Calculating Uncertainty of an intrinsic function

[sam400]

Homework Statement

I need to calculate the value and the uncertainty of the equation \begin{equation} S = 2d \sin(\theta) \end{equation} (one side of the Bragg formula), however, the final answer is strange, so I would like to know what I'm doing wrong.

Homework Equations

My $\theta$ value: $17.5^{\circ}$
My $d$ value: $4.28 \pm 0.03$ cm
As far as I know, for an intrinsic function, the uncertainty is the root mean squared of the differential, so
\begin{equation} \Delta S = \sqrt{\left(\frac{\partial S}{\partial a} \Delta a \right)^{2} + \left(\frac{\partial S}{\partial b} \Delta b \right)^{2} } \end{equation}

The Attempt at a Solution

After plugging in the numbers and taking the derivatives, the formula that I get is:

\begin{equation}\Delta S = 2 \sqrt{( \sin\theta \Delta d)^{2} + (d \cos\theta \Delta \theta)^2} \end{equation}

For S, I get about 2.60 cm, which is more or less what I expected, but for $\Delta S$, I am getting a value of about 2, which means a relative error of 77%, so there is probably something wrong with what I did.

 

[vela]

What value did you use for $\Delta \theta$?

 

[BvU]

Most frequent error in this context is using degrees instead of radians.

 

[sam400]

Oops, I just realized I have been using degrees for my values. Now my error is more sensible. I had 0.3 degrees for my angle uncertainty, but it wasn't converted to radians.

 

[HalfLostAndSearching]

First of all, it is important to note that the uncertainty of an intrinsic function is not always the root mean squared of the differential. It depends on the specific function and how the variables are related. In this case, the uncertainty of the function S depends on the uncertainties of both d and theta, so the formula you have used is correct.

However, there may be a mistake in your calculation. It would be helpful if you could provide the full calculation so that I can check for any errors. Additionally, it would be helpful to know the units of your values for d and theta. If they are in different units, you may need to convert them to the same units before plugging them into the formula for uncertainty.

Another possibility is that your values for d and theta are not the true values, but rather estimated values with associated uncertainties. In that case, the uncertainty of S would also depend on the uncertainties of d and theta, which may explain the larger value you are getting for $\Delta S$. Again, providing the full calculation and units would be helpful in determining the issue.

In any case, it is important to carefully check your calculations and units to ensure accuracy in your results. If you are still having trouble, I suggest seeking help from a colleague or professor who is familiar with the specific equation and variables you are working with.