A nonuniform rod O M of length l has linear mass density that varies with the distance x from left end of the rod according to λ = λ 0 x 3 L 3 ; Where λ 0 is constant, is kept along x -axis and is rotating about an axis A B , which is perpendicular to the rod as shown in the figure. What is the value of x so that moment of inertia of rod about axis A B I A B is minimum?

Step-by-step solution

I A B = ∫ y 2 d m = λ 0 ℓ 3 ∫ - x l - x ( y + x ) 3 y 2 d y ∫ ( y + x ) 3 y 2 d y = y 2 ( y + x ) 4 4 - ∫ 1 2 y ( y + x ) 4 d y = y 2 ( y + x ) 2 4 - y 10 ( y + x ) 5 + 1 10 ( y + x ) 5 dy = y 2 ( y + x ) 2 4 - y ( y + x ) 5 10 + ( y + x ) 2 60 ∫ - x 0 ( - x y 2 ( y + x ) 3 d x = ( ℓ - x ) 2 ( ℓ ) 4 4 - ( ℓ - x ) ℓ 5 10 + ℓ 6 60 I AB = λ 0 ( ℓ - x ) 2 4 ℓ - 1 10 ( ℓ - x ) ℓ 2 + 1 60 ℓ 3 = λ 0 ℓ ( ℓ - x ) 2 4 ℓ - 1 10 ( ℓ - x ) + 1 60 ℓ 2 d l A B d x = 0 ⇒ - ( ℓ - x ) 2 + ℓ 10 = 0 ⇒ 5 x = 4 ℓ ⇒ x = 4 l 5