Centre of Mass: Discrete Particles

This lesson covers finding the centre of mass of a system of discrete particles. The centre of mass (also called the centre of gravity in a uniform gravitational field) is the single point at which the entire mass of the system can be considered to act.

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1 Definition

For $n$n particles with masses $m_1, m_2, \ldots, m_n$m1​,m2​,…,mn​ at positions $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$(x1​,y1​),(x2​,y2​),…,(xn​,yn​):

$$\boxed{\bar{x} = \frac{\sum m_i x_i}{\sum m_i}, \quad \bar{y} = \frac{\sum m_i y_i}{\sum m_i}}$$xˉ=∑mi​∑mi​xi​​,yˉ​=∑mi​∑mi​yi​​​

In vector form:

$$\bar{\mathbf{r}} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i} = \frac{\sum m_i \mathbf{r}_i}{M}$$rˉ=∑mi​∑mi​ri​​=M∑mi​ri​​

where $M = \sum m_i$M=∑mi​ is the total mass.

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2 Physical Interpretation

• If the system is supported at the centre of mass, it balances.
• The system moves as if all external forces act at the centre of mass.
• In a uniform gravitational field, the centre of mass coincides with the centre of gravity.

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3 Worked Examples

Example 1 — Two Particles on a Line

Masses 3 kg at $x = 2$x=2 and 5 kg at $x = 6$x=6. Find $\bar{x}$xˉ.

Solution