I hope that the two components of the velocity of \(m_{2} \) that I have marked are self-explanatory; the speed of \(m_{2} \) is given by \( v^2_{2} = l^2_{1}\dot{\theta}^2_{1} + l^2_{2}\dot{\theta}^2...I hope that the two components of the velocity of \(m_{2} \) that I have marked are self-explanatory; the speed of \(m_{2} \) is given by \( v^2_{2} = l^2_{1}\dot{\theta}^2_{1} + l^2_{2}\dot{\theta}^2_{2} + 2 l_{1}l_{2}\dot{\theta}_1 \dot{\theta}_2 \cos(\theta_2- \theta_1) \). Equating the two expressions for the ratio \( \theta_2/\theta_1 \), or putting the determinant of the coefficients to zero, gives the following equation for the frequencies of the normal modes:
