Optimizing Error: Minimizing Rx for L1=50cm

[LogicX]

Homework Statement

Show that this uncertainty expression is minimized for L1=50cm:

Uncertainty in Rx= R3 (1/L1 + L2/L1^2)σL

(that is R sub x, L sub 1, σ sub L etc.)

Homework Equations

I don't know.

The Attempt at a Solution

This is the uncertainty for sliding a contact up and down a 100cm stick. So L1 and L2 are related by L1+L2=100cm

I don't know if the above expression is even right (though that's what was given to me). Let's say instead I made L1= 55 cm (meaning L2 is 45cm). Guess what! the uncertainty is lower.

Of course, I'm supposed to somehow derive this uncertainty expression on my own. But of course, as with everything in this bull**** 1 credit hour physics lab course, they just throw a bunch of strange questions at us, most of which we are completely unequipped to handle. Like, here, use calculus to derive this uncertainty expression where these two variable are dependent on each other! And the foreign TA just talks really loudly when I ask him how to do it, I think as his defense mechanism for not knowing anything. So I just leave even more confused and wondering how he got away without actually answering any questions. And then they tell us to study those questions for what is going to be on the final. Oh and of course it makes sense that the exams for this LAB are worth 90% of the grade, whereas the 4 hour long weekly lab reports are worth just 10%. Makes complete sense. For one credit hour. I just want to get my A and never think about this stupid course ever again.

Sorry that you had to experience my anger about physics lab.

 

[praharmitra]

Given any function f(x) do you know how to minimize it?

 

[LogicX]

Given any function f(x) do you know how to minimize it?

Yes. Take the derivative set it equal to zero, solve, plug back into original function. But they have already told me that the mimimum arises when L1=50cm. But if I put a different value in for L1, I can get a lower value for the uncertainty, making me think they gave me the wrong expression.

I'm pretty sure I only have to worry about the inner part of the function, (1/L1 + L2/L1^2), since the other stuff is just constant. Minimize that part and the entire expression will be minimized. Am I supposed to be setting L2=100-L1?

EDIT: Just tried that, I keep getting all my L's to cancel out, leaving me with -200=1.

 

[praharmitra]

I agree with you. something is amiss. the part inside the brackets is actually just 100/L1^2. What is L in your original expression?

 

[LogicX]

I agree with you. something is amiss. the part inside the brackets is actually just 100/L1^2. What is L in your original expression?

The original expression for Rx is Rx=(L2/L1)R3

R3 is a constant. (This lab was measuring an unknown resistance using a wheatstone bridge)

I could normally easily find the uncertainty expression for this equation but my book says that it "differs from the error equation we would normally obtain because the errors in L1 and L2 are not independent, i.e. L1+L2=100cm. Thus, if one makes a positive error in L1, this would result in a negative error in L2."

I've never encountered something like this before, and there is nothing in any of my books about this type of error. So I'm kind of confused.

I appreciate the help by the way.

 

[praharmitra]

ya, what the book is basically saying is that you should put L2 = 100 - L1, then minimize the function of L1. Then the answer would indeed be 50cm.

However, what I don't see is how you went from this expression (your last post) to the first expression (first post)?

 

[LogicX]

ya, what the book is basically saying is that you should put L2 = 100 - L1, then minimize the function of L1. Then the answer would indeed be 50cm.

However, what I don't see is how you went from this expression (your last post) to the first expression (first post)?

The second equation I gave is how you actually find the unknown resistance. The first expression was the equation for the propagation of uncertainty of the second equation, which was somehow derived using calculus.