State what is meant by centre of gravity

What is the Centre of Gravity? ⚖️

The centre of gravity (CG) is the point at which the total weight of an object can be considered to act. In other words, if you were to suspend the object from a single point, that point would be the CG. Mathematically, for a system of point masses $m_i$ located at position vectors $\mathbf{r}_i$, the CG is given by:

$$\mathbf{r}_{CG} = \frac{\sum_i m_i \mathbf{r}_i}{\sum_i m_i}$$

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

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

Key point: The CG is the same as the centre of mass for objects in a uniform gravitational field. It is the point where the weight can be treated as a single force acting downwards.

Analogy: The Seesaw 🎯

Imagine a seesaw with two kids of different weights sitting on either end. The seesaw balances when the torque on both sides is equal. The point where the seesaw pivots is effectively the CG of the combined system of the seesaw and the kids. If you moved the heavier child closer to the pivot, the CG would shift toward that side, making the seesaw tip.

Real‑World Example: Hanging a Book 📚

A book hanging from a string will hang so that its CG is directly below the point of suspension. If you cut the book in half, each half will still hang with its own CG below the string. This demonstrates that the CG is independent of the shape of the support.

Exam Tips for IGCSE 📑

• Remember: CG is the point where the weight acts; it can be found using the weighted average of positions.
• For simple shapes (rectangles, cylinders), use symmetry: the CG lies along the symmetry axis.
• When a body is in a uniform gravitational field, CG = centre of mass.
• In questions, look for phrases like “point of suspension” or “balance point” to identify the CG.
• Practice drawing free‑body diagrams: mark the CG and the weight vector to reinforce the concept.

Quick Check: CG of a Uniform Rod

A uniform rod of length $L$ and mass $M$ has its CG at its midpoint:

$$\mathbf{r}_{CG} = \frac{L}{2}$$

So if you hang it from one end, it will tip until the midpoint is directly below the suspension point.