Errors in a Product

solved Uncertainty

Q. How sure can we be of the product of $a$ ± $Δ a$ and $b$ ± $Δ b$ ?

Inputs

first measurement

a

16.2

its error

Δ a

0.1

second measurement

b

10.1

its error

Δ b

0.1

Outputs

product

p

163.6

relative error in product

δ_{p}

1.6%

error in product

Δ p

2.63

How it's solved

Solved with Uncertainty

Given

$a$ , $Δ a$ , $b$ , $Δ b$ yours to vary

Find $p$ , $δ_{p}$ , $Δ p$ .

Working

product of measurements $p = a \cdot b$
$p = 16.2 \cdot 10.1$
$p = 163.62$
first error over first $δ_{a} = \frac{Δ a}{a}$
$δ_{a} = \frac{0.1}{16.2}$
$δ_{a} = 0.00617283951$
second error over second $δ_{b} = \frac{Δ b}{b}$
$δ_{b} = \frac{0.1}{10.1}$
$δ_{b} = 0.0099009901$
relative errors add $δ_{p} = δ_{a} + δ_{b}$
$δ_{p} = 0.00617283951 + 0.0099009901$
$δ_{p} = 1.6%$
error from relative error $Δ p = δ_{p} \cdot p$
$Δ p = 0.0160738296 \cdot 163.62$
$Δ p = 2.63$

Conclusion

No surer than what went into it: the relative errors add, δp = δa + δb. So the least precise of the measurements decides how precise the product can be.

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 :errors_in_a_product do4  describes "%{error} of the product of %{first} ± %{first_error} and %{second} ± %{second_error}?"5  given :first6  given :first_error7  given :second8  given :second_error9  asks error: "How sure can we be"10end

solution.rb

1require_relative "../questions/errors_in_a_product"2require_relative "../science/units_and_measurement/uncertainty"34solve :errors_in_a_product, using: Uncertainty do5  take first: :a, first_error: :Delta_a, second: :b, second_error: :Delta_b6  find_value :p7  find_value :delta_p8  take error: :Delta_p9  find_value :error1011  conclusion "No surer than what went into it: the relative errors add, δp = δa + δb. So the " \12             "least precise of the measurements decides how precise the product can be."13end

playground.rb

1require_relative "../solutions/errors_in_a_product"23play_with :errors_in_a_product do4  title "Errors in a Product"56  input :a,       1.0..50.0, step: 0.1,  default: 16.2, as: "first measurement"7  input :Delta_a, 0.0..2.0,  step: 0.01, default: 0.1,  as: "its error"8  input :b,       1.0..50.0, step: 0.1,  default: 10.1, as: "second measurement"9  input :Delta_b, 0.0..2.0,  step: 0.01, default: 0.1,  as: "its error"1011  output :p12  output :delta_p, in: "%"13  output :Delta_p14end

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