Centre of Gravity and Stability

Why It Matters

The centre of gravity lets an extended body be treated more simply in force and moment analysis. Instead of drawing the weight of every small part of the body, the total weight can often be represented as one force acting at one point. This is especially useful for stability and toppling questions.

Definition

The centre of gravity is the point through which the resultant weight of a body may be considered to act.

Near Earth’s surface, if the gravitational field is approximately uniform, the weight of a body is:

$$\vec{W}=m\vec{g}$$

In magnitude form:

$$W=mg$$

For a homogeneous symmetric body in a uniform gravitational field, the centre of gravity lies on the axes or planes of symmetry. For an irregular or non-uniform body, it depends on the distribution of mass.

Key Representations

In a free-body diagram for an extended object, weight is drawn as a downward force acting through the centre of gravity.

The turning effect of the weight about a pivot is:

$$\vec{\tau }=\vec{r}\times \vec{W}$$

where $\vec{r}$ is the position vector from the pivot to the centre of gravity.

The magnitude is:

$$|\vec{\tau }|=Wd$$

where $d$ is the perpendicular distance from the pivot to the line of action of the weight.

In a two-dimensional scalar calculation, choose a clockwise or anticlockwise sign convention before writing moment equations.

Stability and Toppling

A body is stable against toppling when the line of action of its weight passes through the base of support. If the line of action falls outside the base, the weight produces a moment that topples the body.

Stability is increased by:

• lowering the centre of gravity;
• widening the base of support;
• keeping the line of action of the weight well within the base.

This explains why tall narrow objects topple more easily than low wide objects.

Common Exam Points

• Weight acts vertically downward through the centre of gravity.
• The centre of gravity is a point of action for the resultant weight, not necessarily a physical particle.
• For toppling, compare the line of action of weight with the base of support.
• Moment calculations must use the perpendicular distance to the line of action of weight.
• A body can have translational equilibrium but still topple if the resultant moment is not zero.

Links

• Prerequisite: forces
• Prerequisite: equilibrium moments and couples
• Prerequisite: vectors
• Related: dynamics
• Related: force diagrams and resolution
• Related: equilibrium moments and couples
• Misconception: vector direction consistency