Question

How do you find the center of mass of a rod?

Answer 1

Step 1: Given that

1. Suppose a rod of mass $M$ and length $L$ is lying along the $x$-axis with one end at $x=0$ and the other at $x=L$.
2. Mass per unit length of the rod: $\mathrm{I}=\frac{\mathrm{M}}{\mathrm{L}}$
3. Hence, $dm(themassoftheelementdxsituatedatx=xis)=Idx$
4. The coordinates of the element $dx$ are $(x,0,0)$.

Step 2: Calculate the center of mass of a rod

The rod's x-coordinate for its center of mass (COM) will be:

$$\begin{gathered} \Rightarrow {\mathrm{x}}_{\mathrm{COM}}=\frac{{\int }_0^{\mathrm{L}}\mathrm{x}d\mathrm{m}}{{\int }_0^{\mathrm{L}}d\mathrm{m}} \\ \Rightarrow {\mathrm{x}}_{\mathrm{COM}}=\frac{{\int }_0^{\mathrm{L}}\mathrm{x}\left(\mathrm{\lambda }\mathrm{dx}\right)}{{\int }_0^{\mathrm{L}}\mathrm{\lambda }\mathrm{dx}} \\ \Rightarrow {\mathrm{x}}_{\mathrm{COM}}=\frac{1}{\mathrm{L}}{\int }_0^{\mathrm{L}}\mathrm{xdx} \end{gathered}$$

Step 3: Integrate the center of mass

Integrate the center of mass with respect to $x$

$$\begin{gathered} \Rightarrow {\mathrm{x}}_{\mathrm{COM}}=\frac{1}{\mathrm{L}}{\left[\frac{{\mathrm{x}}^2}{2}\right]}_0^{\mathrm{L}} \\ \Rightarrow {\mathrm{x}}_{\mathrm{COM}}=\frac{\mathrm{L}}{2} \end{gathered}$$