Question

A rod of length L is placed along the X-axis between x = 0 and x = L. The linear density (mass/length) rho of the rod varies with the distance x from the origin as rho = 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

(a) To find the S.I units of "a" and "b" is S.I. unit of 'a' = kg/m   and S.I unit of 'b' = kg/m² .

(b) To find the mass of the rod in terms of a, b and L is aL + (bL²2) .

Explanation:

Given data  :

A rod of length L is set between x = 0 and x = L along the X-axis.

ρ = mass / length = a + bx

(a) To find the S.I units of "a" and "b" :

     S.I. unit of 'a' = kg/m

     S.I unit of 'b' = kg/m²

That is, because of the principle of dimensional homogeneity.  

(b) To find the mass of the rod in terms of a, b and L :

Consider a small length dimension ' dx ' at a distance x from its origin

Therefore,  

                 dm = mass of element = ρ dx = (a+b) x

                 rod’s mass = m = ∫dm = ₐ∫ᵇ (a+bx) dx  

                                    = ₐ[ax + (bx²2)]ᵇ  

Here, a = 0 , b = L  

                       = aL + (bL²2) .