Question

Derive an expression for the position of centre of mass of a system of n particles and for continuous mass distribution.

Answer 1

System of n particles:

i. Consider a system of n particles of masses m₁, m₂ …, mn having position vectors \(\overrightarrow{r}_1\), \(\overrightarrow{r}_2\) ,….., \(\overrightarrow{r}_n\) from the origin O.

The total mass of the system is,

Centre of mass for n particles:

ii. Position vector \(\overrightarrow{r}\) of their centre of mass from the same origin is then given by

iii. If the origin is at the centre of mass, \(\overrightarrow{r}\) = 0

∴ \(\sum^n_1\) m_i \(\overrightarrow{r}_i\) = 0,

iv. In this case, \(\sum^n_1\) m_i \(\overrightarrow{r}_i\) gives the moment of masses (similar to moment of force) about the centre of mass.

v. If (x₁, x₂, …… x_n), (y₁, y₂, ….y_n), (z₁, z₂, … z_n) are the respective x, y and z – coordinates of (r₁, r₂, …….. r_n) then x,y and z – coordinates of the centre of mass are given by,

vi. Continuous mass distribution: For a continuous mass distribution with uniform density, the position vector of the centre of mass is given by,

Where ∫ dm = M is the total mass of the object.

vii. The Cartesian coordinates of centre of mass are