# Models of the Motion of a Mass Suspended from a Spring

PHYSICS EXPERIMENTS —1322-1

Experiment 2

**Models of the Motion of a Mass Suspended from a Spring**

PHYSICS EXPERIMENTS — 13210-1

In this experiment you investigate the behavior of a simple physical system consisting of a mass hanging on the end of a spring. Although mechanically simple, this system is important because it exhibits repetitive motion. The mass oscillates! The frequency of oscillation depends on physical properties of the system; the masses in motion and the spring stiffness. First you will determine whether Hooke’s Law is a good model for your spring stretch as a function of the masses hung on it. Second, we will discuss and test a model that predicts the oscillation period. Sometimes physics models may not apply to our system and if so, we will determine why it failed. Finally, we could explore alternate models that could improve its predictions.

Preliminaries.

**Part A. Static Determination of Spring Constant.**

A spring with spring constant k hangs vertically with a mass m attached to the lower end, as in Figure 1.

Figure 1. Hanging a mass on a spring

An external force F applied to a springstretches the spring a distance x. Hooke’s Law states that, for an ideal spring, the applied force and stretch are linearly related

F = kx(eq. 1)

where k is the spring constant, a measure of spring stiffness. The stretched spring exerts an (equal and opposite) elastic restoring force, Fs = -kx, in the opposite direction. If the mass is stationary, in equilibrium, the elastic restoring force must balance the mass’s weight,

mg = kx(eq. 2)

where g is the acceleration due to gravity.

Procedure.

•Clamp a long vertical pole to the edge of the lab table. Connect a horizontal bar near the top of the pole and attach one end (the narrower one) of a spring to the end of the horizontal bar. The spring should hang with its narrower end on top so that, as it stretches, the coil spacing is uniform in width.

•Hang at least five different weights on the spring Measure the amount of spring stretch for each. **Make certain not to damage the spring by stretching it beyond its elastic limit.**

•Graph 1. Make a graph with the "spring stretch" on the vertical axis and "hanging mass" on the horizontal axis.

Q1. What is the spring constant from the graph? Use a fit line, the slope or y-intercept, and your Hooke’s Law physics model (Eq. 2) to determine the spring constant. You will use this later.

**Part B. Dynamic Model of Simple Harmonic Motion.**

If the mass in the system of Figure 1 is pulled away from its equilibrium position and then released, it will oscillate up and down about its equilibrium position. Motion in which the restoring force is proportional to the displacement, but oppositely directed, is called **simple harmonic motion**.

You should consult your physics text to find a physics model that should apply to this system. We want to predict the period of the spring as a function of different, controllable variables.

Q2. What experimental parameterscould you test in an experimental setup that might change the period? List them all.

Q3. Write down your physics model that you found in your book. Which of the variablesdoes your model depend on? Which variables does your model predict will not affect the period?

Procedure.

Let’s test your model!

●Attach 100 g total mass to the unattached end of the spring, using the weights mounted on the weight hanger. Record the total hanging mass m. Do not forget to add the hanger mass! Pull the mass several centimeters from equilibrium and release. Start the timer and time five oscillations. Divide by five to determine the period of oscillation. Repeat several times to insure consistency and determine precision of data. Record.

●Add mass in 50 g increments up to about 300 g and determine the period for each hanging mass. Be very careful that the spring is not stretched excessively, where it will be permanently deformed and not able to snap back!

●Graph 2. Graph the data as a “scatter plot” in Excel with the "period" on the vertical axis and "hanging mass" on the horizontal axis.

●Now compute the period as a function of hanging mass using your physics model for each mass you tested. Plot this model’s prediction on Graph 2 as a line connected your computed period values. You should use the spring constant you determined from Part A in your model.

Q4. How did you do? Does your model accurately describe the data?

Q5. Is your model more or less accurate at for small masses or for large masses?

Revised model

Your model probably does not look very accurate. Back to the drawing board!

Q6. Why was your model inaccurate? Try to use your answer from Q5 to inform you why your model is inaccurate.

Q7. Come up with a new model (equation) that may do a better job of computing the period vs. the hanging mass. This should not be a random polynomial or other mathematical function. Rather, you should try to come up with a physical explanation that improves on the ideal mass on a spring equation that you used in Graph 2. You should still use the spring constant from Part A in your model. We will discuss possible ideas in class.

**Graph 2 continued**. Compute the period vs. mass using your new model and plot on Graph2 and label it in a legend as “Model 2”. Compare the new prediction to the old. Is it an improvement? There may be a new parameter in your model that is a “free variable”. You could try changing this to a few values to find the best choice to fit your data.

Turn in your answers to question Q1-7 along with your two graphs, where Graph 2 should have data, a plot of the Model 1 prediction and a plot of the Model 2 prediction. We’ll skip the rest of the lab writeup this week so we have enough time to finish.

This is how “real” research in physics is done! In physics we almost always try to find a model of some physical system. We derive this model using math and physics and use it to predict data in an experiment. Often, our first try is not very good. We then try to figure out how we could improve things and come up with a revised model. Sometimes, the model still doesn’t work well. This is still a result! The only thing that is “bad” physics is if you can’t tell if your model is good or if it’s bad. If you can’t tell, the model and the experiment are pretty useless and you don’t have much to talk about. Back to the drawing board!

rev. 1/2017