Lab Spring.SHM

College Physics Lab
PH 144 Return to lab main Simple Harmonic Motion
Spring-Mass System

Purpose:
To investigate the properties of an oscillating spring-mass system. To verify that a spring-mass oscillator behaves like a simple harmonic oscillator.

Discussion:
If a mass m on a spring is displaced from the equilibrium position (x = 0) to a new position x, Hooke's law states that the spring will exert a restoring force on the mass F_r= -kx. Furthermore, if the mass is released, it will execute simple harmonic motion with a period of t = 2p(m/k)^1/2 As is demonstrated in the textbook, these two conclusions are correct even if the spring is hanging vertically. However, the equilibrium position is determined by the position where the mass m will hang in static equilibrium, not the position of the spring before the mass is attached.

A spring-mass system can be made to oscillate with many different amplitudes. Examination of the equation involving period suggests that the period of oscillation only depends on the mass attached to the spring, m, and the spring constant, k. Specifically the period should not depend on the amplitude of the oscillation.

The above conclusions strictly hold for systems involving massless springs. Real springs have mass. The mass of the spring has an influence on the oscillation of the system. However, since the mass of the spring is distributed along the length of the spring, it is not the whole mass of the spring, m_s, which figures into determination of the period of oscillation. Rather the influence of the spring's mass can be represented by an effective mass, m_eff. The period of oscillation of the system is predicted to be

The quantity m_eff is less than m_s; and, in the case of a uniform spring (mass distributed uniformly along its length), m_eff is predicted to be m_s/3.

Materials:
spring, mass set, meterstick, stopwatch

Procedure:

1. Measure the mass m_s of the large tapered spring.
2. Determine the spring constant of the spring.
Do this by investigating the correspondence between the amount of mass hanging from the spring and the amount that the spring stretches in response to the mass. (The mass of the spring, m_s, plays no role here.) Analyze the data graphically in order to determine the spring constant. Please don't overstretch the spring. The thin part of the spring should be at the top.

3. Investigate the dependence of the oscillation period on the oscillation amplitude.
Determine accurately the period of oscillation of the system for different amplitudes of oscillation. The amplitude of oscillation is determined by the distance between the point where the mass is in equilibrium and the point of release of the mass when the oscillation is initiated. (You should be able to explain conceptually why this is so.) Describe the relationship between period and amplitude that you found.

4. Investigate the period-mass relationship.
Determine the period of oscillation of your system for several different masses. Verify that your data are consistent with the predicted period mass relationship,

Construct a graph of (period)² vs mass. Justify your conclusions. Employ % difference calculations where appropriate.

5. Investigate the effective mass, m_eff, of the spring in your system.
Determine m_eff from your graph. Compare the value you obtained with the value m_s/3 that is predicted for springs having a uniform distribution of mass. Employ % difference calculations where appropriate.

6. Select another spring. Compare qualitatively the stretchiness of the spring to your first spring.
Make a qualitative prediction. Sketch a graph of (period)² vs mass for this system. How does its slope compare to that of the first spring? How should be the y-intercept of the line compare to the y-intercept for the first spring? Explain your answers.
Measure the spring constant for this spring.
Investigate the period - mass relationship for this spring as you did for the first spring.

Question:
If you combined your first and second springs so that they were both hanging from the same point and the hanging mass was attached to both, how would you expect the period of oscillation of the spring to be affected? Explain your reasoning.