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 $3 \;kg$.
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 $3 \;kg$.
Choosing the $X$ and $Y$ axes as shown in Figure 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 $(1 / 2,1 / 2)$ $(3 / 2,1 / 2),(1 / 2,3 / 2)$ 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
$X=\frac{[1(1 / 2)+1(3 / 2)+1(1 / 2)] kg m }{(1+1+1) kg }=\frac{5}{6} m$
$Y=\frac{[1(1 / 2)+1(1 / 2)+1(3 / 2)] \quad kg m }{(1+1+1) kg }=\frac{5}{6} m$
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