Improving Accuracy in Spring Constant Measurements: Tips and Considerations

[Swamifez]

A student hangs masses on a spring and measures the spring's extension as a function of the applied force in order to find the spring constant k. Her measurements are:

Mass(kg): 200, 300, 400, 500, 600, 700, 800, 900
Extension (cm): 5.1, 5.5, 5.9, 6.8, 7.4, 7.5, 8.6, 9.4

There is an uncertainty of 0.2 inces in each measurment of the extension. The unccertainity in the masses is neglible. For a perfect string, the extension delta L of the spring will be related to the applied force by the relation kDelta L=F, where F=mg, and Delta L= L-L_0, L_0 is the unstretched length of the spring. Use these data and method of the least squares to find the spring constant k, the unstretched length of the spring L_0, and their uncertainties. Find Chi^2 for the fit and associated probability.

 

[dacruick]

what is Chi^2?

 

[Swamifez]

(χ2) or http://en.wikipedia.org/wiki/Pearson's_chi-square_test

 

[jrlaguna]

Do you know how to do the fit? Just least squares... Write the chi², just by writing down the sum of the squares of the distances from each point to the fitting line. Now, minimize for the fitting parameters.

For the associated probability, you have to put the resulting chi^2 into the appropriate chi^2 distribution... but I don't know yet where your real problem is...

 

[paranormal]

I would first like to acknowledge that the student has done a good job in collecting data and recognizing the importance of accounting for uncertainties in measurements. The use of the least squares method is also appropriate in this case.

However, there are a few areas of concern with this experiment. Firstly, it is important to ensure that the masses used are accurately measured and that the spring is not stretched beyond its elastic limit. This could affect the accuracy of the data and the calculated values for the spring constant and unstretched length.

Additionally, the uncertainty in the extension measurements is quite high at 0.2 inches. This could be due to limitations in the measurement equipment or human error. It would be beneficial for the student to try and reduce this uncertainty in future experiments.

Moving on to the data analysis, the use of the equation kΔL = F is appropriate for a perfect spring. However, it is important to note that real springs do not behave perfectly and may have some amount of non-linearity. This could affect the accuracy of the calculated values for the spring constant and unstretched length.

To address the uncertainly in the data and account for any potential non-linearity, it would be beneficial for the student to take multiple measurements at each mass and calculate the average extension. This would also help in reducing the uncertainty in the extension measurements.

Using the least squares method, the student can calculate the values for the spring constant and unstretched length, along with their uncertainties. The Chi^2 value can also be calculated to assess the goodness of fit and determine the associated probability.

Overall, while the student has done a good job in designing and conducting the experiment, there are some areas that could be improved upon for more accurate results. It is important for scientists to continuously evaluate and improve their methods in order to obtain reliable and meaningful data.