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 $\sigma=\frac{\sigma_0 x}{a b}$ (where $\sigma_0$ is a constant), would be $\qquad$

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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=(σ_0 x)/(a b) (where σ_0 is a constant), would be

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Correct answer: A. ((2)/(3) a, ~b 2 )

aligned & d m=σ d A & aligned d & =σ(d x)(d y)=(σ_0 x)/(a b)(d x)(d y) x_ c o m & =(∫ x d m)/(∫ d m)=(∫ x ((σ_0 x))/(a b)(d x)(d y))/(∫_0 (σ_0)/(a b)(d x)(d y)) & =(∫_0^a x^2 d x ∫_0^b d y)/(∫_0^b x d x ∫_0^b d y)=(2 a)/(3) y_ c o m & =(∫ y d m)/(∫ d m)=(∫ y((σ_0 x)/(a b))(d x)(d y))/(∫ (σ_0 x)/(a b)(d x)(d y)) aligned aligned =(∫_0^a x d x ∫_0^b y d y)/(∫_0^a x d x ∫_0^b d y)=(b)/(2) i.e., r _ com ((2 a)/(3),(b)/(2) )

The center of mass of a thin rectangular plate (fig - x ) with...

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