You are given a rod of length 'L'. The linear mass density varies as λ′. λ=a+bx. Here a & b are constants and the mass of the rod increases as x increases. Find the mass of the rod.
We cannot calculate the mass of the rod by simply multiplying linear mass density '′λ′ will the length of the rod as ′λ′ is not constant.
Hence we have to divide entire rod length into a number of elements and add the mass of all elements.
This step can be written in terms of integration.
If we take an element dx on the rod at a distance x from the left end of the rod.
Mass of this element dm=λ.dx
Hence total mass M=∫dm⇒M=L∫0(a+bx)dx
M=L∫0a dx+L∫0bx dx=aL∫0dx+bL∫0x dx=a[x]L0+b[x22]L0⇒M=aL+bL22