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JEE Main202528 Jan 2025Morning ShiftPhysicsCenter of Mass, Momentum and CollisionActual

The center of mass of a thin rectangular plate (fig - x ) with sides of length a and b , whose mass per unit area ( ) varies as = ₀ x a b (where ₀ is a constant), would be

Options

  1. A( 2 3 a, ~b 2 )
  2. B( a 2 , ~b 2 )
  3. C( 1 3 a, ~b 2 )
  4. D( 2 3 a, 2 3 b )

Correct answer

A. ( 2 3 a, ~b 2 )

Step-by-step solution

aligned & d m= d A & aligned d & = (d x)(d y)= ₀ x a b (d x)(d y) x_ c o m & = x d m d m = x ( ₀ x ) a b (d x)(d y) ₀ 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 ( ₀ x a b )(d x)(d y) 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 )

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