Question

A small body of mass m is fixed to the middle of a stretched string of length $2\ell$. In the equilibrium position the string tension is equal to $T_0$. Find the angular frequency of small oscillations of the body in the transverse direction. The mass of the string is negligible, the gravitational field is absent.

Answer 1

The force $F$, acting on the weight deflected from the position of equilibrium is $2T_0\sin \theta$. Since the angle $\theta$ is small, the net restoring force, $F=2T_0\frac{x}{l}$
or, $F=kx$, where $k=\frac{2T_0}{l}$
So, by using the formula,
${\omega }_0=\sqrt{\frac{k}{m}},{\omega }_0\sqrt{\frac{2T_0}{ml}}$