# Spring Force and Oscillations

You have heard a lot about ideal springs in your textbook. These theoretical creatures follow very simple mathematical rules. Specifically,

1. Hooke's Law: the force a spring exerts is proportional to the distance it has been displaced from rest:
```             F  =  -k * x
```
where
```             F       is force exerted by spring (Newtons)
x       is distance spring is displaced from rest (meters)
k       is the "spring constant"
```

2. simple harmonic oscillation: when a spring is moved from its rest position, then released, it oscillates according to
```            x(t)  =   A sin (omega * t)
```
where
```            x(t)     is the position of the end of the spring (meters)
A        is the amplitude of the oscillation (meters)
omega    is the frequency of the oscillation (radians/sec)
t        is time (seconds)
```

So, this is the theory. But do real springs follow these rules? Today, your job is to decide if one particular set of real springs act like ideal ones. You will test each of the two laws described above.

#### Hooke's Law

First, Hooke's Law: place several different masses on the end of a spring, and measure how far it stretches in response. The spring must exert a force equal to the force of gravity Is the size of the stretch really just a constant times the force exerted on the spring by a mass?

Make a graph which shows the amount by which your spring stretches as a function of the mass added to it. Use at least 5 different masses, and make two rounds of measurements. Does your spring obey Hooke's Law? If yes, derive the spring constant (including uncertainty therein). If not, explain how the real spring does behave.

#### Simple harmonic motion

Second, harmonic oscillations. As your lab manual explains, it is possible to start with the equation for simple harmonic motion and derive another relationship for ideal springs:

```	                  2
2       4 * pi
T    =   ------- * m
k
```
where
```
T         is the period oscillation (seconds)
m         is the mass added to a spring (kg)
k         is the spring constant (N/kg)
```

Does your spring really oscillate in this fashion?

Add five different masses to your spring, and measure its period of oscillation in each case. You must figure out a good way to measure the period. Write down the results. Now, to analyze the results, it would be easiest if you could find an equation like this:
```          (stuff on y-axis)  =   k * (stuff on x-axis)
```
Figure out such an equation, plot your data, and use the graph to answer the question: "Does my spring obey the rules?" If it does, calculate the spring constant from the graph. If not, explain how it actually does behave.

#### What do I have to submit?

This week, I want to emphasize making and analyzing graphs. They really are very useful for answering scientific questions. Therefore, I expect you to give me:

• A neat table of all your measurements, including headings and all appropriate units and uncertainties
• Two graphs, one for Hooke's Law and one for SHM. I think a good graph
• covers nearly the entire extent of a sheet of graph paper with data points
• has labels on each axis, which include units
• places errorbars on each plotted datum
• has a title at the top which is not just a rehash of the axes, but describes in some way the experiment; that is, not "Displacement versus mass", but something like "A test of Hooke's Law"
• One or two paragraphs of text which explain the results of each experiment. I expect you to write complete sentences, follow standard rules of grammar, and spell correctly every word you write. The text may be handwritten neatly or typed.
• One paragraph which mentions sources of error in the experiment. You must state the single largest source of error and its magnitude as a percentage.

I will deduct a full letter grade from any report which includes the phrase "human error."