Driven Harmonic Oscillator

Introduction

Harmonic oscillators play a very important role in physics, and so it is necessary that you understand them well. In this experiment, you will be able to explore damped and driven harmonic motion with the Driven Harmonic Motion Analyzer (DHMA). The analyzer displays the frequency at which it is driving the mass-spring system and measures the amplitude and period of the oscillations. The operation and setup procedures are described in detail in the manual. Be sure to follow the setup procedures carefully before performing each of these experiments.

Equipment

·        Pasco ME-9210A Driven Harmonic Motion Analyzer, which includes:

·        DHMA unit

·        Support column

·        Drive cord

·        Four springs of two different spring constants

·        Mass bar

·        Magnetic damping unit

·        Damping rod

·        Masses

·        Instruction manual

·        Liquid damping kit, with beaker and liquid damping rod

Procedure

Follow the procedures in the manual.

1. Spring constants: Determine the spring constants of the four springs which are provided with the experiment. Springs of the same color have approximately the same spring constant. Using the weights provided and the displacement-measuring capabilities of the DHMA, determine the spring constants of the blue and red springs.

2. Natural frequency: Measure the period of oscillation of the undamped spring-mass system and use this to determine the natural frequency. Measurements should be taken for both springs at different values of the mass. Compare the experimentally determined values with those calculated from theory and discuss any differences.

3. Amplitude vs time: Without any driving force, measure the amplitude of the oscillations vs time for the following cases: undamped, light magnetic damping, heavy magnetic damping, and liquid damping with water. Make a plot of the data, and from the plot, determine the values of the damping coefficients of the damping media.

4. Amplitude and phase vs frequency: Using the motor of the DHMA, determine the amplitude of the oscillations vs frequency for all the cases listed above. In addition, do the same using heavy syrup as damping medium. Make plots of all these. Change the spring constant and mass for the first two cases and determine how this changes the plot. For the case of heavy magnetic damping, measure the phase vs the frequency.

5. Beats: For this experiment, it is necessary to use both 50 gram weights. Determine the resonant frequency of the system, and then drive the system at a frequency of ± 0.1 Hz from the resonant frequency. You will observe that beats occur. Measure the amplitude of the beats as a function of time. Take care to do this quickly, as beats in this system are a transient effect and die off rapidly. Determine the frequency of the beats and explain how they are related to the driving frequency.