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 changes with the distance $x$ from the origin $\rho =a+bx$. (1) Find the SI units of a and b. (2) Find the mass of the rod in terms of a, b and L.

Answer 1

Step 1: Given data

1. The length of the length is $L$.
2. The density of the rod is $\rho =a+bx$.

Step 2: Concept and formulae

1. According to unit analysis, the unit of the left-hand side of an equation is the same as the unit of the right-hand side.
2. The mass of a rod of line charge density $\rho$ is defined by the form, $m={\int }_0^L\rho dx$, here, L is the length of the body.

Step 3: Units of a and b

As we know that

$unitof\rho =unitof\left(a+bx\right)$.

We know the SI unit of linear density is $kg/m$.

So the unit of a is $kg/m$ and unit of $bx$is $kg/m$.

So, the unit of a is $kg/m$ and the unit of b is $kg/m^2$

Step 4: Finding the mass of the rod

The mass of the rod is

$$\begin{gathered} m={\int }_0^L\rho dx \\ orm={\int }_0^L\left(a+bx\right)dx \\ orm={\int }_0^{L}adx+{\int }_0^{L}bxdx \\ orm=a{\left[x\right]}_0^L+b{\left[\frac{x^2}{2}\right]}_0^L \\ orm=aL+b\frac{L^2}{2} \end{gathered}$$

Therefore the mass of the rod is $m=aL+b\frac{L^2}{2}$.