2.12 Centre of mass of a rigid complex bodies

Position of centre of mass of a rigid complex bodies can also be known by finding the position of centre of mass of individual sections.

Let us understand this concept with the help of an example.

Question 7. Consider 4 identical rods each of length $L$ are arranged as shown in the figure. Find the position of the centre of mass.

Solution: We can find the position of the centre of mass of the whole system by following some steps.

1. Identify the individual sections in the system

2. Find the position of centre of mass of the individual section

3. Find the co-ordinates of the centre of mass in the Cartesian co-ordinate system

4. Apply the formula of the COM for the system of particles

We know that the centre of mass of rod lies at its centre.

So, the position of centre of mass of all the rods is as shown in the figure.

Position of centre of mass of individual rods are,

Rod 1: ${\overrightarrow r _1} = \left( {\frac{L}{2},0} \right)$

Rod 2: ${\overrightarrow r _2} = \left( {L,\frac{L}{2}} \right)$

Rod 3: ${\overrightarrow r _3} = \left( {\frac{L}{2},L} \right)$

Rod 4: ${\overrightarrow r _4} = \left( {0,\frac{L}{2}} \right)$

As all the rods are identical, let the mass of each rod (let) $M$.

So, the centre of mass $(COM)$ of the system is given by, $$\begin{equation} \begin{aligned} {\overrightarrow r _{COM}} = \frac{{M{{\overrightarrow r }_1} + M{{\overrightarrow r }_2} + M{{\overrightarrow r }_3} + M{{\overrightarrow r }_4}}}{{4M}} \\ {\overrightarrow r _{COM}} = \frac{{{{\overrightarrow r }_1} + {{\overrightarrow r }_2} + {{\overrightarrow r }_3} + {{\overrightarrow r }_4}}}{4} \\ {\overrightarrow r _{COM}} = \frac{{\left( {\frac{L}{2}\widehat i + 0\widehat j} \right) + \left( {L\widehat i + \frac{L}{2}\widehat j} \right) + \left( {\frac{L}{2}\widehat i + L\widehat j} \right) + \left( {0\widehat i + \frac{L}{2}\widehat j} \right)}}{4} \\ {\overrightarrow r _{COM}} = \left( {\frac{L}{2}\widehat i + \frac{L}{2}\widehat j} \right) \\\end{aligned} \end{equation} $$ or $${r_{COM}} = \left( {\frac{L}{2},\frac{L}{2}} \right)$$