Question

a rod of uniform thickness is placed along x-axis with one of its end at origin.If length of rod is L and its linear mass density is proportional to x then find distance of its centre of mass from origin

Answer 1

let linear density be ​[ρ which is proportional to x. So ρ = K x So the mass of the piece of thickness dx at a distance x will be K x dx As all the masses lie in the same line i.e x axis, the centre of mass can be calculated as ∫K x dx * x / ∫Kx dx Integrating within limits 0 to L [(K x^3 / 3) / (K x^2 /2)]L0 Plugging the limits the centre of mass will be at 2 L/3 from the origin

Answer 2

heres the answer ☺ using the formula for centre of mass of rigid continuous body