi (yi - \bar y) = 0$

Always sketch the composite area, label segment centroids, and choose a convenient coordinate origin (often at symmetry lines or lower-left corner) to minimise negative coordinates.
Exploit symmetry: If the area is symmetric about an axis, the centroid lies on that axis; only one coordinate needs calculation.
Keep units consistent; convert $\text{mm}^2$ to $\text{cm}^2$ if mixing units will occur in tabulation.
Double-check arithmetic in the $Ai xi$ and $Ai yi$ summations; small sign errors move the centroid dramatically.