Question

Find the position of the centre of mass of $T$ - shaped lamina of negligible thickness as shown in figure. Assume the origin to be at the intersection of axes and take the mass density of the lamina, $\sigma =1{\text{kg/m}}^2$

• A. $(0,6.5)\text{m}$
• B. $(0,6)\text{m}$
• C. $(0,5.28)\text{m}$
• D. $(0,4.55)\text{m}$

Answer 1

The correct option is C $(0,5.28)\text{m}$

The $T$ - shaped lamina is divided into two pieces of mass $m_1$ and $m_2$


$m_1=\sigma \times$ Area of part $1$
$=1\times (6\times 2)$
$=12\text{kg}$
$m_2=\sigma \times$ Area of part $2$
$=1\times (8\times 2)$
$=16\text{kg}$
Taking origin at point $\text{O}$,
Centre of mass of part $1$ is $(x_1,y_1)=(0,3)\text{m}$
Centre of mass of part $2$ is $(x_2,y_2)=(0,7)\text{m}$

$x$ - coordinate of center of mass will be 0 $(\because$ lamina is symmetric about $y$ - axis$)$ i.e $x_{com}=0\text{m}$
and $y$ - coordinate of centre of mass is
$y_{com}=\frac{m_1{y}_1+m_2{y}_2}{m_1+m_2}$
$=\frac{12\times 3+16\times 7}{12+16}=\frac{148}{28}\text{m}$
= $5.28\text{m}$
So, position of centre of mass of $T$- shaped lamina is $(x_{com},y_{com})=(0,5.28)\text{m}$