Question

The mass per unit length of a non uniform rod of length L varies as \[{\text{m} = {\lambda x}}\]. Where \[\lambda\] is a constant. The center of mass of the rod will lie at
 (A) \[\dfrac{{\text{2}}}{{\text{3}}}{\text{L}}\]
 (B) \[\dfrac{3}{2}{\text{L}}\]
 (C) \[\dfrac{1}{2}{\text{L}}\]
 (D) \[\dfrac{4}{3}{\text{L}}\]

Answer 1

Hint The center of mass of a body is the summation of mass of the body multiplied by their respective distances divided by the total mass of the body

Complete step-by-step solution
We know that position of center of mass:
 \[{{\text{X}}_{{\text{cm}}}}{\text{ = }}\dfrac{{\smallint {\text{xdm}}}}{{\smallint {\text{dm}}}}\]
Where x is the distance of dm mass from one end of the body.
In the question, we are given that the body is non uniform and its mass distribution is given by \[{\text{m} = {\lambda x}}\], differentiating the equation we get,
 \[{\text{dm} = {\lambda dx}}\],
  \[{{\text{X}}_{{\text{cm}}}}{\text{ = }}\dfrac{{\smallint {\text{x}\lambda dx}}}{{\smallint {\lambda dx}}}\]
 \[{{\text{X}}_{{\text{cm}}}}{\text{ = }}\dfrac{{\dfrac{{{\text{|}}{{\text{x}}^{\text{2}}}{\text{|}}_{\text{0}}^{\text{L}}}}{{\text{2}}}}}{{{\text{|x|}}_{\text{0}}^{\text{L}}}}{\text{ = }}\dfrac{{{{\text{L}}^{\text{2}}}}}{{{\text{2L}}}}{\text{ = }}\dfrac{{\text{L}}}{{\text{2}}}\]

Therefore, the correct answer is option D.

Note Whenever we have to find the center of mass of a non uniform section, we have to follow this procedure. In case you are given masses of different bodies and their respective location, then you have to use the summation symbol but if the body is continuous, we have to use the integration.