Errors

[Firepanda]

1. The problem statement, all variables and given/known data

I have read previous topics about errors, however none have been suited for my problems. Basically, this is for my a level physics coursework, which includes more than this example, but if i crack this then im sure i can do the rest. :)

I am using hookes law (F=kx) to find a spring constant, i have my spring constant but finding the error in it is a challenge.

I have readings which are as follows:

Force (N) Extension (m)
0 0
0.987 0.027
1.962 0.066
2.940 0.104
3.912 0.142

extension error = ±0.0005m
mass balance error = ±0.000005kg
gravity estimation error = ±0.005m/s².


I can find the spring constant from this by drawing a graph and finding the gradient. But what is the error in the spring constant (27.087)?


3. The attempt at a solution

I tried to find the error in each seperate reading, but i couldn't adapt this to find a single error for the spring constant.

Any help with be much appreciated

Thankyou

[mezarashi]

I'm not sure what kind of exercise this is for you? Are you trying to find the experimental error from the graph or do you want to calculate the error based on the extension errors, mass error, and gravity errors given.

Normally, errors are given in percentage, because with absolute values, the final error depends completely on what your mass is, and what your extension is. Then you'd have to use some calculus to determine the total error.

[Flux = Rad]

OK, so I did Adv Higher instead of A level but this is how they told us to do it:

The errors you've got written down are used to draw the error bars on your graph. You have to combine the errors in g and m to get an absolute error in force. Since they don't teach you the general method (using calculus) till 1st year university, you have probably been told to combine errors of multiplied parameters using:

(\frac{\Delta A}{A})^2 = (\frac{\Delta B}{B})^2 + (\frac{\Delta C}{C})^2

for a formula of the form A = BC.

Once you've got those you can plot your points with error bars. The way we then found the error in our gradient was using the "centroid method" which involved drawing parallelograms with points etc. I've forgotten the details but you've probably been taught something similar?

[andrevdh]

Use error bars on your graph to determine the uncertainty in the slope

http://www.rit.edu/~uphysics/graphing/graphingpart1.html

[Firepanda]

To find the gradient on my graph i used Sxy/Sxx as adapted from my statistics module to find the perfect gradient.

I can live without finding errors for the linear graph is its too complicated without error bars. But ive also plotted a graph which is curved, is it possible to plot error bars for those points and will the errors be the same for each point?

Thanks in advance

[Firepanda]

no problem guys, ive talked with other people on using error bars for my curves, i can do it all now :D thx for the info and links anyway

[loom91]

Hmm, this is a purely statistical problem. Why do they set such problems in Physics? I think the way to do this is by doing a multivariate Taylor expansion and then discarding non-linear terms (assuming error is small), which gives you the error of the function as a linear combination of the errors in the variables, where the coefficients are the partial derivatives evaluated at the measured value. I may be wrong in the details, I never pay much attention to my high-school statistics.

Basically try to write the spring constant symbolically in terms of the parameters and then Taylor expand about the mean values (which I think you can get by minimising mean square deviation, though I may be wrong, we don't study curve-fitting till the next grade).

I'm sorry I can't be more specific, but you can find detailed derivations of various results in error analysis in any book on statistics, it's a major part of statistics. But it will probably make for a more solid learning if you sat down and tried to figure out the formulas yourself (if you have an interest in maths) instead of looking them up and memorising them.

Molu

[Firepanda]

Hi guys!

Just a really quick question :)

If my percentage error in Velocity is 2%, then i square Velocity what is my % error now? 4%?

EDIT: sorry i was beign a dumbass it seems, i add the errors not square :D 2+2=4

[Flux = Rad]

Actually, you don't square or add the errors but your answer is still right. You multiply the percentage error by the power (for the one variable case). More generally,

if

A = BC^n

then

(\frac{\Delta A}{A})^2 = (\frac{\Delta B}{B})^2 + n^2 (\frac{\Delta C}{C})^2

[loom91]

Let y be a function of a quantity v. The correct value of v is v_0, but there is small error in our measurement. What then is the error in y? If we assume y to be analytic, then

y(v) = y(v_0) + (\frac{dy}{dv})_{v_0} (v - v_0) + ...

If y = v^2 then \frac {dy}{dv} = 2v. Discarding higher order terms (assuming error to be small), we get

y(v) = y(v_0) + 2v_0 (v - v_0)
or, y(v) - y(v_0) = 2 v_0 (v - v_0)
or, \frac {y(v) - y(v_0)}{y(v_0)} = 2 \frac {v - v_0}{v_0}
or, E_R(y) = E_R(v^2) = 2 E_R(v)

Therefore, relative error behaves just like logarithms.