Question

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.

Answer 1

ρ = mass/length = a + bx

So, the SI unit of ρ is kg/m.

(a)
SI unit of a = kg/m
SI unit of b = kg/m²
(From the 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 given below:


dm = mass of the element
= ρdx
= (a + dx) dx
∴ Mass of the rod = ∫ dm
$$\begin{gathered} ={\int }_0^L\left(a+bx\right)dx \\ ={\left[ax+\frac{b{x}^2}{2}\right]}_0^L \\ =aL+\frac{b{L}^2}{2} \end{gathered}$$