Error analysis through differentiation

[Arnav]

I was required to do the error analysis for an experiment with the aim of finding the relative density of a body.

 

In the experiment, the weight of the body in air war found to be w₁ = 8.00 +- 0.05 N while the weight of the body in water was found to be w₂ = 4.00 +- 0.05 N.

 

Using the relation, ρ = weight in air / (weight in air - weight in water) where ρ denotes relative density, I did the error analysis as shown in the attachment. 

 

However, my friend did the error analysis as this: 

ρ = w₁/(w₁- w₂)

Δρ/ρ = Δw₁/w₁ + Δ(w₁- w₂)/(w₁- w₂) 

Δρ/ρ(max) = Δw₁/w₁ + (Δw₁+Δw₂)/(w₁- w₂)

Δρ/ρ(max) = 0.05/8 + (0.05+0.05)/(8-4)

Δρ/ρ(max) = 0.03125

 

Now, which one of us is right? who did it wrong?and how? in which of tge ways should we report the measurement? 

2.00 +- 1.88% or 2.00 +- 3.12% ?

 

[studiot]

I make it 2 + 1.923%, -1.829% (brute force and Excel)

You have come up against the difficulty of evaluating the partial derivatives of a function with a reciprocal in it.

A good strategy in this case (as in most cases of small changes) is to remember the definition of a derivative

[math]\frac{{\partial F}}{{\partial {x_1}}} = \frac{{F\left( {{x_1} + \Delta {x_1}} \right) - F\left( {{x_1}} \right)}}{{\Delta {x_1}}}[/math]

 

Alternatively you can calculate the extremes of the function itself by spreadsheet, as I have done, or otherwise