1. Understanding Centre of Mass in Regular Shapes
Many everyday objects have regular and symmetrical shapes. For such bodies, finding the centre of mass becomes significantly easier because their mass is distributed uniformly. Instead of calculating the position of every tiny particle, we make use of symmetry to determine the COM quickly and accurately.
Before diving into the formulas, it's helpful to build an intuitive understanding of where the COM lies for objects we commonly see—like sticks, discs, rings, and spheres.
1.1. Why Symmetry Helps
When an object's shape and mass distribution are symmetrical, the centre of mass lies along the line or point of symmetry. This allows us to determine the COM simply by observation and basic reasoning, without requiring detailed calculations.
1.1.1. Simple Visual Examples
- A uniform rod balances exactly at its midpoint because mass is evenly spread across its length.
- A circular disc has its COM at the geometric centre due to radial symmetry.
- A hollow ring (like a bracelet) also has its COM at its centre because the mass is distributed equally around the circle.
2. Centre of Mass of a Uniform Rod
A uniform rod has the same mass per unit length. Therefore, the centre of mass lies exactly at its midpoint. This is one of the simplest examples of using symmetry to locate the COM.
2.1. Mathematical Expression
If the rod lies along the x-axis from \( x = 0 \) to \( x = L \), the position of the centre of mass is:
\( x_{cm} = \dfrac{L}{2} \)
This position remains the same regardless of the rod’s mass, as long as it is uniformly distributed.
3. Centre of Mass of a Disc
A disc has both radial and circular symmetry. All parts of the disc are equally spaced around the centre, so the centre of mass must lie at the geometric centre of the disc.
3.1. Reasoning with Symmetry
If you divide a disc into equal sectors, each section has an identical opposite section. Their individual masses balance each other, meaning the COM must be at the centre point where all axes of symmetry intersect.
4. Centre of Mass of a Ring
A ring is a hollow circular object where all the mass is distributed at a fixed distance from the centre. Just like a disc, a ring has circular symmetry, but the mass lies on the circumference rather than the entire area.
4.1. Position of the COM
The centre of mass of a ring lies at the geometric centre, which is the centre of the circular path formed by the ring. Even though no actual material is present at the centre, the COM still lies at that point due to symmetry.
5. Centre of Mass of a Sphere
Spheres are among the most symmetrical objects in physics. All points on a sphere’s surface are the same distance from its centre, and the mass is uniformly distributed throughout the volume.
5.1. COM of a Solid Sphere
In a solid sphere, the centre of mass is located at the geometric centre because every direction has identical mass distribution.
5.2. COM of a Hollow Sphere
For a hollow sphere or spherical shell, the centre of mass is also at the geometric centre. Although the mass exists only on the surface, symmetry ensures the COM remains at the centre.
6. Centre of Mass of Composite Bodies
Sometimes, objects are made by combining simple shapes. In such cases, the COM can be found using the weighted average formula for different components. This requires calculating the COM of each simple shape and then combining them.
6.1. Formula for Composite Systems
For two parts with masses \( m_1 \) and \( m_2 \) located at positions \( x_1 \) and \( x_2 \), the combined COM is:
\( x_{cm} = \dfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)
This method is commonly used in engineering and design to determine balance points.
7. Summary of COM for Common Shapes
The centre of mass for regular shapes can often be identified by symmetry. The following list gives the COM for commonly used bodies:
- Uniform Rod: At the midpoint.
- Disc: At the geometric centre.
- Ring: At the centre of the circle.
- Solid Sphere: At the centre of the sphere.
- Hollow Sphere: At the geometric centre.
Understanding these basic cases prepares you for more complex systems involving rotation, torque, and equilibrium in later topics.