What's wrong with my calculation here? (propagation of uncertainty)

[Mrq]

Homework Statement

I have to calculate the propagation of uncertainty for my error analysis lab but I am getting a percentage error of about 22-34 percent, just because the partial derivative of gravitational acceleration with respect to time has a large value. Could somebody please help? I have attached the calculation table with this thread.

Homework Equations

The Attempt at a Solution

 

[FactChecker]

I can't understand the question. You should explain a lot more about what the problem is, what the data means, and what your calculations were.

 

[Mrq]

Basically I am trying to calculate the uncertainty in g with the help of the standard deviations of "l" and "T", the experiment is about calculating the value of g with the help of a simple pendulum. These are the standard deviations
Standard deviation in l = 0.1
Standard deviation in 'T' = 0.2
I am trying to calculate the uncertainty propagated in g due to the deviations in the measurement of l and T using the formula.

That's the general formula.
The partial derivatives of g with respect to time has the value -1638 while the partial derivative of g with respect to the length is 28.6. (-8pi^2*l/T^3 and 4pi^2/T^2 with l=33.5 cm and T=1.17s)
This is for the first measurement where the average length was 33.5 cm and the average time period was 1.17 seconds.
The value of g I got was 964 cm/s^2.
Now when I put the values of the partial derivatives and the standard deviations in the above formula I get a propagated deviation of about 387 cm/s^2 which indicates an error of about 33%.

 

[Ray Vickson]

Homework Statement

I have to calculate the propagation of uncertainty for my error analysis lab but I am getting a percentage error of about 22-34 percent, just because the partial derivative of gravitational acceleration with respect to time has a large value. Could somebody please help? I have attached the calculation table with this thread.

Homework Equations

The Attempt at a Solution

Your attached document is completely incomprehensible. You need to explain what the quantities are that you are measuring, and the supposed theoretical relations between them. Are you measuring distance vs. time? Are you measuring velocity vs. time? Are you measuring velocity vs. distance? Are you measuring something else? You may know what the numbers mean, but nobody else in the world does.

 

[mjc123]

Your answers look right if your standard deviations are what you say they are. An error of ca. 20% in T, with T² in your formula, would give an error of roughly 40% in g. Are you sure of your standard deviation for T? What is your standard deviation for an individual measurement of t? How would you calculate the standard deviation of T from this?

 

[Mrq]

Your attached document is completely incomprehensible. You need to explain what the quantities are that you are measuring, and the supposed theoretical relations between them. Are you measuring distance vs. time? Are you measuring velocity vs. time? Are you measuring velocity vs. distance? Are you measuring something else? You may know what the numbers mean, but nobody else in the world does.

I am really sorry about that. The experiment is actually "measuring g with the help of a simple pendulum". The table contains different measurements of length the pendulum and the time it took for ten vibrations, it's time period and the value of g which has been calculated using these parameters. The final column has the fractional propagated uncertainty in 'g'. It's actually the propagated deviation in the value of g.

 

[Mrq]

Your answers look right if your standard deviations are what you say they are. An error of ca. 20% in T, with T² in your formula, would give an error of roughly 40% in g. Are you sure of your standard deviation for T? What is your standard deviation for an individual measurement of t? How would you calculate the standard deviation of T from this?

The time measurements were 11.5, 11.7 and 11.9 respectively with 11.7 as the average. The standard deviation was 0.2.
I used this formula standard deviation of T, T= sqrt((11.9-11.7)^2+(11.5-11.7)^2+(11.7-11.7)^2)/sqrt(3-1)= 0.2
Sorry for not using LaTeX.

 

[Ray Vickson]

The time measurements were 11.5, 11.7 and 11.9 respectively with 11.7 as the average. The standard deviation was 0.2.
I used this formula standard deviation of T, T= sqrt((11.9-11.7)^2+(11.5-11.7)^2+(11.7-11.7)^2)/sqrt(3-1)= 0.2
Sorry for not using LaTeX.

As a percentage, the standard deviation is $100 \times 0.2 / 11.7 \approx 1.71$, so the (relative) uncertainty in $T$ is about 1.71%, not 20%.
The values of $T^2$ go from $11.5^2 =132.25$ to $11.9^2 = 141.61$, with a "central" value of $11.7^2 = 136.89$. The standard deviation of $T^2$ is nearly $(141.61 - 132.25)/2 = 4.68$, so the (relative) uncertainty in $T^2$ is about $100 \times 4.68/136.89 \approx 3.42$, so about 3.42%, not 40%.

 

[mjc123]

The time measurements were 11.5, 11.7 and 11.9 respectively with 11.7 as the average. The standard deviation was 0.2.
I used this formula standard deviation of T, T= sqrt((11.9-11.7)^2+(11.5-11.7)^2+(11.7-11.7)^2)/sqrt(3-1)= 0.2
Sorry for not using LaTeX.

That's not how it's done. If the standard deviation of a single measurement of t is s, then the s.d. of t_m, the mean of 3 measurements, is s/√3. The standard deviation of T = t_m/10 is s/(10√3) = 0.0115, or about 1% of T.

 

[Mrq]

That's not how it's done. If the standard deviation of a single measurement of t is s, then the s.d. of t_m, the mean of 3 measurements, is s/√3. The standard deviation of T = t_m/10 is s/(10√3) = 0.0115, or about 1% of T.

Thanks a lot!