Any oscillation of a coupled system can be expressed as a superposition of its normal modes. The normal modes and their frequencies were observed and measured in the previous experiment. The in-phase frequency (\( \omega_{1} \)) is equal to the frequency of the un-coupled oscillator and t he frequency of the out-of –phase (\(\omega_{2} \)) depends on the coupling length -

$$\omega_{2}^{2} = \frac{2k\omega_{0}^{2}}{mgL_{CM}(I^{2}) + \omega_{c}^{2}}$$

Where,

\( k \) - spring constant of the coupling spring

\( \omega_{0} \) – frequency of the uncoupled oscillator

\( L_{CM} \)– distance between hanging point and center of mass

\( m \) – Mass of the pendulum

\( l \)– coupling length

The coupling strength is given by -

$$\frac{\omega_{2}^{2} - \omega_{1}^{2}}{\omega_{2}^{2} +\omega_{1}^{2}}$$

The pendulum used in the experiment is refered to as physical pendulum. Its natural frequency is given by

$$\omega_{0} = \sqrt{\frac{mgL_{CM}}{I}}$$

where \(𝙸 \)is the moment of inertia about the point of suspension.