Physics IA Uncertainty Question

[sportsfan998]

Does anyone know how to calculate uncertainty involving sine functions.

 

For example if I divide: sin(50.0 +/- 0.5) / sin(30.0 +/- 0.5), what would be the uncertainty?

[Award Winning Boss]

If you're dividing or multiplying and you have uncertainties, you add them together. 

 

If you're adding or subtracting, you add them together. 

 

If the number is raised to a power, you multiply the uncertainty with the power. 

[Vioh]

Does anyone know how to calculate uncertainty involving sine functions.

 

For example if I divide: sin(50.0 +/- 0.5) / sin(30.0 +/- 0.5), what would be the uncertainty?

 

There are several methods of calculating the uncertainty for functions like sine. I'll show your 3 methods, but I reckon that there are many more.

 

Method 1: Take a look at this link under section (5e) http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart2.html

 

Method 2 (Using derivatives):

 

As you can guess, is simply the uncertainty of the sine function itself, and is the uncertainty in the angle. Also, there are 2 things that you must keep in mind:

• This method only works if for small uncertainty in the angle (i.e. when in small). This is because derivatives only work for small changes in the input, i.e. when tends to zero.
• This method only works if all the angle measurements (and their uncertainties) are in RADIAN (not degrees). This is because calculus only works in radians

Method 3 (The "Range divided by 2" method):

 

Simplifying the equation using compound identity for sine function, we have:

 

So the uncertainty must be:

 

Discussion:

As already stated in the link, method 1 is a bit of an oversimplification (because it doesn't take into account the other side of the uncertainty), so method 2 and method 3 would be more appropriate to use in most cases.

Now, if you look at the final equation derived from method 2 and method 3, you will see that they are "approximately" equivalent for small value of when you use radian instead of degree. This is because for small radian value of , you can approximate:

So in conclusion, the safest method to use is method 3 because it works for both small and big value of

 

Example:

We'll now go back to the example you gave using method 3. The uncertainty of the first term is:

The uncertainty of the second term is:

 

I'll let you figure out the uncertainty for the final result yourself because the rest is just simple propagation of error that you learn in IB