Linearising an Inverse Square Law Graph for Gamma Radiation

[Bairdo97]

1. Homework Statement

A piece of work I am doing for college (UK college that is) has me investigating the inverse square law for gamma radiation. I have collected data and the graph comes out looking right. I want to create a linearised graph of the data to investigate the results further. If I did this, what process would I run the results through, and what would the graph (e.g. gradient and points where the graph crosses the axes) tell me?

To cut it short, if one was to linearise an inverse square law graph, how would one do it, and what would this linearised graph show?

Homework Equations

Intensity = S/(4πr^2)

The Attempt at a Solution

 

[Chestermiller]

Welcome to physics forums.

There are several different ways of doing this. What are your thoughts so far?

Chet

 

[Ray Vickson]

1. Homework Statement

A piece of work I am doing for college (UK college that is) has me investigating the inverse square law for gamma radiation. I have collected data and the graph comes out looking right. I want to create a linearised graph of the data to investigate the results further. If I did this, what process would I run the results through, and what would the graph (e.g. gradient and points where the graph crosses the axes) tell me?

To cut it short, if one was to linearise an inverse square law graph, how would one do it, and what would this linearised graph show?

Homework Equations

Intensity = S/(4πr^2)

The Attempt at a Solution

Why do you want to linearise something that is not linear? What can you do with a linearised graph that you cannot do with a more accurate non-linear graph? Frankly, the whole exercise strikes me as unscientific and inappropriate, and unless an instructor has ordered you to do it (I hope not!) you should not do it.

 

[Chestermiller]

Why do you want to linearise something that is not linear? What can you do with a linearised graph that you cannot do with a more accurate non-linear graph? Frankly, the whole exercise strikes me as unscientific and inappropriate, and unless an instructor has ordered you to do it (I hope not!) you should not do it.

He's not actually talking about linearizing the relationship about some point. He is talking about plotting the data in such a way that the results all fall on a straight line. In this way, he can determine, from the slope and intercept, the parameter values (provided the equation is a good representation of the data). One way, for example, would be to plot I vs 1/r².

 

[Ray Vickson]

He's not actually talking about linearizing the relationship about some point. He is talking about plotting the data in such a way that the results all fall on a straight line. In this way, he can determine, from the slope and intercept, the parameter values (provided the equation is a good representation of the data). One way, for example, would be to plot I vs 1/r².

OK, in that case he would be doing something quite standard.

He should realize, though, that a least-squares straight-line fit (say) to a transformed problem is not necessarily a least-squares fit to the original, nonlinear relationship. If the errors in the fit are "small" it won't make much difference, but if they are "large" the two fits could disagree significantly. (In other words, if one transforms the data, fits a straight line, then reverse-transforms the fit, one might not always obtain a nonlinear formula that is close to the one obtained by doing a least-squares fit directly on the original data.) It all depends on the details of the functions and the transformations performed.

 

[Chestermiller]

OK, in that case he would be doing something quite standard.

He should realize, though, that a least-squares straight-line fit (say) to a transformed problem is not necessarily a least-squares fit to the original, nonlinear relationship. If the errors in the fit are "small" it won't make much difference, but if they are "large" the two fits could disagree significantly. (In other words, if one transforms the data, fits a straight line, then reverse-transforms the fit, one might not always obtain a nonlinear formula that is close to the one obtained by doing a least-squares fit directly on the original data.) It all depends on the details of the functions and the transformations performed.

By the way, the example I gave in post #3 is definitely not the way I would plot the data. I would be thinking more in terms of log-log plots.

Chet