Centre of Mass of Point-Size Bodies

The centre of mass is a conceptual point where the total mass of a system can be thought to be concentrated. For point-size bodies, which are idealized objects with mass but no volume, the centre of mass is determined by the distribution of mass and the positions of these bodies in space.

Understanding Centre of Mass

The centre of mass (COM) is the point at which the weighted relative position of the distributed mass sums to zero. This is the point where if the body is supported, it would balance perfectly.

Formula for Centre of Mass

For a system of point masses, the centre of mass can be calculated using the following formula:

$$ \vec{R}_{\text{COM}} = \frac{\sum m_i \vec{r}_i}{\sum m_i} $$

Where:

• $\vec{R}_{\text{COM}}$ is the position vector of the centre of mass
• $m_i$ is the mass of the $i$-th point mass
• $\vec{r}_i$ is the position vector of the $i$-th point mass
• The summation is over all point masses in the system

Centre of Mass in Different Dimensions

In one dimension, the centre of mass is simply the weighted average of the positions of the point masses along the line.

In two or three dimensions, the centre of mass is found by taking the weighted average of the positions in each dimension separately.

Table of Differences and Important Points

Property	Description
Definition	The centre of mass is the point where the total mass of the system can be considered to be concentrated.
Calculation	Calculated using the weighted average of the positions of the point masses.
Dimensionality	Can be calculated in one, two, or three dimensions.
Physical Significance	The motion of the centre of mass of a system of particles is the same as if all the mass were concentrated at that point.

Examples

Example 1: Centre of Mass of Two Point Masses

Consider two point masses, $m_1$ and $m_2$, located at positions $x_1$ and $x_2$ on the x-axis. The centre of mass is given by:

$$ x_{\text{COM}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} $$

If $m_1 = 2 \text{ kg}$, $m_2 = 3 \text{ kg}$, $x_1 = 1 \text{ m}$, and $x_2 = 4 \text{ m}$, then:

$$ x_{\text{COM}} = \frac{2 \cdot 1 + 3 \cdot 4}{2 + 3} = \frac{2 + 12}{5} = \frac{14}{5} = 2.8 \text{ m} $$

Example 2: Centre of Mass in Three Dimensions

For three point masses $m_1$, $m_2$, and $m_3$ at positions $\vec{r}_1$, $\vec{r}_2$, and $\vec{r}_3$ in three-dimensional space, the centre of mass is:

$$ \vec{R}_{\text{COM}} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2 + m_3 \vec{r}_3}{m_1 + m_2 + m_3} $$

If $m_1 = 1 \text{ kg}$, $m_2 = 2 \text{ kg}$, $m_3 = 3 \text{ kg}$, $\vec{r}_1 = \langle 1, 0, 0 \rangle \text{ m}$, $\vec{r}_2 = \langle 0, 2, 0 \rangle \text{ m}$, and $\vec{r}_3 = \langle 0, 0, 3 \rangle \text{ m}$, then:

$$ \vec{R}_{\text{COM}} = \frac{1 \cdot \langle 1, 0, 0 \rangle + 2 \cdot \langle 0, 2, 0 \rangle + 3 \cdot \langle 0, 0, 3 \rangle}{1 + 2 + 3} = \frac{\langle 1, 4, 9 \rangle}{6} = \langle \frac{1}{6}, \frac{2}{3}, \frac{3}{2} \rangle \text{ m} $$

The centre of mass of this system is at the point $(\frac{1}{6}, \frac{2}{3}, \frac{3}{2})$ in Cartesian coordinates.

Conclusion

The centre of mass is a fundamental concept in physics that simplifies the analysis of motion for systems of particles. By understanding how to calculate and interpret the centre of mass, one can predict the behavior of a system under various forces and conditions. It is particularly important in the study of mechanics and dynamics, where it plays a crucial role in the conservation of momentum and the analysis of collisions.