How Do You Calculate Uncertainty in Physics Equations?

[tmilford]

Homework Statement

I am writing a report where I got a value of distance of let's L= 0.250 +/- 0.005 m. Then I need to use this value to calculate another distance value and an angle. How would I incoporate the uncertainity of this value into my other calculated values?

Homework Equations

The equations I am using are

0.11 = L tan A

d sin A = 600 x 10^-9

Where d is the distance I am trying to incorporate the uncertainity into and A is the angle I am trying to get the uncertainity of.

The Attempt at a Solution

I tried directly using the uncertainity value in the equations but I got a huge uncertainity value for my angle and a really small value for the distance, d, so I don't know that's the correct way of doing it.

 

[HallsofIvy]

If L tan A= 0.11 and L= 0.25+- .005, then
$$tan A= \frac{0.11}{L}$$

L may be as small as 0.25- 0.005= 0.245. In that case,
$$tan A= \frac{0.11}{0.245}= 0.4490$$
Determine A from that.

L may be as large as 0.25+ 0.005= 0.255. In that case,
$$tan A= \frac{0.11}{0.245}= 0.4314$$
Determine A from that.

Those two values of A give the possible range for A.

 

[tmilford]

Okay I understand what you did, but there isn't an exact value for the uncertainity for let's the say the angle. The difference between the max value and actual value is different than the difference between the min value and the actual value.

I am using a table to record these inputs and thought I could simply put a value down for the uncertainity not the range of two numbers, do you think I could average the two out or is there another way to do it?

 

[gneill]

Okay I understand what you did, but there isn't an exact value for the uncertainity for let's the say the angle. The difference between the max value and actual value is different than the difference between the min value and the actual value.

I am using a table to record these inputs and thought I could simply put a value down for the uncertainity not the range of two numbers, do you think I could average the two out or is there another way to do it?

You could use error analysis theory. If some result R depends upon a function f(x,y,z,...) of several variables, each with its own independent, gaussian error (+/- δ value), then

$$\delta R = \sqrt{\left(\frac{\partial f}{\partial x}\delta_x\right)^2 + \left(\frac{\partial f}{\partial y}\delta_y\right)^2 + \left(\frac{\partial f}{\partial z}\delta_z\right)^2 + ...}$$

In your case you have

$$A = arctan\left(\frac{0.11m}{L}\right)$$

So, only one variable with an error term (L +/- δ_L). Do the math!

 

[rwisz]

As a scientist, it is important to consider and account for uncertainties in your data and calculations. In this case, the uncertainty in your initial distance value of L= 0.250 +/- 0.005 m should be propagated through to your other calculated values. This means that the uncertainty in L should be taken into account when determining the uncertainty in the other distances and angles you are calculating.

One way to incorporate the uncertainty in L is to use error propagation techniques, such as the propagation of uncertainty formula. This formula takes into account the uncertainties in all the variables used in a calculation to determine the overall uncertainty in the final result. In this case, you would need to use the propagation of uncertainty formula for both the distance, d, and the angle, A, to incorporate the uncertainty in L.

Another approach could be to perform a sensitivity analysis, where you vary the value of L within its uncertainty range and observe the effect on your calculated values. This can give you an idea of how sensitive your results are to the uncertainty in L and can help you determine an appropriate range of values for your final results.

In addition, it is important to properly document and report your uncertainties in your report. This can include stating the uncertainty in your initial distance value, explaining how you incorporated the uncertainty into your calculations, and reporting the final uncertainties in your calculated values.

Overall, it is important to carefully consider and account for uncertainties in your work as a scientist to ensure the accuracy and reliability of your results.