A uniform thin metal plate of mass 10 kg with dimensions is shown… | JEE Main 2024 (8 Apr, Shift 1)

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Composite & Cut-Out Bodies

A uniform thin metal plate of mass $10\text{ kg}$ with dimensions is shown. The ratio of $x$ and y coordinates of center of mass of plate is $\frac{n}{9}$ . The value of $n$ is ________.

Numerical answer type. Work it out, then open the answer.

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Answer: 15

Plate = full rectangle $3\times 2$ (area $6$ , centre $(1.5,\ 1)$ ) minus square $1\le x\le 2,\ 1\le y\le 2$ (area $1$ , centre $(1.5,\ 1.5)$ ).
The shape is symmetric about $x = 1.5$ , so $x_{cm} = 1.5$ .
$y_{cm} = \frac{6(1) - 1(1.5)}{6-1} = \frac{4.5}{5} = 0.9$
$\frac{x_{cm}}{y_{cm}} = \frac{1.5}{0.9} = \frac{5}{3} = \frac{15}{9}$
So $n = 15$ .

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