Introduction:

When you study physics, many problems assume you can treat an object as a point mass. But real objects are made of many particles. To describe how the whole object moves, physicists use the idea of centre of mass. The motion of this point (even for a system of particles) helps simplify many problems, especially when forces act from outside the system.



1. Definition: Centre of Mass (CM)

For a system of discrete particles, the centre of mass is the point at which you can imagine all the mass of the system concentrated, for the purpose of analyzing translational motion. If masses are continuous (like rods, lamina, solid bodies), integrals are used.

Read Also: Angular Momentum (Fixed Axis Rotation) – Class 11 Physics

2. Mathematical Expression: Discrete Particles

If you have n particles with masses m1, m2, ..., mn and position vectors r1, r2, ..., rn, then:

Rcm = (1/M) Σ mi ri, where M = Σ mi.

For the x, y, z components: Xcm = Σ mixi / M, Ycm = Σ miyi / M, Zcm = Σ mizi / M.

3. Motion: Velocity and Acceleration of CM

Velocity of CM: Vcm = (1/M) Σ mi vi

Acceleration of CM: Acm = (1/M) Σ mi ai

These are derived by differentiating the position equation with respect to time.

4. External Forces and Motion of CM

Internal forces cancel due to Newton’s third law. External forces determine the motion:

M Acm = Fext

This means the system behaves like a single particle of mass M placed at CM under the effect of external forces.

5. Special Cases & Examples

- Two-particle system: simplest calculation of CM.

- Uniform rigid bodies (rods, discs, spheres): CM often lies at the geometric centre.

- Explosions or splitting: if no external force, CM continues its motion unaffected.

6. Why It Matters (Utility)

The CM simplifies problem solving. It helps in understanding conservation laws and is widely used in astrophysics, engineering, and rigid body motion problems.



8. Sample Problems (Sketches)

- Find CM of two masses m1, m2 on a line.

- If two masses move with known velocities, find the velocity of the CM.

- A uniform rod: find its CM.

- System breaks apart: track the motion of CM.

Conclusion:

The concept of centre of mass and its motion is powerful. It allows us to deal with complex systems as if they were simple point masses (for translational motion). External forces alone decide the motion of the centre of mass.