It is stated that a straight rod has one endpoint at the origin (0,0) and the opposite endpoint at (L,0), with a linear density defined by , where a is a constant and x is the x coordinate.
Thus, the infinitesimal mass is expressed as:
The total mass can be calculated by integrating the above expression as follows:
Consequently,
Now, we can calculate the center of mass, of the rod as:
Now, it follows that
x_{cm}=\frac{1}{\frac{aL^3}{3}}\int_{0}^{L}ax^3dx=\frac{3}{aL^3}\times [\frac{ax^4}{4}]_{0}^{L}
Therefore, the center of mass, is located at: