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:

(1)

Associated with the natural frequency f is the angular frequency ω, which is the rate at which the oscillator goes through phase angle (one full oscillation is equal to 360° or 2π radians in phase angle). Thus the angular frequency is 2π times the natural frequency:

(2)

The physical properties of the oscillator determine the frequency. In the case of the mass-on-a-spring oscillator, the angular frequency is

(3)

where k is the spring constant (stiffness) of the spring and m is the mass.

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, v_max. 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:

(4)

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:

(5)

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.