Question

Spring Constants & Periods

By varying the mass placed on a spring, the period of oscillation can be changed. When multiple data sets are collected and plotted, the spring constant can be determined.

Before data was collected, the minimum and maximum masses were determined by visually watching the harmonic motion. For the minimum, a mass was chosen such that the resulting simple harmonic motion had an appreciable amplitude without significant dampening over ten periods. For the maximum, the mass was selected such that the harmonic motion did not appear to exceed the linear elasticity of the spring. The suitable range of mass was determined to be between 25 grams and 45 grams.

The same spring was used throughout the experiment. Several different masses were hung from the spring, and the spring was stretched approximately 2 cm from its equilibrium point and allowed to oscillate with each mass. A stopwatch was used to determine the total time for ten (10) periods.

Plot #1 -- Period vs Mass

Although the constraints of the experimental set-up did not allow collecting data using masses smaller than 25 grams, a data point near (0,0) can be approximated. If the spring was an ‘ideal’ spring with zero mass, the data point (0, 0) would be a valid data point. Using the provided data set, and the data point (0,0), create a scatter plot of Period vs Mass using Excel.

If plotted correctly, the resulting plot will not be linear. If there is any doubt that the plot is not linear, fit the plot with a linear fit.

Nonlinear plots such as this can be difficult to fit properly. Even if a proper fit can be applied using software, interpreting the resulting fit coefficients can be challenging.

Manipulating the parameters placed on each axis such that a plot results in a linear relationship can be of great value. This can be accomplished even if an equation does not appear to be linear. Linear plots are typically easier to interpret for most individuals, and information from the linear fit is typically more readily understood. This can be especially true if the plots are being plotted by hand on graph paper.

After manipulating   T=2πmk   

into                                                   T=(2πk)m ,

it should be mathematically evident that T∝m , and plotting T vs m will clearly not result in a linear plot.

Plot #2 -- T vsm

For this plot and Plot #3, do not include (0,0) as a data point. Since a real spring has mass, a portion of the spring's mass would impact the period. Therefore, the spring's mass would create a non-zero y-intercept, and a data point of (0, 0) is not a valid data point. The data point (0,0) was only used in Plot #1 to emphasize the non-linearity of the plot. The curvature of the T vs m plot could not be properly demonstrated without data points that utilized small masses in the range of 1 to 20 grams.

Plot  T vsm , and determine the spring constant.

In addition to the scatter plot, include a data table that includes the data given and the columns that were used to calculate a single period and m. Finally, type the value of the spring constant to three decimal places with units in a cell. Highlight the cell with a border and yellow background.

Plot #3

Create a scatter plot in Excel such that the following conditions are met:

1. The plot will be linear.
2. The parameter plotted on the y-axis is related to the period.
3. The slope is equal to 1k.

Utilize the same data used to create Plot #2. [Do not use (0, 0)].

In addition to the scatter plot, include a data table that includes the data given and the columns that were used to calculate the parameters placed on the y-axis and x-axis. Finally, type the value of the spring constant to three decimal places with units in a cell. Highlight the cell with a border and orange background.

Total Mass Hung from Spring (g)	Total time for 10 periods (s)
25	5.01
30	5.54
35	5.92
40	6.36
45	6.75

Answer 1

And the graph of the above mentioned plots is as follow :

X-axis Y-axis

Total Mass Hung from Spring (g)	Total time for 10 periods (s)
25	5.01
30	5.54
35	5.92
40	6.36
45	6.75

Plot m v/s T :

And

Spring constant can be calculated by the slope of T v/s m1/2.

( slope of T v/s m1/2 ) = 2*pi / (k)1/2

So, k =