1. How the Centre of Mass Moves
The centre of mass (COM) of a system behaves like a single particle that represents the overall motion of the entire system. Even if every particle inside the system moves in a complicated way, the motion of the COM follows simple and predictable laws.
This is extremely helpful because analysing the COM is much easier than tracking each particle's movement. A moving car, a thrown stick, or even a group of dancers — all have a centre of mass that follows a smooth path governed by external forces.
1.1. Averaging Out Individual Motions
Each particle contributes to the position of the centre of mass based on its mass. Heavier particles pull the centre of mass closer to them. As particles move, their individual motions combine to produce a single, averaged motion for the COM.
1.1.1. Everyday Examples
- A juggler’s clubs rotate in the air, but the COM of each club follows a simple arc.
- A diver flips in mid-air, yet the COM travels in a smooth parabolic path.
- When you walk, your body parts move differently, but your COM shifts steadily forward.
2. Mathematical Description of COM Motion
The motion of the centre of mass follows Newton’s laws of motion. If \( \vec{R}_{cm} \) is the position vector of the COM, then the total external force acting on the system determines its acceleration:
\( M \vec{a}_{cm} = \sum \vec{F}_{ext} \)
Here, \( M \) is the total mass of the system, and \( \vec{F}_{ext} \) represents external forces like gravity, friction, or applied pushes.
2.1. Internal Forces Don’t Affect COM
Internal forces — forces between particles of the same system — cancel out because they occur in equal and opposite pairs. As a result, they have no effect on the motion of the centre of mass. This simplifies analysis dramatically.
2.2. External Forces Govern Motion
If the sum of external forces is zero, then the acceleration of the centre of mass is zero. In such cases, the centre of mass moves with constant velocity:
\( \vec{a}_{cm} = 0 \Rightarrow \vec{v}_{cm} = \text{constant} \)
3. Understanding the COM Path
The path traced by the centre of mass depends only on external forces. Even if the object rotates, vibrates, or breaks apart, the COM maintains a regular and predictable motion. This behaviour allows us to treat complex bodies as simple particles for analysing their overall motion.
3.1. Examples of COM Path in Different Situations
- A thrown stick: While the stick spins, its COM moves along a smooth parabola determined by gravity.
- A rocket ejecting fuel: The COM moves smoothly even when mass is lost, provided we consider the system of rocket + ejected fuel.
- Ice skaters: During complex motion, their COM follows a smooth trajectory that reflects their overall movement.
4. Relation Between COM Motion and Total Momentum
The motion of the centre of mass is closely linked to the total momentum of the system. The total momentum \( \vec{P} \) is related to the COM velocity by:
\( \vec{P} = M \vec{v}_{cm} \)
This means that studying the COM velocity directly gives us the total momentum of the system, making it a powerful tool in analysing multi-particle problems.
4.1. When COM Moves with Constant Velocity
If no external force acts on the system, the centre of mass will continue moving with constant velocity. This result forms the basis of several physical principles, including the conservation of momentum.
5. Why COM Motion Simplifies Complex Problems
Tracking each particle in a system is complicated, but studying the centre of mass offers a clean and simple way to understand the system’s behaviour. The COM behaves like a single particle whose motion is governed only by external forces.
This insight is used in topics like collisions, rotational motion, explosion problems, and rigid body dynamics, making the concept of centre-of-mass motion central to mechanics.