Physics Lab Propagation of Error Issue

In summary, the homework statement is about calculating the uncertainty in a function from the lab manual for the critical angle of refraction. The equations that were provided ask for the derivative of the inverse tan function to be unitless. However, when multiplied by Δa, the units of cm^-1 do not cancel. This causes confusion for the student because they do not understand how to treat a derivative as a fraction. In the end, the student is provided with a summary of what the derivative is and what units it would have.
  • #1
Browntown
18
0
Homework Statement
Physics Lab Propagation of Error Issue
Relevant Equations
∂f/∂b=∂/∂b (tan^(-1)⁡(a/4b))= 1/(1+(a/4b)^2 )×(-a)/(4b^2 )=1/(1+((5.922 cm)/(4×1.766 cm))^2 )×(-5.922 cm)/(4×(1.766 cm)^2 )
Homework Statement: Physics Lab Propagation of Error Issue
Homework Equations: ∂f/∂b=∂/∂b (tan^(-1)⁡(a/4b))= 1/(1+(a/4b)^2 )×(-a)/(4b^2 )=1/(1+((5.922 cm)/(4×1.766 cm))^2 )×(-5.922 cm)/(4×(1.766 cm)^2 )

Hello,

I'm trying to find the uncertainty in a function from the lab manual for the critical angle of refraction and since this is my first time doing such a thing I'm a bit confused.
When I take the derivative of the inverse tan function provided and input my values, the units of cm don't seem to cancel and I can't figure out how to fix that
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  • #2
It seems to me that you want ##\Delta f = \Delta \tan^{-1}(\frac{AB}{4t})## to be unitless. What happens when you multiply ##\frac{\partial f}{\partial a}## by ##\Delta a##?
 
  • #3
Δa is in units of cm so technically when they're multiplied they cancel out (cm^-1 x cm) but I thought anytime you take the inverse of a trig function you get an angle?
 
  • #4
That is true. Then you differentiated that angle with respect to a variable with units of cm. What would you expect to be the units of the derivative?
 
  • #5
Would the derivative of a variable with units in cm be then cm/s? instantaneous velocity?
 
  • #6
Consider the definition of derivative:

$$\frac {df}{da} = \lim_{\Delta a \rightarrow 0} {\frac {f(a +\Delta a)- f(a)}{\Delta a}}$$

With ##f(a)## an angle and ##a## a distance, what would be the units of ##\frac {df}{da}##?
 
  • #7
I'm sorry but I honestly have no clue..
 
  • #8
Browntown said:
I'm sorry but I honestly have no clue..
OK. Let's back up one step. if ##f(a)## is unitless and ##a## has units of cm, what are the units of $$\frac {f(a +\Delta a)- f(a)}{\Delta a}$$
 
Last edited:
  • #9
Would they just cancel?
 
  • #10
I'm sorry, my lab report is due tomorrow and I think the stress of trying to get it done is just not letting me think properly.
 
  • #11
Here's the bottom line: for the purpose of determining its units, you can treat a derivative like a fraction. So if ##f## is unitless and ##a## has units of cm, then ##\frac {df}{da}## has units of ##\frac {\text {unitless}}{\text {cm}} = \text {cm}^{-1}##
 
  • #12
So that means that even though its the inverse tangent function, because it's the derivative of it, it can have units of cm^-1?
 
  • #13
Browntown said:
So that means that even though its the inverse tangent function, because it's the derivative of it, it can have units of cm^-1?
Yes.
 
  • #14
Awesome! Thank you so much!
 

Related to Physics Lab Propagation of Error Issue

1. What is propagation of error in a physics lab?

Propagation of error in a physics lab refers to the process of calculating the uncertainties in the final result of an experiment based on the uncertainties in the individual measurements and the mathematical operations used to obtain the final result.

2. Why is propagation of error important in a physics lab?

Propagation of error is important in a physics lab because it allows us to account for the uncertainties in our measurements and ensure that our final result is as accurate and reliable as possible. It also helps us identify which measurements have the greatest impact on the final result, allowing us to prioritize improving their accuracy.

3. How is propagation of error calculated in a physics lab?

Propagation of error is calculated using the principles of error propagation, which involves using partial derivatives and the rules of calculus to determine the uncertainty in the final result based on the uncertainties in the individual measurements. This is typically done using a formula or equation specific to the type of experiment being performed.

4. What are some common sources of error in a physics lab?

Some common sources of error in a physics lab include limitations of measurement instruments, human error in taking measurements, environmental factors such as temperature and humidity, and systematic errors caused by flaws in the experimental setup or procedure.

5. How can propagation of error be minimized in a physics lab?

To minimize propagation of error in a physics lab, it is important to carefully plan and design the experiment, use high-quality and properly calibrated measurement instruments, take multiple measurements to reduce random error, and identify and account for potential sources of systematic error. It is also helpful to use more precise mathematical techniques, such as error propagation by Taylor series, when calculating uncertainties.

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