PHY115 Oscillations and Waves Tutorial 6

Study Notes

Coupled Oscillators and Synchronization

Synchronization of metronomes can be observed when placed on a flexible surface, demonstrating coupled oscillation behaviors.
The model used includes damping forces represented by dashpots, exhibiting why symmetric motion predominates over time.

Coupled Pendulum Resonant Frequencies

A coupled pendulum setup involves three masses and two identical springs to analyze normal modes.
Resonant frequencies of different normal modes can be derived from the properties of the springs and the masses involved.

Triatomic Molecule Dynamics

A triatomic molecule comprises a heavy atom (mass M) flanked by two lighter atoms (mass m) connected by springs of stiffness k.
In equilibrium, atom centers are aligned, allowing for analysis of simple harmonic oscillations along a linear pathway.
The equations of motion for each atom can be established, leading to the calculation of normal mode frequencies.

Forced Coupled Pendulum Dynamics

In a coupled pendulum system, a mass experiences a horizontal forcing function of the form ( F_0 \cos(\omega t) ).
By assuming no damping, the amplitudes of the normal modes can be deduced and compared to the individual mass displacements, ( x_a ) and ( x_b ).

Resonant Frequency of Mass-Spring Systems

When two masses ( m_1 ) and ( m_2 ) are coupled by a spring of stiffness k, the resonant frequency expression can be derived from their interactions.

Vertical Oscillation of Equal Masses

Equal masses in a pendulum system oscillate vertically, allowing for the derivation of the system's resonant frequency.
The amplitude ratios of the two masses during resonant modes can also be calculated.

Normal Frequencies in Mass-Spring Systems

Analysis of a specific mass-spring system leads to the deduction of its normal frequencies and identification of the normal modes.

Coupled LC Oscillator System

A system of two identical LC oscillators coupled through a common capacitance ( C ) allows for examination of current flow directions.
Normal coordinates and frequencies of this coupled oscillator system can be identified, providing insights into their dynamic behavior.

Description

In this tutorial, students will explore the synchronization of metronomes through a coupled oscillator model. This hands-on activity is designed to deepen understanding of oscillations and waves concepts discussed in class. Prepare to apply theoretical knowledge to practical scenarios in physics.