Centres of Mass

This section explores the concept of the centre of mass for uniform and composite bodies, including its position and applications.

Position of the Centre of Mass

Uniform Bodies

For a uniform solid body, the centre of mass is located at the geometric centre of the body. For a simple geometric shape, this is straightforward to determine.

For example:

• Thin rod of uniform mass and length L: The centre of mass is at a distance of $\frac{L}{2}$ from either end.
• Thin plate of uniform density and area A: The centre of mass is at the geometric centre of the plate.
• Thin cylinder of uniform density and length L and radius r: The centre of mass is at a distance of $\frac{L}{2}$ from either end and at the geometric centre of the circular cross-section.

Composite Bodies

For a composite body made up of several simpler shapes, the centre of mass is the weighted average of the centres of mass of each individual component.

The position of the centre of mass ($x_{cm}$) for a composite body is given by:

$$x_{cm} = \frac{m_1 x_{cm1} + m_2 x_{cm2} + ... + m_n x_{cmn}}{m_1 + m_2 + ... + m_n}$$

where $m_i$ is the mass of the $i$-th component and $x_{cmi}$ is the position of the centre of mass of the $i$-th component.

Applications to Uniform Bodies

Thin Rod

Consider a thin rod of uniform mass $M$ and length $L$ pivoted at one end. The distance of the pivot from the centre of mass is:

$$d = \frac{L}{2}$$

This is a useful property in determining the balance point of a lever.

Thin Plate

A thin uniform plate of mass $M$ and area $A$ is supported at one edge. The distance of the support from the centre of mass is:

$$d = \frac{h}{3}$$

where $h$ is the height of the plate.

Applications to Composite Bodies

Example 1: Two Identical Plates

Consider two identical plates, each of mass $M$ and area $A$, placed side by side. The centre of mass of the combined system is:

$$x_{cm} = \frac{M \cdot \frac{A}{2} + M \cdot \frac{A}{2}}{2M} = \frac{MA}{A} = \frac{A}{2}$$

This is the geometric centre of the combined plates.

Example 2: A Plate with a Hole

Consider a uniform plate of mass $M$ and area $A$ with a circular hole of radius $r$ cut out. The centre of mass of the remaining plate is:

The mass of the remaining plate is $M - \frac{\pi r^2}{A}M = M(1 - \frac{\pi r^2}{A})$.

The centre of mass is the weighted average of the centre of mass of the original plate and the centre of mass of the hole.

$$x_{cm} = \frac{M(1 - \frac{\pi r^2}{A}) \cdot \frac{A}{2} + M \cdot 0}{M(1 - \frac{\pi r^2}{A})} = \frac{A}{2}$$

The centre of mass remains at the geometric centre of the original plate because the hole is in the centre.

Table Summarizing Centre of Mass Position

Body Type	Position of Centre of Mass
Thin Rod (uniform mass)	$\frac{L}{2}$ from either end
Thin Plate (uniform density)	Geometric centre
Thin Cylinder (uniform density)	$\frac{L}{2}$ from either end and geometric centre of circular cross-section
Composite Body	Weighted average of the centres of mass of its components

The concept of the centre of mass is fundamental in mechanics and is used to simplify the analysis of the motion of objects. It allows us to treat a complex system as if all its mass were concentrated at a single point.