Question

Two physical pendulums perform small oscillations about the same horizontal axis with frequencies ω₁ and ω₂. Their moments of inertia relative to the given axis are equal to I₁ and I₂ respectively. In a state of stable equilibrium the pendulums were fastened rigidly together. What will be the frequency of small oscillations of the compound pendulum?

Answer 1

When the two pendulums are joined rigidly and set to oscillate, each exert trques on the other, these torques are equal and opposite. we write the law of motion for the two pendulums as

When ± G is the torque of mutual interactions. we have written the restoring forces on each pendulum in the absence of the other as - ω₁²I₁θ and -ω₂²I₂θ  respectively.then