Question

Consider a rectangular plate of dimensions $a\times b$. If the plate is considered to be made up of four rectangles of dimensions $\frac{a}{2}\times \frac{b}{2}$ and we now remove one (the lower right) out of the four rectangles, find the position where the centre of mass of the remaining system will be (considering the center of the rectangular plate as the origin)

• A. $\frac{a}{12},\frac{-b}{12}$
• B. $\frac{a}{12},\frac{b}{12}$
• C. $\frac{-a}{12},\frac{-b}{12}$
• D. $\frac{-a}{12},\frac{b}{12}$

Answer 1

The correct option is D $\frac{-a}{12},\frac{b}{12}$

The rectangular plate is shown in the figure of which one part is removed. We can find the x and y coordinates of the centre of mass of this system, taking origin at the centre of the plate. The coordinates of the three remaining rectangles are (a/4, b/4), (-a/4, +b/4) and (-a/4, -b/4). By geometry, masses of these rectangles can be taken as M/4. Now x-coordinate of the centre of mass:
$x_{CM}=\frac{\frac{Ma}{16}-\frac{Ma}{16}-\frac{Ma}{16}}{3\frac{M}{4}}=-\frac{a}{12}$
and y-coordinate of the centre of mass:
$y_{CM}=\frac{\frac{Mb}{16}-\frac{Mb}{16}-\frac{Mb}{16}}{3\frac{M}{4}}=\frac{b}{12}$