Question

For a uniform square sheet from which another smaller square is cut out as shown in the figure, find out the distance of centre of mass of the remaining part from the $y-$axis. Consider the origin as shown in figure.

• A. $1.96\text{cm}$
• B. $16.66\text{cm}$
• C. $36\text{cm}$
• D. $5\text{cm}$

Answer 1

The correct option is B $16.66\text{cm}$

Considering the removed smaller square part $(\text{side}=20\text{cm})$ as the negative mass superimposed on the larger square $(\text{side}=40\text{cm})$

Area of larger square
$A_1={40}^2$
& Area of smaller square
$A_2={20}^2$


Their respective $x$ coordinates of COM are given by:
$x_1=20\text{cm}$
& $x_2=20+10=30\text{cm}$

The required distance of COM from $y$ axis is the $x_{\text{CM}}$

$x_{\text{CM}}=\frac{A_1{x}_1-A_2{x}_2}{A_1-A_2}=\frac{\left(\right({40}^2\times 20)-({20}^2\times 30\left)\right)}{({40}^2-{20}^2)}$

$\Rightarrow x_{\text{CM}}=\frac{50}{3}=16.66\text{cm}$

The cut is symmetric along a line passing through the centre of bigger square and parallel to $x-$ axis, so COM of system will lie along that line. i.e $y_{\text{CM}}=20\text{cm}$

Distance of COM from y-axis $=16.66\text{cm}$