The Same Error Twice

solved Uncertainty

Q. Two masses, 1.02 g $a = 1.02$ g and 9.89 g, are each weighed to within 0.01 g $Δ a = 0.01$ g. What is the relative error in the lighter one?

How it's solved

Solved with Uncertainty

Given

$a = 1.02$ g
$Δ a = 0.01$ g

Find $δ_{a}$ .

Working

$a = 1.02$ g $= 0.00102$ kg
$Δ a = 0.01$ g $= 1 \times 10^{−5}$ kg
first error over first $δ_{a} = \frac{Δ a}{a}$
$δ_{a} = \frac{1 \times 10^{−5}}{0.00102}$
$δ_{a} = 1.0%$

Conclusion

The same error matters more in a smaller measurement: what counts is the error over the measurement.

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 :the_same_error_twice do4  kind :scq5  describes "Two masses, %{lighter} and 9.89 g, are each weighed to within %{error}. What is the relative " \6          "error in the lighter one?"7  given lighter: "1.02 g"8  given error: "0.01 g"910  option 0.0111  option 0.00112  option 0.113  option 0.0214end

solution.rb

1require_relative "../questions/the_same_error_twice"2require_relative "../science/units_and_measurement/uncertainty"34solve :the_same_error_twice, using: Uncertainty do5  take lighter: :a, error: :Delta_a6  answer find_value(:delta_a, in: "%", within: 0.001)78  conclusion "The same error matters more in a smaller measurement: what counts is the error over " \9             "the measurement."10end

playground.rb

1require_relative "../solutions/the_same_error_twice"23play_with :the_same_error_twice do4  title "The Same Error Twice"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