Question

Linear mass density (mass per unit length) of a rod depends on the distance from one end (say A) as ${\lambda }_x=ax+\beta .Hereaand\beta .$ are constants. The moment of inertia of this rod about an axis passing through A and perpendicular to the rod. (Length of the rod is l) is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$\text{At a distance x from A, consider an element of length dx. Linear mass density at the element is =}ax+\beta \therefore Massoftheelementisdm=(ax+\beta )dx\text{Moment of inertia of this element is}dl=dm.x^2=x^2(ax+\beta )dxordl=(a{x}^3+\beta x^2)dx\therefore I_A={\int }_1^0(a{x}^3+\beta x^2)dx=\frac{a{l}^4}{4}+\beta \frac{l^3}{3}$