Inversely proportional or a linear relationship?

[maxim07]

Homework Statement

For an experiment investigating standing waves, the frequency is inversely proportional to the length. If you plotted a graph of frequency against inverse length and got a straight line with a constant gradient through the origin it would suggest inverse proportion, but if you got a straight line with an constant gradient that had a y-intercept (due to a systematic error) what would you call it?

Relevant Equations

F = k x 1/l

Would you call it an inverse relationship or a linear relationship, what is the correct terminology?

 

[haruspex]

Homework Statement:: For an experiment investigating standing waves, the frequency is inversely proportional to the length. If you plotted a graph of frequency against inverse length and got a straight line with a constant gradient through the origin it would suggest inverse proportion, but if you got a straight line with an constant gradient that had a y-intercept (due to a systematic error) what would you call it?
Relevant Equations:: F = k x 1/l

Would you call it an inverse relationship or a linear relationship, what is the correct terminology?

I'm not aware of any standard term, but 'inverse affine' seems reasonable.
Trouble is, that does not distinguish between f(L+c) is constant and (f+c)L is constant.
What you get graphically with those depends on which you choose to invert. E.g. if it is f(L+c) = constant and you happen to plot f against 1/L you will not get a straight line.

 

[maxim07]

Okay so maybe it would be fine to say that it shows that there is an inverse relationship between f and l, and clarify that this means that as l increase f decreases, but not proportionally because of the y-intercept.

 

[haruspex]

Okay so maybe it would be fine to say that it shows that there is an inverse relationship between f and l, and clarify that this means that as l increase f decreases, but not proportionally because of the y-intercept.

It was not clear to me whether your problem statement was as given to you or arose from a lab you are writing up.
If the latter, if I interpret correctly, the graph from your lab gives a nonzero frequency at infinite length. I would consider making an argument that at infinite length the frequency must be zero, so we are free to force the line through the origin.

 

[maxim07]

Yeah, sorry, its for a write up. I don’t think I am allowed to force the line through the origin as I need to identfy any systematic errors in the experiment. Other variables were kept constant, so we can say

f ∝ 1/l
so f = k/l

for a graph of f against 1/l, 1/l = x so you get
f = kx

so this would give a straight line through the origin, but if there is a y-intercept, I’ve used a graphing tool to check that this can come from a translation in the y (f in this case) direction of

f = kx + c

so in the context of the experiment, a positive y intercept could originate from the frequency reading being too large by a constant amount from badly calibrated equipment (y translation).

A translation in the x direction also causes a y-intercept, but since x is a function of l I am not sure if it is the case here

say each length measurement had an errr of 0.2

then we have x = 1/l + 0.2

take l = 1, with no error we get x = 1/1 = 1, with an error we get x = 1/1 + 0.2 = 1.2

Then with no error f = k, with an error f = 1.2k which changes the gradient but does not shift the line, so I am assuming an error in length would have no effect.

Is my working correct? And would this apply to all straight line graphs, is it only an error in the y direction that would cause an intercept? Thanks.

 

[haruspex]

Yeah, sorry, its for a write up. I don’t think I am allowed to force the line through the origin as I need to identfy any systematic errors in the experiment. Other variables were kept constant, so we can say

f ∝ 1/l
so f = k/l

for a graph of f against 1/l, 1/l = x so you get
f = kx

so this would give a straight line through the origin, but if there is a y-intercept, I’ve used a graphing tool to check that this can come from a translation in the y (f in this case) direction of

f = kx + c

so in the context of the experiment, a positive y intercept could originate from the frequency reading being too large by a constant amount from badly calibrated equipment (y translation).

A translation in the x direction also causes a y-intercept, but since x is a function of l I am not sure if it is the case here

say each length measurement had an errr of 0.2

then we have x = 1/l + 0.2

take l = 1, with no error we get x = 1/1 = 1, with an error we get x = 1/1 + 0.2 = 1.2

Then with no error f = k, with an error f = 1.2k which changes the gradient but does not shift the line, so I am assuming an error in length would have no effect.

Is my working correct? And would this apply to all straight line graphs, is it only an error in the y direction that would cause an intercept? Thanks.

Any imperfections in the readings, i.e. leading to anything other than a perfect straight line fit, is likely to make the line miss the origin. It doesn’t necessarily imply a systematic error. OTOH, if you were to get a perfect fit through the datapoints and it still misses the origin that would be a systematic error.
There are statistical methods to tease this apart. Inserting an extra constant into an equation, in this case a constant offset, has to improve the goodness of fit by a certain margin to justify it. Note that this reverses the burden of proof from what you are assuming: it's not that you have to justify putting the line through the origin but that you have to justify not doing so.
But that's surely too advanced for what is expected here.

 

[maxim07]

Yeah I agree that random, not just systematic errors, could make it miss the origin as the line would not be perfect. I think I will be able to call it an inverse relationship.

what if it were a Charles‘s law experiment, where the length of an air column was measured as it changed with temperature, and there was a systematic error in length. A graph of length (because it is proportional to volume) against temperature in °C can be plotted and absolute zero would be
-c/m. Would you have to account for the systematic error by adding/subtracting the error (depending on whether it made the length to big or too small) or does it not matter? I‘m assuming it does because it affects c, unless it is only the change in length that matters?

 

[Mister T]

If you have a straight line then you can conclude that the relationship is linear. If that straight line passes through the origin, then you have a direct proportion.

I would advise you to draw a best fit line that passes through the origin, but your instructor may have a different opinion.

 

[haruspex]

Yeah I agree that random, not just systematic errors, could make it miss the origin as the line would not be perfect. I think I will be able to call it an inverse relationship.

what if it were a Charles‘s law experiment, where the length of an air column was measured as it changed with temperature, and there was a systematic error in length. A graph of length (because it is proportional to volume) against temperature in °C can be plotted and absolute zero would be
-c/m. Would you have to account for the systematic error by adding/subtracting the error (depending on whether it made the length to big or too small) or does it not matter? I‘m assuming it does because it affects c, unless it is only the change in length that matters?

Sorry, but I'm unsure what you are asking.
If you plot length against °C it will miss the origin in a systematic manner, but I would not call that an error. The improvement of fit by allowing it to miss the origin would be so great that it would clearly justify a constant in the equation.