Question

Seven homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced with respect to the one in contact by L/10. Find the x-coordinate fo the centre of mass relative to the origin shown.
Figure

Answer 1

Let OX be the X-axis and point O (0, 0) be the origin of the system.

The mass of each brick is M.
The length of each brick is L.
Each brick is displaced with respect to another in contact by a distance $\frac{\mathrm{L}}{10}$.

∴ The X-coordinate of the centre of mass is given as,
$$\begin{gathered} {\mathrm{X}}_{\mathrm{cm}}={\frac{1}{7m{\displaystyle \frac{\mathrm{L}}{10}}}}^{\left\{\frac{m\mathrm{L}}{2}+m\left(\frac{\mathrm{L}}{2}+\frac{\mathrm{L}}{10}\right)+..........+m\left(\frac{\mathrm{L}}{2}\right)\right\}} \\ {\mathrm{X}}_{\mathrm{cm}}=\frac{{\displaystyle \frac{\mathrm{L}}{2}}+{\displaystyle \frac{\mathrm{L}}{2}}+{\displaystyle \frac{\mathrm{L}}{10}}+{\displaystyle \frac{\mathrm{L}}{5}}+{\displaystyle \frac{\mathrm{L}}{2}}+{\displaystyle \frac{3\mathrm{L}}{10}}+{\displaystyle \frac{\mathrm{L}}{2}}+{\displaystyle \frac{\mathrm{L}}{5}}+{\displaystyle \frac{\mathrm{L}}{2}}+{\displaystyle \frac{\mathrm{L}}{10}}+{\displaystyle \frac{\mathrm{L}}{2}}}{7} \\ =\frac{7{\displaystyle \frac{\mathrm{L}}{2}}+5{\displaystyle \frac{\mathrm{L}}{10}}+2{\displaystyle \frac{\mathrm{L}}{5}}}{7}=\frac{22\mathrm{L}}{35} \end{gathered}$$

The X-coordinate of the centre of mass relative to the origin is $\frac{22\mathrm{L}}{35}$.