Question

Consider two simple pendulums P and Q of equal length L and masses m1 and m2 respectively. Suppose the bobs of the pendulum are connected to each other by a weightless spring of force constant k such that in the equilibrium position, the spring is unstretched. Calculate the normal mode frequencies, and draw labelled diagrams corresponding to each mode.

Answer 1

Normal mode frequencies can be calculated by the following two equations:

$$\omega_1=\sqrt{\frac{g}{L}},\\ \space\\ \omega_2=\sqrt{\frac{m_1m_2g+kL(m_1+m_2)}{m_1m_2L}}.$$

The diagram corresponding to the first mode:

The diagram corresponding to the second mode: