The center of mass of a thin rectangular plate (fig - x) with sides of length a and b, whose mass per unit area (σ) varies as σ = (σ₀ x)/(ab) (where σ₀ is a constant), would be __________. (a) (2/3a, 2/3b) (b) (2/3a, b/2) (c) (1/3a, b/2) (d) (a/2, b/2)

Answer: (b). σ ∝ x only, so along y the plate is uniform: ycm = b/2. xcm = (∫₀a x · x dx)/(∫₀a x dx) = (a³/3)/(a²/2) = 2a/3 So the answer is (b) (2/3a, b/2).

The center of mass of a thin rectangular plate (fig - x) with sides… | JEE Main 2025 (28 Jan, Shift 1)

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Continuous Bodies: Standard Results & Variable Density

The center of mass of a thin rectangular plate (fig - x) with sides of length $a$ and $b$ , whose mass per unit area $(\sigma)$ varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be __________.

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Answer: (b)

$\sigma \propto x$ only, so along $y$ the plate is uniform: $y_{cm} = \frac{b}{2}$ .
$x_{cm} = \frac{\int_0^a x \cdot x\,dx}{\int_0^a x\,dx} = \frac{a^3/3}{a^2/2} = \frac{2a}{3}$
So the answer is (b) $\left(\frac{2}{3}a, \frac{b}{2}\right)$ .

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