A metal rod of length L having uniform cross-section area A and variable linear mass density $λ_{x} = λ_{0} (1 + \frac{x}{L})$ , x is measured from the end A. Rod is placed on a smooth horizontal surface. A constant force $F_{0}$ is applied as shown, find the Elastic energy stored is the rod.

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TestHub · Accepted Answer

M = ∫ 0 L ​ λ 0 ​ ( 1 + L x ​ ) d x = 2 3 λ 0 ​ L ​ ; a = M F 0 ​ ​ = 3 λ 0 ​ L 2 F 0 ​ ​ Tension in element at x: T = m x ​ a = λ 0 ​ ( x + 2 L x 2 ​ ) 3 λ 0 ​ L 2 F 0 ​ ​ = 3 L 2 F 0 ​ ​ ( x + 2 L x 2 ​ ) E = ∫ 2 A Y T 2 d x ​ = ∫ 0 L ​ 9 A Y L 2 2 F 0 2 ​ ​ ( x + 2 L x 2 ​ ) 2 d x = 135 A Y 19 F 0 2 ​ L ​ Hence (D).

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