S4.3 Motion of the centre of mass
[A] Centre of mass and Newton’s 2nd law
- Let’s begin by writing the centre of mass vector (Key Point 4.1) in the following way:
- The centre of mass will evolve with time. Its velocity is defined (Key Point 1.1) by differentiating Equation 4.1 term-by-term with respect to time.
The acceleration of the centre of mass is given by another differentiation (Key Point 1.2)
Using Newton’s 2nd law (Key Point 2.2) we can write this as
where
is the force acting on particle 1, etc.
- The forces
,
, etc, acting on the particles are of two types:
- those acting on the system from outside: external forces
- forces acting within the system: internal forces
- By Newton’s 3rd law
(Key Point 2.3), the internal
forces form action-reaction pairs and cancel from the sum
in Equation 4.2. All that is left are the external
forces.
Key Point 4.3
Newton’s 2nd law for a system of particles. The motion of the centre of mass is governed by external forces only
- This gives us another way of viewing the centre of mass:
Key Point 4.4
The centre of mass of a system of particles is that point that moves as though all of the mass were concentrated there and all external forces were applied there.
[B] Consequences
Key Point 4.3 is the key result. It tells us that:- if the centre of mass has zero velocity and no external forces act, then regardless of the motion of the individual bodies in the system, the position of the centre of mass remains fixed (Q5.5).
- linear momentum is conserved. We will learn more about linear momentum in subsequent sections.
Learning Resources
![]() | HRW Chapter 9.3 |
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