Pendulum equation/expression, manipulation Help

[tdog13]

Homework Statement

Converting the simple pendulum expression for g in terms of the angular frequency ω instead of the period T, where ω=2π/T, yields g=ω2L.
Derive an expression for the error (Δg) in g by first setting g=A⋅B.
A=
B=
Therefore in terms of A, ΔA, B, and ΔB:
Δg/g=

Converting back to the original variables ω and L:
ΔA/A=
ΔB/B

Thus, an expression for the error Δg in g is:
Δg=

Homework Equations

The simple pendulum expression I have is T = 2pi √(L/g)

The Attempt at a Solution

I am in my first year physics class. Already on the first week the professor requires questions like this to be completed prior to my lab. The problem is... we haven't learned ANYTHING yet in class that is relevant to this question. I am not sure where to even begin, I am having difficulty comprehending and picturing what the question is asking. It would be great if anyone can provide any insight or lead me in the correct path

 

[DrClaude]

Have a look at http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart2.html

 

[tdog13]

Have a look at http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart2.html

So in the first part.. would A= w^2 and B = L ...?

 

[haruspex]

So in the first part.. would A= w^2 and B = L ...?

Yes.

 

[tdog13]

Yes.

Thank you and would be or is that total wrong lol
Δg/g= ((dA/A)^2))+((dB/B)^2))

 

[haruspex]

Thank you and would be or is that total wrong lol
Δg/g= ((dA/A)^2))+((dB/B)^2))

No.
The link DrClaude gave you lists cases a to f. Which one matches the format g = A.B?

 

[tdog13]

No.
The link DrClaude gave you lists cases a to f. Which one matches the format g = A.B?

I thought I replied to this, I but I eventually got it...

I am currently trying to do another one that deals with the formula for moment of intertia.
I = .5mr^2.
Would ΔI = √(.5(Δm/m))^2 + (2(Δr/r))^2 ?

 

[haruspex]

I thought I replied to this, I but I eventually got it...

I am currently trying to do another one that deals with the formula for moment of intertia.
I = .5mr^2.
Would ΔI = √(.5(Δm/m))^2 + (2(Δr/r))^2 ?

No.
To make it clearer, insert parentheses to set the order of precedence of the operations:
(.5)(m)(r²).
Work from the outside in. The outermost operations are all multiplications and can be done in any order. Which formula in the list deals with products?

 

[tdog13]

No.
To make it clearer, insert parentheses to set the order of precedence of the operations:
(.5)(m)(r²).
Work from the outside in. The outermost operations are all multiplications and can be done in any order. Which formula in the list deals with products?

Link from first post indicating is rules is not working but I'm reading something else and
since .5 is a constant... it is factored out.
(m) would be (Δm/m)
(r) would be 2(Δr/r)
Since it is multiplication, putting it all together it would be... √ (Δm/m) + (2(Δr/r)) ...?

 

[haruspex]

Link from first post indicating is rules is not working but I'm reading something else and
since .5 is a constant... it is factored out.
(m) would be (Δm/m)
(r) would be 2(Δr/r)
Since it is multiplication, putting it all together it would be... √ (Δm/m) + (2(Δr/r)) ...?

Yes, that link is broken, and I cannot remember exactly what it said.
There is a possible confusion here, between the actual error and mean square error.

Suppose the actual errors are ΔI, Δm, Δr. We can write I+ΔI=.5 (m+Δm)(r+Δr)². Ignoring second order small quantities we can deduce ΔI/I=Δm/m+2Δr/r.

If we want the expected (i.e. mean square) error then we need to know something about the distribution of the source errors. These are often taken to be Gaussian, for convenience, though in practice they are often more like uniform. But even then we need to know the standard deviations. These will not usually be the same for the different quantities, m and r in this case.
Look at the f=AB line in the table of example formulas at https://en.m.wikipedia.org/wiki/Propagation_of_uncertainty. You will see the standard deviations of A and B feature in the expression.

Perhaps, in something of an abuse of notation, the Δ symbols in your post are intended to represent standard deviations, i.e. where you have written Δx you mean σ_x, where x stands variously for I, m, r. With that understanding, we have
(σ_I/I)²=(σ_m/m)²+(2σ_r/r)²
This looks similar to your expressions in posts 7 and 9, but does not quite match either. Can you see the differences?

 

[tdog13]

Yes, that link is broken, and I cannot remember exactly what it said.
There is a possible confusion here, between the actual error and mean square error.

Suppose the actual errors are ΔI, Δm, Δr. We can write I+ΔI=.5 (m+Δm)(r+Δr)². Ignoring second order small quantities we can deduce ΔI/I=Δm/m+2Δr/r.

If we want the expected (i.e. mean square) error then we need to know something about the distribution of the source errors. These are often taken to be Gaussian, for convenience, though in practice they are often more like uniform. But even then we need to know the standard deviations. These will not usually be the same for the different quantities, m and r in this case.
Look at the f=AB line in the table of example formulas at https://en.m.wikipedia.org/wiki/Propagation_of_uncertainty. You will see the standard deviations of A and B feature in the expression.

Perhaps, in something of an abuse of notation, the Δ symbols in your post are intended to represent standard deviations, i.e. where you have written Δx you mean σ_x, where x stands variously for I, m, r. With that understanding, we have
(σ_I/I)²=(σ_m/m)²+(2σ_r/r)²
This looks similar to your expressions in posts 7 and 9, but does not quite match either. Can you see the differences?

Yeah I see the difference between the 2 approaches. I wish I fully understand what you are conveying but I am only a first year health major forced to take physics and have never done so before or statistics for the matter so the language is kinda foreign ro me haha
However, umm perhaps you can provide me some insight based on this? http://www.physics.brocku.ca/Labs/MathBasics/errorrules.pdf This is what we're basically supposed to use... If it helps we did a laboratory on with a rotational disc spinning and calculated the acceleration (a) in regards to I = mr^2(g/a-1). We found I and now need the +/- for the I.

 

[haruspex]

Yeah I see the difference between the 2 approaches. I wish I fully understand what you are conveying but I am only a first year health major forced to take physics and have never done so before or statistics for the matter so the language is kinda foreign ro me haha
However, umm perhaps you can provide me some insight based on this? http://www.physics.brocku.ca/Labs/MathBasics/errorrules.pdf This is what we're basically supposed to use... If it helps we did a laboratory on with a rotational disc spinning and calculated the acceleration (a) in regards to I = mr^2(g/a-1). We found I and now need the +/- for the I.

That link clarifies that we are looking for the expected error. In the table, there are two formulae you need to use, one for product (AB) and one for exponent (Aⁿ). Unfortunately, in neither post 7 nor post 9 did you apply them correctly.
Let's take it in stages. First, let B stand for r², so we have I=.5 mB. Apply the product rule. What do you get for ΔI/I?

 

[tdog13]

That link clarifies that we are looking for the expected error. In the table, there are two formulae you need to use, one for product (AB) and one for exponent (Aⁿ). Unfortunately, in neither post 7 nor post 9 did you apply them correctly.
Let's take it in stages. First, let B stand for r², so we have I=.5 mB. Apply the product rule. What do you get for ΔI/I?

ΔI/I= √ (Δ.5/.5)^2 + (Δm/m)^2 + (ΔB/B)^2?

And B would then be (2ΔB/B)^2... so

ΔI/I= √ (Δm/m)² + (2ΔB/B)²?

 

[haruspex]

ΔI/I= √ (Δ.5/.5)^2 + (Δm/m)^2 + (ΔB/B)^2?

And B would then be (2ΔB/B)^2... so

ΔI/I= √ (Δm/m)² + (2ΔB/B)²?

Yes, except that in the second step the B's should have become r's.