Motion of Centre of Mass

subtopicmedium~25 min study10 MCQ

Analyzing the translational motion of the centre of mass under the influence of external forces.

Practice 10 questions Back to syllabus~15 min

What is Motion of Centre of Mass?

The unique point where the weighted relative position of the distributed mass sums to zero. It is the average position of all parts of the system, weighted according to their masses.

Key formula / rule: Equation of Motion for Centre of Mass

Key points

Understand how the centre of mass of a system moves under external forces.
Apply Newton's second law to the centre of mass of a system.
Differentiate between the effects of internal and external forces on the centre of mass.
Analyze situations where the centre of mass remains at rest or moves with constant velocity.

Common exam trap

Confusing internal forces with external forces when analyzing the motion of the centre of mass.

Definitions

Term: Centre of Mass (CM)
Meaning: The unique point where the weighted relative position of the distributed mass sums to zero. It is the average position of all parts of the system, weighted according to their masses.

Term: External Force
Meaning: A force acting on a system of particles that originates from outside the system.

Term: Internal Force
Meaning: A force acting between particles within a system.

Learning objectives

Understand how the centre of mass of a system moves under external forces.
Apply Newton's second law to the centre of mass of a system.
Differentiate between the effects of internal and external forces on the centre of mass.
Analyze situations where the centre of mass remains at rest or moves with constant velocity.

Formulae

Name: Equation of Motion for Centre of Mass
Note: $F_ext$ is the net external force, $M$ is the total mass of the system, and $a_CM$ is the acceleration of the centre of mass.
Expression: $F_ext = M a_CM$

Name: Position of Centre of Mass
Note: This formula defines the position vector of the centre of mass, where $m_i$ is the mass of the i-th particle and $r_i$ is its position vector.
Expression: $R_CM = \frac{\sum m_i r_i}{M}$

Name: Velocity of Centre of Mass
Note: This formula defines the velocity vector of the centre of mass.
Expression: $V_CM = \frac{\sum m_i v_i}{M}$

Name: Acceleration of Centre of Mass
Note: This formula defines the acceleration vector of the centre of mass.
Expression: $a_CM = \frac{\sum m_i a_i}{M}$

Prerequisites

Newton's Laws of Motion
Concept of Centre of Mass
Linear Momentum
Force and Acceleration

Common mistakes

Confusing internal forces with external forces when analyzing the motion of the centre of mass.
Assuming the centre of mass is always at rest, even when external forces are present.
Incorrectly applying Newton's laws to individual particles instead of the system's centre of mass.

Keywords

Centre of Mass
System of Particles
External Force
Internal Force
Newton's Second Law
Translational Motion
Conservation of Momentum

Practice preview

Consider a system of two particles of masses 2 kg and 3 kg. If an external force of 10 N acts on the system, what is the acceleration of the centre of mass?…
easy
A system of particles is in motion. If the centre of mass of the system is observed to be accelerating, what can be concluded about the forces acting on the system?…
medium
A system of particles is subjected to external forces. If the net external force is zero, which of the following is true about the centre of mass?…
medium

Practice 10 questions