OSCILLATOR REVIEW
PRELAB
PURPOSE
To study the properties of different oscillators and the interrelationships between their physical properties and their motion.
RELEVANT FORMULAS
Frequency and Period:
Angular Frequency of Mass/Spring:
Kinematic Relations for SHM:
DISCUSSION
Simple harmonic motion is observed whenever an object is subject to a linear restoring force. A good example is the mass-on-a-spring oscillator. If the mass is displaced from the equilibrium position, it will oscillate back and forth at a fixed rate called the natural frequency f. The rate is just the inverse of the time that it takes for the mass to complete one full cycle of oscillation (one period T). We can thus relate the period and the frequency by:
The physical properties of the oscillator determine the frequency. In the case of the mass-on-a-spring oscillator, the angular frequency is:
(3)
The maximum distance that the mass covers from the equilibrium position is called the amplitude A. Its position thus varies from a minimum of −A to a maximum of +A and back. This distance can be related to the maximum velocity that the mass reaches during the course of one oscillation, vmax. Of course these two maxima occur at different points during the course of one oscillation, with the velocity reaching its maximum at the precise instant that the mass is passing through the equilibrium position (x = 0) after having reached its minimum value at −A. Since this is one-fourth of a complete cycle before the point when the position is a maximum, we can say that the difference in phase angle is , with the velocity leading the position. The relation between these two quantities is:
In a similar way the maximum acceleration of the mass occurs at the time when the position of the mass is at −A. The minimum acceleration occurs when the position of the mass is at +A. Thus these two events occur at a time interval of one-half a cycle, or a phase angle difference of 180°, with the acceleration leading the position. The relation between the two magnitudes is:
In this experiment, you will analyze simulated data on a mass-on-a-spring oscillator with specific physical properties, and use the spreadsheet to analyze the motion.