College Physics (4th Edition)

Since the value of $k$ is proportional to $m$, the angular frequency of a pendulum is independent of the mass. $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{m~g}{L~m}} = \sqrt{\frac{g}{L}}$
For a simple pendulum, $\omega = \sqrt{\frac{g}{L}}$ Suppose that $\omega$ has the form $\sqrt{\frac{k}{m}}$. We can find an expression for $k$: $\frac{k}{m} = \frac{g}{L}$ $k = \frac{mg}{L}$ Since the value of $k$ is proportional to $m$, the angular frequency of a pendulum is independent of the mass. Note that $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{m~g}{L~m}} = \sqrt{\frac{g}{L}}$