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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.
Solution
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
\[= \int_0^L \left( a + bx \right) dx\]
\[ = \left[ ax + \frac{b x^2}{2} \right]_0^L \]
\[ = aL + \frac{b L^2}{2}\]
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