Question

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

• A. $(5 / 6) m ,(5 / 6) m$
• B. $(3 / 4) m ,(3 / 4) m$
• C. $(5 / 8) m ,(5 / 8) m$
• D. $(3 / 5) m ,(3 / 5) m$

Answer 1

Answer: (A)

We can think of the $L$-shape to consist of $3$ squares each of length $1\, m$ as shown in figure.
The mass of each square is $1\, kg$ as the lamina is uniform. The centres of masses $C_{1}, C_{2}$ and $C_{3}$ of the squares are (by symmetry) their geometric centres and have coordinates $(1 / 2,1 / 2),(3 / 2,1 / 2)$ and $(1 / 2,3 / 2)$, 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, $X=\frac{[1(1 / 2)+1(3 / 2)+1(1 / 2)] kg - m }{(1+1+1) kg }$
$=\frac{5}{6} m$
$Y=\frac{[1(1 / 2)+1(1 / 2)+1(3 / 2)] kg - m }{(1+1+1) kg }$
$=\frac{5}{6} m$