Question

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

Thanks for asking the question!

ANSWER::

ρ = mass/length = a + bx

(a) S.I. unit of 'a' = kg/m
      S.I unit of 'b' = kg/m²
i.e. due to the principle of homogeneity of dimensions

(b) Consider a small element of length 'dx' at a distance x from the origin 

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

Mass of the road = m = ∫dm = ₐ∫ᵇ (a+bx) dx = ₐ[ax + (bx²/2)]ᵇ = aL + (bL²/2)

Here a = 0 , b = L 

Hope it helps!