$3.2.10.$ Measurements of the circular frequency of oscillations of a body of mass $m$, fixed in the middle of a tensioned string with length $2 l$, gave the value $\omega$. Find the force of tension of the string.
The total external force acting on the system $$ F=2T\sin\varphi $$ Newton's second law $$ m\ddot{x}(t)+2T\sin\varphi=0 $$ Let's use the approximation for small angles $(\varphi \ll 1)$: $$ \sin\varphi\approx\varphi=\frac{x}{l} $$ Harmonic oscillation equation $$ \ddot{x}(t)+\frac{2T}{ml}x(t)=0 $$ Where does the period of oscillation come from? $$ \omega^{*2}=\frac{2T}{ml}\Rightarrow \boxed{T=\frac{ml\omega^{*2}}{2}} $$
$$T=\frac{ml\omega^{*2}}{2}$$