Concept of Centre of Mass

Centre of Mass (CM)

The unique point in a system of particles or a body where the weighted average position of all the mass is located. It is the point at which the system would balance if it were pivoted there.

System of Particles

A collection of discrete particles, each with its own mass and position, considered together as a single entity for analysis.

Translational Motion

Motion in which all points of a body move in the same direction and at the same speed. The centre of mass describes the translational motion of the entire system.

Position Vector of Centre of Mass (Discrete System)

Where mᵢ is the mass of the i-th particle and rᵢ is its position vector. M is the total mass of the system.

\vec{R}_{CM} = \frac{\sum_i=1^{n} m_i \vec{r}_i}{\sum_i=1^{n} m_i} = \frac{1}{M} \sum_i=1^{n} m_i \vec{r}_i

Velocity of Centre of Mass (Discrete System)

Where vᵢ is the velocity of the i-th particle.

\vec{V}_{CM} = \frac{d\vec{R}_{CM}}{dt} = \frac{\sum_i=1^{n} m_i \vec{v}_i}{\sum_i=1^{n} m_i} = \frac{1}{M} \sum_i=1^{n} m_i \vec{v}_i

Acceleration of Centre of Mass (Discrete System)

Where aᵢ is the acceleration of the i-th particle. Using Newton's second law (Fᵢ = mᵢaᵢ), A<0xE2><0x82><0x9C><0xE1><0xB5><0x80> = (Σ Fᵢ) / M = F<0xE2><0x82><0x91><0xE2><0x82><0x99><0xE1><0xB5><0x9C> / M.

\vec{A}_{CM} = \frac{d\vec{V}_{CM}}{dt} = \frac{\sum_i=1^{n} m_i \vec{a}_i}{\sum_i=1^{n} m_i} = \frac{1}{M} \sum_i=1^{n} m_i \vec{a}_i

Centre of Mass (Continuous Body)

Where dm is an infinitesimal mass element and r is its position vector. Integration is performed over the entire body.

\vec{R}_{CM} = \frac{\int \vec{r} dm}{\int dm} = \frac{1}{M} \int \vec{r} dm