This lab describes an activity with a spring-mass system, designed to explore concepts related to modeling a real world system with wide applicability.
The goals of this project are:
- Deepen your understanding of linear second order homogeneous differential equations.
- Study a prototype model application for the harmonic oscillator, via a spring-mass system.
- Practice with data collection, analysis and interpretation.
For this lab you will need:
- A “spring”. This can be a real spring, or, as a substitute, you can use a rubber band, or several smaller bands tied together.
- A “mass”. The mass could be a roll of coins, a coffee mug (don’t use your favorite one), a stapler, a can of food, or any attachable small dense object.
- Measuring devices: a measure tape for measuring distances, and a kitchen scale to measure masses. A timing device, such as a smart phone. A regular watch may be hard to use. You will have to decide which units to use, and adjust all constants accordingly.
- (Optional) A video-recording device, such as a video camera, cell phone or tablet.
The spring-mass system is good model problem, since there are many related application problems for which the same mathematical equations apply.
This activity will instruct you on how to measure and model your own system, connecting the physical behavior of the model system with the mathematical features first introduced in Chapter 5 of Trench’s text.
The figure below demonstrates the situation when the mass is hung on a spring of length , stretching it by a distance . Any movement of the mass can be measured from this equilibrium position. We make the arbitrary choice to call the upward direction positive.
To set up the system, the spring should hang from above by some fixture such as a hook. Place the spring near a wall, and attach the measuring tape behind the spring. If using a recording device, set it up some distance away, in such a way that both the spring-mass system and the measuring tape are visible. The image below illustrates a possible setup.
The model for the spring-mass system has the following parameters:
- The mass, .
- The relaxed length of the spring, .
- The spring constant, .
- The acceleration of gravity .
- The coefficient of friction .
The dynamic variable in the model is the displacement from the equilibrium position, which we denote by . We assume that is positive in the upward direction.
In this first exploration, we assume that the friction is negligible. That is, as a first approximation, we assume that .
Recall that we model the intensity of the force exerted by the spring by Hooke’s law: where the spring deformation is represented by in Figure fig:spring-mass-system.
Using this information, answer the following questions:
Let’s now denote by the -coordinate corresponding to the relaxed spring.
Newton’s Second Law states that:
Now set up the experiment. As an initial goal, let’s try to establish the period of the oscillation.
This was added by Felipe.
One more line by Paul here.