Errors in calculating the acceleration of gravity

[Puchinita5]

So I did a lab where we are calculating the acceleration due to gravity using a variety of methods. For two of the methods, we have an object attached to a magnet, then the magnet is turned off so the object is in free fall and it makes a spark/mark every 1/60th of a second on a piece of tape. And then we use the distances of the marks on the tape to calculate gravity. The first method, we make a plot of velocity versus time, because V = V0 + gt, so the slope of the line will be g. The second method we plot Distance versus t^2 since D(t) = .5 g t^2. and the slope is 1/2g.

My question is, I'm supposed to think of which method seems better. Is it accurate to say that the first method using v vs t is probably more reliable because, maybe errors would get propagated through the t^2 in the second method, but not for the first method since it's just t?

I also notice that in my D vs t^2 plot, there is a slight curve at the beginning, I'm guessing this is because the magnet when it's closer at the beginning of the drop. But i don't see this curve in the other plot, is this again, just because the t^2 is magnifying the errors?

 

[mfb]

The second method certainly gets problems if the initial time measurement is not accurate, or if the magnet does not switch of instantly (it does not...). Plotting the square root of distance against time would avoid the timing problem, but then you get problems if the initial distance is not accurate.

The velocity method will have larger fluctuations between the measurements, but the overall slope should be better, I agree.

 

[haruspex]

You have not explained how you determine the velocity values for the plot.

 

[Puchinita5]

basically, we are taking the distance between two points, and diving by 1/60 to get the velocity between those points. The first point is at first taken to be at time 0, but then when we plot V vs t, we take the y intercept and add this to all the time values in excel. Then plot position versus time^2. Why is there a difference between plotting position versus time, and fitting a polynomial, and position versus time^2, and fitting a line?
I guess I just don't understand why all three plots don't find the same value of g and what the source of the discrepancy is.

 

[haruspex]

we are taking the distance between two points, and diving by 1/60 to get the velocity between those points.

That gives you the average velocity over that interval, but that is not the velocity at either of those times. What time did you plot it against?

 

[Puchinita5]

We basically just plot against relative times, since we know that every spark happened every 1/60 seconds, so the time column in excel is just 0, 1/60, 2/60, 3/60 etc... and then when we plot the average velocities vs time, we fit a line, and then the y intercept we add back to each of the times, since that gives the time the first spark should have happened. Then we can plot D vs t^2 using the position of each of the sparks.
so at first, we have a column of positions s1, s2, s3, and we assume s1 was at t=0, then s2 was at t= 1/60, etc... but then we add the y intercept, so s1 at t=yint, s2 at t= 1/60 + y int etc.

 

[Puchinita5]

wait, hmmm the whole adding y intercept thing isn't right. we add the time at which the best fitting line intersects V = 0, sorry, my brain is fried.

 

[Puchinita5]

But I guess, either way, assuming all my data is correct, and I have a column in excel for position, velocity, time, and time-squared, what would be the advantages or disadvantages of plotting
1) velocity vs time and calculating g from the slope of the best fitted line,
2) position vs t and calculating g from a polynomial fit
3) position vs t^2 and calculating g from the slope of the best fitted line

Because I would assume they would all give the same answer but they don't, so which would be most reliable?

 

[haruspex]

We basically just plot against relative times, since we know that every spark happened every 1/60 seconds, so the time column in excel is just 0, 1/60, 2/60, 3/60 etc... and then when we plot the average velocities vs time, we fit a line, and then the y intercept we add back to each of the times, since that gives the time the first spark should have happened. Then we can plot D vs t^2 using the position of each of the sparks.
so at first, we have a column of positions s1, s2, s3, and we assume s1 was at t=0, then s2 was at t= 1/60, etc... but then we add the y intercept, so s1 at t=yint, s2 at t= 1/60 + y int etc.

You have not answered my question.
You have a distance at time 1/60 and a distance at time 2/60. From that you compute an average speed for the interval [1/60, 2/60]. But do you plot that as the speed at time 1/60, the speed at time 2/60, or something else?

 

[Puchinita5]

I'm taking the average velocity between two points and assigning it as the velocity of the second point. So if I have
s0 is at 0cm,
s1 is at 1.2 cm
s2 is at 2.6 cm

then velocity
v0 = 0
v1 = (s1-s0)/(dt) = (1.2 - 0)/(1/60.) = 72 cm/s
v2 = (s2-s1)/(dt) = (2.6 - 1.2)/(1/60) = 84 cm/s

So then I have in excel:
s v t
0 0 0
1.2 72 1/60
2.6 84 2/60

etc. with more values.

 

[haruspex]

I'm taking the average velocity between two points and assigning it as the velocity of the second point.

Ok, but you understand that it is not correct, right? Can you think of a better way?

 

[Puchinita5]

Ok, but you understand that it is not correct, right? Can you think of a better way?

This is what the lab instructions tell me to do, so I can't really change it.

 

[haruspex]

This is what the lab instructions tell me to do, so I can't really change it.

Ok, but if you understand the error in doing that then you have a good answer to your original question.

 

[Puchinita5]

Ok, but if you understand the error in doing that then you have a good answer to your original question.

Hmmm, well I know that the actual velocity at each point won't be the average over a segment. But that would make me think that plotting distance over time would give a better estimate, except my distance versus time^2 plot gives me a value that isn't as close to the known value of g than v versus t. But I feel like i know the distances much better than I know the velocities. I only know the relative times, and perhaps even with the correction that we add in, it's just much more uncertain? In which case, squaring it will enhance the error on it?

 

[haruspex]

Hmmm, well I know that the actual velocity at each point won't be the average over a segment. But that would make me think that plotting distance over time would give a better estimate, except my distance versus time^2 plot gives me a value that isn't as close to the known value of g than v versus t. But I feel like i know the distances much better than I know the velocities. I only know the relative times, and perhaps even with the correction that we add in, it's just much more uncertain? In which case, squaring it will enhance the error on it?

I don't see any good reason the D v. T² should be less accurate. We know that your method of determining v at a given time is wrong, and it may be that this is somehow compensating for an error in the data, giving the appearance of greater accuracy.

Setting aside what you were told to do, for the moment, what would be a more appropriate way to plot velocity against time? If you do that, do you see the same result as for D v. T²?

Perhaps you could post the actual data.

 

[Puchinita5]

Hmmm I can't seem to find a good way to paste data here. Is there a way to add a table? :( So, maybe I should plot total distance over total time traveled? Instead of the little increments. Hmm, maybe that is what he meant, and I just misunderstood, it's not a super well written lab manual.

 

[haruspex]

Hmmm I can't seem to find a good way to paste data here. Is there a way to add a table? :( So, maybe I should plot total distance over total time traveled? Instead of the little increments. Hmm, maybe that is what he meant, and I just misunderstood, it's not a super well written lab manual.

Think about this: if the distance at time t₁ is s₁ and the distance at time t₂ is s₂, and you calculate the speed as v=(s₂-s₁)/(t₂-t₁), at what time will the speed actually be v?

 

[haruspex]

I can't seem to find a good way to paste data here

Click on the UPLOAD button at bottom right. .txt and .xlsx are suitable choices.

 

[Puchinita5]

Okay here is my data. I'm still so confused. Sorry for being so slow with this!

 

[Puchinita5]

so i think when i was adjusting the time to adjust for the initial falling time, from the y intercept of the v vs t plot, i had definitely done that wrong, and now my values are close... 997 cm/s^2 for the v vs t method, 978 for the p vs t polynomial fit method, and 921 for the p vs t^2 linear method.
So it does seem that the least accurate method is when we square time, then the velocity method, then the polynomial fit method (which coincidentally, wasn't actually asked in the lab, I just wondered why we couldn't do it).

 

[haruspex]

Okay here is my data. I'm still so confused. Sorry for being so slow with this!

OK, I see what the problem is.

There are some very doubtful datapoints near the start. Notice it seems to be faster in the first time interval than the second!
The plot of velocities becomes steadily less influenced by these errors, since it only looks at what is happening currently. The plot of total distance remais affected by those errors much longer.
This suggests to me that (other than the error I have already pointed out in plotting velocity against time) the primary sources of error in the experiment are uncertainties in the first fraction of a second: when did the clock start exactly, where is the descent measured from, how long did it take for the magnetic field to drop to zero?
Given that, the more accurate method is the one less sensitive to such errors. I was able to get much better looking graphs simply by adding .034s to all times and setting the initial position as 0.45cm.

 

[haruspex]

so i think when i was adjusting the time to adjust for the initial falling time, from the y intercept of the v vs t plot, i had definitely done that wrong, and now my values are close... 997 cm/s^2 for the v vs t method, 978 for the p vs t polynomial fit method, and 921 for the p vs t^2 linear method.
So it does seem that the least accurate method is when we square time, then the velocity method, then the polynomial fit method (which coincidentally, wasn't actually asked in the lab, I just wondered why we couldn't do it).

Bear in mind that because in reality there is drag, a value further from the known g does not necessarily indicate a less accurate method.

 

[mfb]

The parabola fit with three free parameters should give the best estimate, I get 9.79 with it.
Adding a drag term could improve it even more, but I'm not sure if the measurement is sensitive to it - if not, fitting it would increase your fit uncertainty too much.

The methods differ in the way uncertainties are treated, so they get different results even if you fix the issue with the time and starting point determination.