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`.

Options

A. `((5)/(6)m,(5)/(6)m)`
B. `(1 m, 1 m)`
C. `((6)/(5)m, (6)/(5)m)`
D. `(2 m, 2 m)`

Answer

A

Explanation

(a) Choosing the `x` and `y` axes as shown in the figure. The coordinates of the vertices of the L-shaped lamina is as shown in the figure. Divided the L-shaped lamine into three squares each of side `1 m` and mass `1 kg` (because the lamina is uniform). By symmetry, the centres of mass `C_1,C_2` and `C_3` of the squares are their geometric centres and have coordinates `C_1((1)/(2),(1)/(2)), C_2((3)/(2),(1)/(2))` and `C_3((1)/(2),(3)/(2))` respectively.
The coordinates of the centre of mass of the L-shaped lamina is.
`X_(CM) =(m_1 x_1 + m_2x_2 + m_3 x_3)/(m_1 + m_2 + m_3)`
=`(1 xx(1)/(2) + 1 xx (3)/(2) + 1 xx(1)/(2))/(1 + 1 + 1) =(5)/(6) m`
`Y_(CM) = (m_1 y_1 + m_2 y_2 + m_3 y_3)/(m_1 + m_2 + m_3)`
=`(1 xx(1)/(2) + 1 xx (1)/(2) + 1 xx (3)/(2))/(1 + 1+1) = (5)/(6) m`.
.