Question

You are given a rod of length $L$. The linear mass density is $\lambda$ such that $\lambda =a+bx$. Here $a$ and $b$ are constant and the mass of the rod increases as $x$ decreases. Find the mass of the rod.

Answer 1

We are not calculated the mass of the rod simply multiplying linear mass density $\lambda$ will the length of the rod as $\lambda$ is not constant.
Hence, we have to divide entire rod length into a number of elements and all the mass of all elements.
$M=d{m}_1+d{m}_2+d{m}_3+....$
This step can be written in terms of integration.
$M=\int dm$
If we take an element $dx$ on the rod at distance $x$ from the left end of the rod.
Mass of this element $dm=\lambda .dx$
$dm=(a+bx)dx$
Hence, total mass,
$M=\int dm$
$={\int }_0^L(a+bx)dx$
$={\int }_0^{L}adx+{\int }_0^{L}bxdx$
$=a{\int }_0^{L}dx+b{\int }_0^{L}xdx$
$=a[x{]}_0^L+b{\left[\frac{x^2}{2}\right]}_0^L=aL+\frac{b{L}^2}{2}$.