A uniform thin rod AB of length L has linear mass density μ x = a + b x L , where x is measured from A. If the CM of the rod lies at a distance of 7 12 L from A, then a and b are related as:

Question

A uniform thin rod AB of length L has linear mass density μ x = a + b x L , where x is measured from A. If the CM of the rod lies at a distance of 7 12 L from A, then a and b are related as:, 2 a = b, a = 2 b, a = b, 3 a = 2 b

MARKS App · Accepted Answer

Consider an infinitesimal small element of length d x and mars density λ x c m = ∫ d m x ∫ d m = ∫ λ d x x ∫ λ d x = ∫ 0 L a + b L x x d x ∫ 0 L a + b L x d x = ∫ 0 L a x d x + ∫ 0 L b L x 2 d x ∫ 0 L a d x + ∫ 0 L b L x d x = a x 2 2 0 L + b L x 3 3 0 L a x 0 L + b L x 2 2 0 L = a 2 L 2 - 0 2 + b 3 L L 3 - 0 3 a L - 0 + b 2 L L 2 - 0 2 = a 2 L 2 + b L 2 3 a L + b L 2 = a 2 L + b L 3 a + b 2 ∵ x c m = 7 L 12 = a 2 + b 3 L a + b 2 ⇒ 7 12 = 3 a + 2 b 6 2 a + b 2 ⇒ 7 12 = 3 a + 2 b 3 × 1 2 a + b ⇒ 7 2 a + b = 4 3 a + 2 b ⇒ 14 a + 7 b = 12 a + 8 b ⇒ 2 a = b

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