Determining uncertainty in an experimental value of g

[tw336]

Homework Statement

In an experiment to determine the acceleration due to gravity close to the Earth’s surface g, the time t taken for a steel ball, released from an electromagnetic connector, to fall a vertical distance y was measured. The results are given in Table 1.
a) Use the results to calculate an experimental value of g and compare this to the expected value.

Relevant Equations

g=2*(x/t^2)

Hello there! This is my first post, so I apologise for any faux pas I am about to commit.
I have recently bumped into a few situations where I'm uncertain about my uncertainties. Especially where the value is a product of multiple variables.
Please see the attatched table, where g is a function of time and distance. I have used kinematic equations to solve for g (g=2*(x/T^2)). What would be the correct/appropriate method for calculating the uncertainty in the mean of g?
Can I look to find the standard deviation and error, and call the error the uncertainty in my mean? The data is not normally distributed.
Or do I need to propogate the uncertainties as a function of distance and time? g=f(T,d)

Any help would be greatly appreciated,

Thanks in advance,

T

 

[BvU]

Hello tw , !

First thing to do is make a plot. In your case you expect a straight line when you plot t² as a function of x.

 

[Gaussian97]

Well, you have an instrumental error in the computation of $x$ and $t$, propagating errors you can find how these errors affect your values of $g$, and therefore consider them as the instrumental error of the measures of $g$. Then you compute the statistical error of the values of $g$ and combine them as usual in quadrature.

 

[kuruman]

And your value for g should be consistent with the significant figures of your input variables. How many is that?

 

[Cutter Ketch]

Can I look to find the standard deviation and error, and call the error the uncertainty in my mean

If they were all the same measurement, this would be almost correct. The standard deviation would indicate the error in each measurement. The error in the mean would then be smaller by the square root of the number of measurements. That assumes a normal distribution (but is often right regardless because of the central limit theorem)

Unfortunately, these measurements are all different and are expected to have a different error. You can do what you suggest above, but better would be to have some estimate of the error of each measurement. That would require characterizing the components of the error (timing accuracy, accuracy of the height measurement) and propagating the error to the derived value of g. Then you could calculate a weighted mean and standard deviation.

On the other hand, here you have measurements that were varied in a specific way, a way for which you have a well known model. In that case the best way to analyze the value of g and the error is by fitting the model. The value of g is the best fit straight line, and the uncertainty is determined by how much the least squares error changes if you change the value of g.

 

[BvU]

Did you make the plot ? You will see that the error estimate in the time is grossly overestimated(*) and you have some explaining to do.

(*) to the extent that I suspect a typing error: 0.01 s is much more in agreement with the spread in the data !

In contrast, if the inaccuracy in the measurement of the distance is indeed 0.5 mm, I should expect the data in the table to show one or two more significant digits : e.g. $0.1003 \pm 0.0005$. To comment more, a full description of the setup and the procedure is needed.

 

[tw336]

Did you make the plot ? You will see that the error estimate in the time is grossly overestimated(*) and you have some explaining to do.

(*) to the extent that I suspect a typing error: 0.01 s is much more in agreement with the spread in the data !

In contrast, if the inaccuracy in the measurement of the distance is indeed 0.5 mm, I should expect the data in the table to show one or two more significant digits : e.g. $0.1003 \pm 0.0005$. To comment more, a full description of the setup and the procedure is needed.

 

[tw336]

Thank you kindly for your reply.
Please see an attached table plotting t^2 against distance (x). The sample data given for this example is included also. The error in the period was given as 0.1s, whereas the given error in distance was 0.05cm. When converted to metres the error bars for distance dissappear on this scale.
In this instance I am not awfully concerned with significant figures. Couple of clicks and excel will sort that out, theyre all there, just removed to make the data digestable. It is the method of propogating error that's got me scratching my head.

 

[tw336]

On the other hand, here you have measurements that were varied in a specific way, a way for which you have a well known model. In that case the best way to analyze the value of g and the error is by fitting the model. The value of g is the best fit straight line, and the uncertainty is determined by how much the least squares error changes if you change the value of g.

Thanks so much for your reply, Its greatly appreciated. I can see I need to improve my reading and delve further in calculus to help me come to a solution. Any shunt in the right direction for techniques and tools I need to cover to solve these types of scenarios would be incredibly helpful.
Cheers.
T

 

[BvU]

You see that the errors drawn are bigger than the variance wrt a straight line. A trendline will give you a slope, doing data analysis | regression gives a slope with an error