Simple+Harmonic+Motion+(Spring)

=**Simple Harmonic Motion of a Spring:**= =**A Graphical Analytical Approach of Understanding Energy and Motion**=

**Sang Lee**
= = =__**Purpose**__=

__**Procedure**__
The materials:
 * Metal clamp
 * Spring
 * Weights (specifically eight 100g weights)
 * Labquest 2
 * Labquest 2 motion detector

Firstly, the spring was set up to a metal clamp attached to a desk. Then, the spring was calibrated to 0 to ensure equilibrium prior to adding any mass to the spring.

After, the motion sensor was placed right below the spring to capture the data of position vs. time and velocity vs. time. Be sure to set the motion sensor to the setting of the track (left setting). Below shows the two possible settings for the motion detector and which setting was used for the experiment.



Below shows the setup when the masses were added. Notice the position of the motion detector and how the detector face is set below the spring system.

When the motion detector was prepared, the detector was connected to the Labquest 2 system via DIG 1 (2 can also be used) port located on the top side, next to the power button. The LabQuest App was selected and the graph display was selected by the options located at the top.



Then, masses (varying from 200-800g) were added to the string. The spring was pulled back up to 0 (equilibrium). The Labquest 2 was adjusted to collect data for 5 seconds. Then, the experimental data was collected by pressing the green arrow

When it was deemed sufficient that the data was accurately collecting (no anomalies/fluctuations in the graph) for a second or so, the spring was released from equilibrium, and the data was recorded for 5 seconds.

After the recording was finished, the data and the graph was saved to the Labquest 2 system. A new file was opened for each of the trials for the independent variable of the masses.



The masses varied beginning from 200g and going upwards by 100g until 800g. A 100g mass was added to the spring each time, and the data collection was repeated.

Below is a video of how the system was released from equilibrium: media type="file" key="IMG_0272.MOV" width="320" height="345"

=**__Analysis__**=

After the data was acquired, the graph function of the Labquest 2 displayed the following graphs:

200g:

300g:

400g:

500g:

600g:

700g:

800g:

All of these graphs displayed a similar behavior of the similar periods and amplitude throughout. As the masses increased, there was also an increase in the number of oscillations but a decrease in the amplitude of the curves. This coincides with the equation for the period of a spring:



The graphs and the equation indicate that as mass increases, the period of a spring increases. The knowledge of the inverse relationship between frequency and period was identified with the increases in masses. Initially, the mass increased showed no significant increase in period yet when you compare the 800g graph with the 400g graph, there's a clear change in the frequency and the period of the curves.



The amplitude changed as a result of the variable, but it should technically have no effect on the period of the spring. The amplitude is significant in that it also increased as the mass increased. This coincides with the increased displacement of the spring as mass increases.

Some errors in the experiment can be found in the random spikes of velocity and position which may be the result of an outside interference in the data collection. A recommendation for a repeat of the experiment would be to use a wider range of masses to acquire graphs of a bigger significant different. Another possibility would be to measure the position and the force exerted by the spring to confirm Hooke's law and find the spring constant.

=__**Results**__=

The experiment indicated that there was indeed a relationship between the period of the simple harmonic motion of a spring and mass. The equation of not only the period of the spring but also Hooke's law can be seen in this. Hooke's Law of **F = kx** follows with the results of the experiment. As mass was increased, force was increased. This increased force displaced the spring as the mass increased as shown by the higher amplitudes shown by the increased masses. However, it is important to note that the increases in mass were only by .1 kilograms, possible larger increases may have shown a more clear result.

The results of this experiment verified the knowledge of the simple harmonic motion of a spring. Although there were certain anomalies in the random surge in the graphs, the surge was not significant to compare it to the bigger picture to verify the implications of Hooke's law and the period equation. For example, the position graph of the 400g showed an approximate amplitude of .1 meter while the position graph of the 800g shows an amplitude of almost .3 meters.

In conclusion, the experiment clarified some of the knowledge of simple harmonic motion known beforehand. Although the variables for the experiment was within a very minute range, it still allowed the experiment to affirm the conclusions and previous thoughts of springs. This confirmation is useful in that it is applicable to many aspects of life such as the springs in watches, pens, and several common items. Designers are able to understand this concept and account for how much the mass will affect the motion of the spring depending on the spring constant of the spring.