Question

A circular plate of diameter $a$, is kept in contact with a square plate of side $a$ as shown in figure. The density of the material and thickness are same everywhere. The centre of mass of the composite system will be :

• A. Inside the circular plate
• B. Inside the square plate
• C. At the point of contact
• D. Outside the system

Answer 1

The correct option is B Inside the square plate

Considering origin $(0,0)$ at the point of contact of the given plates.
Due to symmetry about $x-$axis, the centre of mass of system will lie on the $x-$axis.


Area of circular plate, $A_1=\pi {\left(\frac{a}{2}\right)}^2$
x - coordinate of $\text{COM}$, $x_1=-\frac{a}{2}$
Area of square plate, $A_2=a^2$
x - coordinate of $\text{COM}$, $x_2=+\frac{a}{2}$

$\Rightarrow x_{\text{CM}}=\frac{A_1{x}_1+A_2{x}_2}{A_1+A_2}$
$\Rightarrow x_{\text{CM}}=\frac{\pi {\left(\frac{a}{2}\right)}^2\times (-\frac{a}{2})+a^2\left(\frac{a}{2}\right)}{\pi {\left(\frac{a}{2}\right)}^2+a^2}$
$\therefore x_{\text{CM}}=\frac{a(4-\pi )}{2(\pi +4)}$
It clearly shows that $x_{\text{CM}}>0$, hence $\text{COM}$ of composite system lies inside the square plate.