# Why Plot T² Against 1/K for a Straight Line Graph in Oscillation Studies?

• Peter G.
In summary, to assess the relationship between the Constant of a Spring and the Time Period of Oscillations, you need to plot T2 against 1/K. This is because of the derivation of the periodic time formula for a spring oscillation, which can be simplified to the form Y=MX where Y = T2 and X = 1/k. By plotting T2 against 1/K, you can obtain a straight line graph and determine the mass oscillating from the gradient of the graph. Alternatively, a log-log plot can also be used for this purpose. This approach is recommended by an experienced engineer.
Peter G.
So, I have to assess the relationship between the Constant of a Spring and the Time Period of Oscillations.

I have researched the equation that links the two and I know that to have a straight line graph I need to plot:

T2 against 1/K

However, I do not know why; how I get to that conclusion.

I have read about log-log graphs and played around with the concept to see if it would yield the plot I mentioned above. I tried plotting for the log of T against the log of K and I did get a straight line plot. I checked if my gradient was equal to -0.5 and my y-intercept was equal to (log 2∏-0.5 log m) and they were, but I still do not understand why I have to plot T2 against 1/K.

You are asked to plot that because of the derivation of the periodic time formula for a spring oscillation.

The final equation is

$$T = 2 \pi \sqrt{\frac{m}{k}}$$

If you square both sides you will get

$$T^2 = \frac{4 \pi ^2 m}{k}$$

Which is the form Y=MX where Y = T2 and X = 1/k.

Thus plotting Y vs X i.e. T2 vs. 1/k will give you a straight line. So you can get the mass oscillating if you desire from the gradient of the graph you plotted.

I like your idea of using a log-log plot much better. I have over 40 years of experience as a practicing engineer, and that is the way I would do it. Congratulations on thinking out of the box.

Chet

## What is a straight line graph?

A straight line graph is a graphical representation of a linear relationship between two variables. It shows how one variable changes in relation to the other, with the data points forming a straight line on the graph.

## How do you plot a straight line graph?

To plot a straight line graph, you will need two sets of data that have a linear relationship. Then, you will plot the data points on a coordinate plane, with the x-axis representing one variable and the y-axis representing the other. Lastly, you will draw a line that connects all the data points, making sure that the line is as close to all the points as possible.

## What is the formula for a straight line graph?

The formula for a straight line graph is y = mx + b, where m is the slope of the line and b is the y-intercept. The slope is equal to the change in y over the change in x, and the y-intercept is the value of y when x is equal to 0.

## What is the importance of plotting straight line graphs?

Plotting straight line graphs is important because it allows us to visualize and understand the relationship between two variables. It also helps us make predictions and identify patterns in the data, which can be useful in many scientific fields.

## What are some common mistakes when plotting straight line graphs?

Some common mistakes when plotting straight line graphs include not using a proper scale on the axes, not plotting all the data points accurately, and not drawing a straight line that best fits the data points. It is also important to correctly label the axes and provide a title for the graph.

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