The distance of centre of mass from end A of a one dimensional rod A B having mass density ρ = ρ 0 1 - x 2 L 2 kg m - 1 and length L (in meter) is 3 L α m . The value of α is _____ (where x is the distance form end A )

Step-by-step solution

Given here: Mass density, ρ = ρ 0 1 - x 2 L 2   kg   m - 1 Let us suppose there is an element of length d x at a distance x from left must end. Mass of this element is d m = ρ d x = ρ 0 1 - x 2 L 2 The centre of mass of this element is X cm = ∫ xdm ∫ dm = ρ 0 ∫ 0 l x - x 2 L 2 x d x ∫ 0 l ρ 0 1 - x 2 L 2 d x = L 2 2 - L 4 4 L 2 L - L 3 3 L 2 = L 2 4 2 L 3 = 3 L 8 Hence, the value of α = 8 .