Question

A smaller disc is removed from a uniform circular lamina as shown in the figure. Find the position of $COM$ of the uniform lamina, if diameter of the smaller disc is $12\text{m}$.

• A. $(-6,0)\text{m}$
• B. $(-6,-6)\text{m}$
• C. $(-2,0)\text{m}$
• D. $(-2,-2)\text{m}$

Answer 1

The correct option is C $(-2,0)\text{m}$

Let, $A_1$ = area of complete lamina
$A_1=\pi \times {12}^2=144\pi {\text{m}}^2$
$A_2$ = area of smaller disc
$A_2=\pi (12/2{)}^2=36\pi {\text{m}}^2$
Coordinates of $COM$ of complete lamina are $(0,0)$
Coordinates $COM$ of smaller disc are $(6,0)$

Taking removed part as disc with negative area.
$X_{com}=\frac{A_1{X}_1-A_2{X}_2}{A_1-A_2}$
$=\frac{144\pi \times 0-36\pi \times 6}{144\pi -36\pi }=\frac{-216\pi }{108\pi }$
i.e $X_{com}=-2\text{m}$
Also, $Y_{com}=0\text{m}$ (as both $Y_1$ and $Y_2$ are zero)
Therefore, coordinates of $COM$ of lamna are $(-2,0)\text{m}$.