Two Nuggets Apart

solved Uncertainty

Q. The two gold nuggets weigh 20.17 g $a = 20.17$ g and 20.15 g $b = 20.15$ g. How many grams heavier is the one than the other?

Inputs

heavier

a

20.17 g

lighter

b

20.15 g

Outputs

difference

d

How it's solved

Solved with Uncertainty

Given

$a = 20.17$ g
$b = 20.15$ g

Find $d$ .

Working

$a = 20.17$ g $= 0.02017$ kg
$b = 20.15$ g $= 0.02015$ kg
difference of measurements $d = a - b$
$d = 0.02017 - 0.02015$
$d = 0.02$ g
$d = 2 \times 10^{−5}$ kg $= 0.02$ g
$d \approx 0.02$ gto 1 significant figure

Conclusion

A difference keeps the decimal places of what went into it, not its figures: close numbers taken apart lose most of their significant figures.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :two_nuggets_apart do4  kind :numerical5  describes "The two gold nuggets weigh %{heavier} and %{lighter}. %{difference}?"6  given heavier: "20.17 g"7  given lighter: "20.15 g"8  asks difference: "How many grams heavier is the one than the other"9end

solution.rb

1require_relative "../questions/two_nuggets_apart"2require_relative "../science/units_and_measurement/uncertainty"34solve :two_nuggets_apart, using: Uncertainty do5  take heavier: :a, lighter: :b6  take difference: :d7  answer find_value(:difference, in: "g", figures: 1)89  conclusion "A difference keeps the decimal places of what went into it, not its figures: close " \10             "numbers taken apart lose most of their significant figures."11end

playground.rb

1require_relative "../solutions/two_nuggets_apart"23play_with :two_nuggets_apart do4  title "Two Nuggets Apart"5end

The laws it drew on

quantity	written	units	within	from
first measured	$a$	—	—	Uncertainty
error in first	$Δ a$	—	—	Uncertainty
relative error in first	$δ_{a}$	—	—	Uncertainty
second measured	$b$	—	—	Uncertainty
error in second	$Δ b$	—	—	Uncertainty
relative error in second	$δ_{b}$	—	—	Uncertainty
product	$p$	—	—	Uncertainty
error in product	$Δ p$	—	—	Uncertainty
relative error in product	$δ_{p}$	—	—	Uncertainty
total	$t$	—	—	Uncertainty
difference	$d$	—	—	Uncertainty
least count	$l_{c}$	m	—	Uncertainty
scale division	$d_{m}$	m	—	Uncertainty
divisions read past	$N_{d}$	—	—	Uncertainty

δ_{a} = \frac{Δ a}{a}

first error over first

δ_{b} = \frac{Δ b}{b}

second error over second

p = a \cdot b

product of measurements

δ_{p} = δ_{a} + δ_{b}

relative errors add

Δ p = δ_{p} \cdot p

error from relative error

t = a + b

sum of measurements

d = a - b

difference of measurements

l_{c} = \frac{d_{m}}{N_{d}}

division shared out