Q 11-06-091JEE MainJEE Main 2024 (8 Apr, Shift 1)Medium
A uniform thin metal plate of mass $10\ \text{kg}$ with dimensions is shown. The ratio of $x$ and $y$ coordinates of the centre of mass of the plate is $\dfrac n9$. The value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 15
Treat the plate as a $3\times2$ rectangle (area 6, centre $(1.5, 1)$) with the square from $(1,1)$ to $(2,2)$ removed (area 1, centre $(1.5, 1.5)$). The remaining area is 5.
$$x_{cm} = \frac{6(1.5) - 1(1.5)}{5} = 1.5,\qquad y_{cm} = \frac{6(1) - 1(1.5)}{5} = 0.9$$
$$\frac{x_{cm}}{y_{cm}} = \frac{1.5}{0.9} = \frac53 = \frac{15}{9} \;\Rightarrow\; n = 15$$
Solution by Sreeraj P, M.Sc Physics