A rod of length L is placed along the X-axis between x = 0 and x = L. The linear density (mass/length) ρ of the rod varies with the distance x from the origin as ρ = a + bx. (a) Find the SI units of a and b. (b) Find the mass of the rod in terms of a, b and L

Sol. . ρ = mass/length = a+ bx (a) S.I. unit of ‘a’ = kg/m and SI unit of ‘b’ = kg/m2 (from principle of homogeneity of dimensions) (b) Let us consider a small element of length ‘dx’ at a distance x from the origin as shown in the figure. ∴ dm = mass of the element = ρ dx = (a + bx) dx So, mass of the rod = m = ∫ dm = ∫_0^L〖(a+bx)〗 dx = [ax+(bx^2)/2]_0^L = aL + (bL^2)/2