The centre of mass is the single point where the entire mass of an object appears to be concentrated. For symmetric bodies, this point is easy to find because it aligns with the geometric center of the shape.

The Uniform Rod

Imagine a thin, straight rod of length L. Because the mass is distributed equally along its length, the centre of mass lies exactly at its midpoint. You can calculate this by measuring L/2 from either end of the rod. If you try to balance a uniform wooden ruler on your finger, you will notice it stays perfectly horizontal when your finger is placed exactly at this halfway point.

Rings and Discs

A ring is a thin circular wire, and a disc is a flat circular plate. For both, the centre of mass is located at their geometric centre. Even though a ring is hollow and has no physical material in its middle, the centre of mass still exists at the empty centre point. For a disc with radius R, the centre of mass is exactly at the center of the circle on the flat surface. This symmetry means that the forces acting on the object balance out perfectly around this central point.

Spheres

A sphere is a perfectly round 3D object. Whether it is a solid ball or a hollow shell, its centre of mass is at its geometric center. For a solid sphere with radius R, the centre of mass is the central point inside the ball. Because the object is identical in every direction from the middle, the mass distribution is perfectly balanced. This explains why a ball rolls smoothly; it rotates around an axis passing directly through this central point of mass.

• Rod: Centre of mass at L/2.
• Ring/Disc: Centre of mass at the geometric center.
• Sphere: Centre of mass at the interior central point.