Section 3.6: Coupled Oscillations
Coupled oscillations occur when two or more oscillators interact so that energy can be exchanged between them. This phenomenon leads to interesting behaviors such as normal modes and beats.
Key Concepts:
- Two coupled oscillators can oscillate in in-phase or out-of-phase normal modes.
- Beat frequency: \( f_\text{beat} = |f_1 - f_2| \)
- Energy transfers periodically between the coupled systems.
- Mathematical description often uses coupled differential equations.
Example 1
Two identical pendulums of length 1 m are coupled with a weak spring. If the individual pendulum frequency is 1 Hz and the coupling is weak, find the approximate beat period.
Assuming small coupling: \( f_\text{beat} \approx |f_1 - f_2| \). If \( f_1 = 1 \) Hz and \( f_2 = 1.02 \) Hz, then \( f_\text{beat} = 0.02 \text{ Hz} \), and beat period \( T_\text{beat} = 1/f_\text{beat} = 50 \text{ s} \).
Practice Problems
- Two coupled pendulums have normal mode frequencies 2 Hz and 2.1 Hz. Calculate the beat frequency and beat period.
- Two coupled springs have masses 0.5 kg each and spring constant 100 N/m. Write the coupled differential equations.
- Show energy transfer between two coupled oscillators with amplitude 0.2 m in one oscillator initially.
- Explain in-phase and out-of-phase motion for two coupled pendulums.
- A system of two identical coupled LC circuits oscillates. Determine the normal mode frequencies if L = 0.5 H and C = 100 μF.