The centre of mass of a thin rectangular plate (fig x ) with sides of length a and b , whose mass per unit area $(\sigma)$ varies as $\sigma=\frac{\sigma_{0} \mathrm{X}}{\mathrm{ab}}$ (where $\sigma_{0}$ is a constant), would be $\_\_\_\_$

(A) $\left(\frac{2}{3} a, \frac{b}{2}\right)$

(B) $\left(\frac{2}{3} a, \frac{2}{3} b\right)$

(C) $\left(\frac{a}{2}, \frac{b}{2}\right)$

(D) $\left(\frac{1}{3} a, \frac{b}{2}\right)$

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Conservation of Momentum

The centre 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 = \frac{σ _{0} X}{ab}

(where

σ_{0}

is a constant), would be

____

Answer: (A)

The areal density

σ = σ_{0} x / (ab)

depends only on x, so the plate's density is uniform along y; hence

y_{c m} = \frac{b}{2}

For a thin vertical strip at position x, of width dx and height b:

d m = σ d A = \frac{σ _{0} x}{ab} \cdot b d x = \frac{σ _{0}}{a} x d x

x_{c m} = \frac{\int _{0}^{a} x d m}{\int _{0}^{a} d m} = \frac{\int _{0}^{a} x ^{2} d x}{\int _{0}^{a} x d x} = \frac{a ^{3} /3}{a ^{2} /2} = \frac{2 a}{3}

So the center of mass lies at

(\frac{2 a}{3}, \frac{b}{2})

JEE Main · 2025 · 28 Jan 2025 · Shift 1 · Single correct option