The Same Error Twice

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

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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?

The playground

The solution

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 Ruby

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

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

1require_relative "../solutions/the_same_error_twice"23play_with :the_same_error_twice do4  title "The Same Error Twice"5end

The laws

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