State what is meant by centre of gravity

1.5.3 Centre of Gravity

Objective

State what is meant by centre of gravity.

Definition

The centre of gravity (CG) of an object is the point at which the entire weight of the object can be considered to act. It is the unique point where the gravitational force on the object is effectively concentrated.

In a uniform gravitational field the centre of gravity coincides with the centre of mass.

Key Points

• The CG is a single point, not a region.
• It depends only on the distribution of mass within the object.
• In a uniform gravitational field the torque due to weight about any axis is the same as if the total weight acted at the CG.

How to Locate the Centre of Gravity

1. Identify the shape and symmetry of the object.
2. For simple, regular shapes use known positions (e.g., centre of a uniform rod is at its midpoint).
3. For irregular objects, balance the object on a point support; the point of balance is the CG.
4. Alternatively, use the principle of moments: the sum of moments about any point is zero when the weight acts through the CG.

Mathematical Expression

For a system of discrete masses \$mi\$ at positions \$\mathbf{r}i\$, the centre of gravity \$\mathbf{R}_{\text{CG}}\$ is

\$\mathbf{R}{\text{CG}} = \frac{\sumi mi \mathbf{r}i}{\sumi mi}\$

For a continuous body with density \$\rho(\mathbf{r})\$,

\$\mathbf{R}_{\text{CG}} = \frac{\int \rho(\mathbf{r}) \mathbf{r}\, dV}{\int \rho(\mathbf{r}) \, dV}\$

Examples

Common Misconceptions

• Thinking the CG must lie inside the material – for hollow or irregular shapes it can be outside the physical body.
• Confusing centre of gravity with centre of pressure (which relates to fluid forces).

Summary

The centre of gravity is the point at which the total weight of an object can be considered to act. It is crucial for analysing balance, stability, and the effect of forces on objects in a uniform gravitational field.