Obtain the position of the centre of mass of a thin rod of uniform density.

(N/A) homogeneous body is a body with uniformly distributed mass.
Consider a thin rod of length $L$ and total mass $M$. Let the origin be at one end of the rod,and let the $X$-axis lie along the length of the rod.
The linear mass density $\lambda$ is given by $\lambda = \frac{M}{L}$.
Consider a small element of length $dx$ at a distance $x$ from the origin. The mass of this element is $dm = \lambda dx = \frac{M}{L} dx$.
The position of the centre of mass $X_{cm}$ is given by:
$X_{cm} = \frac{1}{M} \int x dm$
Substituting $dm$:
$X_{cm} = \frac{1}{M} \int_{0}^{L} x \left( \frac{M}{L} \right) dx$
$X_{cm} = \frac{1}{L} \int_{0}^{L} x dx$
$X_{cm} = \frac{1}{L} \left[ \frac{x^2}{2} \right]_{0}^{L}$
$X_{cm} = \frac{1}{L} \left( \frac{L^2}{2} - 0 \right) = \frac{L}{2}$
Thus,the centre of mass of a uniform thin rod is at its geometric centre,which is at a distance of $\frac{L}{2}$ from either end.