Question

Find the centre of mass of a uniform $L-$shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is $3kg$.

Answer 1

Choosing the X and Y axes as shown in Fig. we have the coordinates of the vertices of the L-shaped lamina as given in the figure. We can think of the L-shape to consist of $3$ squares each of length $1$m. The mass of each square is $1$kg, since the lamina is uniform. The centres of mass $C_1,C_2$ and $C_3$ of the squares are, by symmetry, their geometric centres and have coordinates $(1/2,1/2),\left(3/2.1/2\right),(1/2,3/2)$ respectively. We take the masses of the squares to be concentrated at these points. The centre of mass of the whole L shape $(X,Y)$ is the centre of mass of these mass points.
Hence $X=\frac{\left[1\right(1/2)+1(3/2)+1(1/2\left)\right]kgm}{(1+1+1)kg}=\frac{5}{6}m$
$Y=\frac{\left[1\right(1/2)+1(1/2)+1(3/2\left)\right]kgm}{(1+1+1)kg}=\frac{5}{6}m$
The centre of mass of the L-shape lies on the line OD. We could have guessed this without calculations. Can you tell why? Suppose, the three squares that make up the L-shaped lamina of Fig. had different masses. How will you then determine the centre of mass of the lamina?