Calculate Uncertainties [& the relevance of..]

OCR B (Advancing Physics) A2 Investigation

I'm doing an investigation into calculating spring constants from SHM formula

Not sure how or whether I need to express the uncertainties in my final value for k.

So, since T=2π√(m/k)
it follows that it can be arranged into cartesian form T²=(4π²m)/k
[y]

whereby the gradient is 1/k
so dx/dy is spring constant k.

The equation of line from Excel graph gives k to be 2.6387..

Do I need to somehow include the uncertainties from measuring Mass? :confused:

Mass of spring used is 0.0014kg +/- 0.00005kg

Also.. The time period.. is number of oscillations over four seconds(in this instance).
I used a stop-clock to count approximately four seconds, uncertainty of +/- 0.005s
Is this actually relevant to my final answer value of k, or should I just include it in my coursework to show i've acknowledged it & know how to work it out.

Reply 1

Stonebridge

Original post by Dreadstone

The equation of line from Excel graph gives k to be 2.6387..

Shouldn't you be drawing this graph by hand with error bars?

Do I need to somehow include the uncertainties from measuring Mass?

Mass of spring used is 0.0014kg +/- 0.00005kg

The uncertainty in the masses is negligible.

Also.. The time period.. is number of oscillations over four seconds(in this instance).
I used a stop-clock to count approximately four seconds, uncertainty of +/- 0.005s
Is this actually relevant to my final answer value of k, or should I just include it in my coursework to show i've acknowledged it & know how to work it out.

You don't measure "the number of oscillations over a time period", you measure the time period for a number of oscillations.
So time (exactly) 20 oscillations.
If you are using a stop clock to do this the uncertainty in the time measurement will be at least ±0.1s as you need to account for your reaction time and the fact that you are timing a moving object. (I would put it nearer ±0.2s). The uncertainty in the T² on the graph will give you error bars on your data.

The uncertainty in your final value from the gradient will be found by drawing your best gradient, using that as the quoted value, and then drawing a "worst fit" gradient to find the uncertainty.

(edited 11 years ago)

Reply 2

Dreadstone

Thank you for your rapid & informative response.

You don't measure "the number of oscillations over a time period", you measure the time period for a number of oscillations.
So time (exactly) 20 oscillations.

- in my original post when I said 'number of oscillations over a time period' it would have been more accurate & less confusing to say 'number of oscillations over a period of time'

I have haphazardly calculated the frequency, rather than the 'time period' T.
Since f=1/T this doesn't present much of a problem.
My apologies for being unclear in my wording.

I must say I do not understand your last sentence.. which is arguably the most crucial part.
I do not understand a 'worst fit' gradient.. I believe that is when you draw a min & max gradient using the error bars for the first & last plotted points..?

How can you determine the uncertainty from that? - two lines?.. is it the deviance of the gradient of these lines from the gradient of the line of best fit? but then there would be two separate values.. the largest deviance? That doesn't seem quite right.
I do not understand.. please, be kind and enlighten me.

Reply 3

Stonebridge

Original post by Dreadstone

Thank you for your rapid & informative response.

- in my original post when I said 'number of oscillations over a time period' it would have been more accurate & less confusing to say 'number of oscillations over a period of time'

I have haphazardly calculated the frequency, rather than the 'time period' T.
Since f=1/T this doesn't present much of a problem.
My apologies for being unclear in my wording.

It's not the wording it's what you are trying to do.
The point of the experiment is to measure the period of the oscillation.
You do this by timing, say, 20 periods and dividing by 20.
You don't do it by measuring how many oscillations take place in a particular interval of time.
It's vital you do it the way I have suggested. You don't measure the frequency and do T=1/f

I must say I do not understand your last sentence.. which is arguably the most crucial part.
I do not understand a 'worst fit' gradient.. I believe that is when you draw a min & max gradient using the error bars for the first & last plotted points..?

How can you determine the uncertainty from that? - two lines?.. is it the deviance of the gradient of these lines from the gradient of the line of best fit? but then there would be two separate values.. the largest deviance? That doesn't seem quite right.
I do not understand.. please, be kind and enlighten me.

Look at this diagram which has been exaggerated to make the point.

When you have the error bars in place (from the uncertainty in T² in your case) draw the "best" line. The one that goes through the centre of most points and inside all the error bars.
This is green on the diagram.
Then draw a "worst" lines as shown, steeper than and shallower than the best line. Both show in the diagram.
Then your experimental value is found from the best line, and the upper and lower limits of uncertainty from the worst lines.

(edited 11 years ago)

Reply 4

Dreadstone

"It's vital you do it the way I have suggested. You don't measure the frequency and do T=1/f"

They are mathematically the same, I cannot understand the difference.
Regardless, this is not a problem in the latter part of my experiment which used an ultrasound to determine the time period, so this is not a problem.

"Then your experimental value is found from the best line, and the upper and lower limits of uncertainty from the worst lines. "

This does not explain how, it states that it is found.

My issue is with that vital detail, how do you determine the upper & lower limits from the 'worst' lines of fit.
The gradient of the 'line of best fit' is 4pi²/k..
- we are supposed to use excel(or any other graph/data program) for all graph work for our coursework - so I can get the equation for each line from that. and rearrange the gradient value m to find k?

However, I have no idea how to add the 'worst lines of fit' on Excel, without having to manually input the intercept?
This seems ludicrous, and would be rather arduous to do for eight graphs as I need to..

Reply 5

Stonebridge

Original post by Dreadstone

In your first post you said

"The time period.. is number of oscillations over four seconds(in this instance).
I used a stop-clock to count approximately four seconds"

My advice is based on this. You said nothing about ultrasound.
As you have been vague about what you have actually done, it's difficult to give you specific advice.

"Then your experimental value is found from the best line, and the upper and lower limits of uncertainty from the worst lines. "

This does not explain how, it states that it is found.

My issue is with that vital detail, how do you determine the upper & lower limits from the 'worst' lines of fit.

Again, you have not given much detail. I'm having to guess a lot here...
If your experimental brief is to find the spring constant by measuring T and plotting T squared against m, and you do this by using the gradient of the graph, then the standard method of finding the uncertainty in the result is to do as I have explained.
Your experimental value is found from using the best line as previously explained. You know how to do this. You then use the worst lines and find the value of k given by these. So if you get k = 7.4 (say) using the best line, and get k=7.2 and 7.6 using the two other lines, your value of k is 7.4 ± 2

- we are supposed to use excel(or any other graph/data program) for all graph work for our coursework - so I can get the equation for each line from that. and rearrange the gradient value m to find k?

However, I have no idea how to add the 'worst lines of fit' on Excel, without having to manually input the intercept?
This seems ludicrous, and would be rather arduous to do for eight graphs as I need to..

Eight graphs? Why?
Again, there is clearly much you have not described about what you are doing.

If you have been told to use Excel to find the gradient of the line, and the value of k from the gradient, I suggest you ask your teachers how they expect you to find the uncertainty in the gradient. You need this to find the uncertainty in k. Does Excel do error bars? If not you need to ask your teachers how they expect you to put error bars on your graph.
You actually only need to draw one of the "worst" lines and can assume the other gives a value equally above or below the best value.
It is a useful (some would say essential) practical physics skill to be able to draw a straight line on a set of points manually and estimate a best and worst gradient.
I'm surprised your teacher has not gone through this with you.