Concept Card: Motion of the Centre of Mass 1. Kinematics & Dynamics of Centre of Mass: The velocity and acceleration of the centre of mass of a system of particles of total mass $M = \Sigma m_i$ are given by the first and second time derivatives of its position vector: $$\mathbf{\vec{v}_{\text{cm}} = \frac{d\vec{r}_{\text{cm}}}{dt} = \frac{\Sigma … Read more
Concept Card: Centre of Mass of a Rigid Body & Continuous Mass Distribution 1. Physical Framework & Integral Formulations: For extended rigid bodies comprising a continuous distribution of matter, the discrete sum $\Sigma m_i \vec{r}_i$ transforms into a definite integral over infinitesimal mass elements $dm$: $$\mathbf{\vec{r}_{\text{cm}} = \frac{\int \vec{r}\,dm}{\int dm} = \frac{1}{M}\int \vec{r}\,dm}$$ where $M … Read more
Concept Card: Centre of Mass of a Two-Particle System 1. Physical Framework & Core Meaning: The centre of mass (CM) of a system of particles is the unique spatial point that moves as though all the mass of the system were concentrated at that point and all external forces were applied directly to it. Internal … Read more