Find the centre of mass of a uniform `L` shaped lamina (a thin flat plate) with dimension as shown in Fig. The mass of the lamina is `3kg`.

Explanation

Refer to Fig Choosing the `X` and Y-axis as shown, the co-ordinates of the vertices of the L-shaped lamina are given in the figure.

As is clear from the figure, the `L` shaped lamina consists of three square, each of side `1m` and mass `1kg ( :'` the lamina is uniform). By symmetry, the centre of mass `C_(1), C_(2), C_(3)` of the squares are their geometic centres and have coordinates `C_(1) ((1)/(2), (1)/(2)), C_(2) ((3)/(2), (1)/(2)),` and `C_(3) ((1)/(2), (3)/(2))`. Taking the masses of the suares to be concentrated at these points, the co-ordinates of the centre of mass are
`x = (m_(1)x_(1) + m_(2)x_(2) + m_(3)x_(3))/(m_(1) + m_(2) + m_(3))`
`= (1 xx(1)/(2) + 1xx (3)/(2) + 1xx (1)/(2))/(1+1+1) = (5)/(6)m`
`y = (m_(1)y_(1) + m_(2)y_(2) + m_(3)y_(3))/(m_(1) + m_(2) + m_(3))`
`= (1 xx(1)/(2) + 1xx (1)/(2) + 1xx (3)/(2))/(1+1+1) = (5)/(6)m`
The centre of mass of L-shaped lamina would lie on the `OD`.