The Most Precise Instrument

solved Uncertainty

Q. Which of these can measure a length most precisely?

How it's solved

Solved with Uncertainty

Working

$d_{m} = 1$ mm, $N_{d} = 20$
division shared out $l_{c} = \frac{d_{m}}{N_{d}}$
$l_{c} = 5 \times 10^{−5}$ m
$d_{m} = 1$ mm, $N_{d} = 100$ $\Rightarrow$ $l_{c} = 1 \times 10^{−5}$ m
$l_{c} = 6 \times 10^{−7}$ m

Conclusion

The finest step an instrument can tell apart decides how precisely it measures. A vernier and a screw gauge share a millimetre out among their divisions; the wavelength of light is finer still.

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_most_precise_instrument do4  kind :scq5  describes "Which of these can measure a length most precisely?"67  option "Vernier callipers whose 20 vernier divisions match 19 main-scale millimetres", as: :vernier8  option "A screw gauge of pitch 1 mm with 100 divisions on its circular scale", as: :screw_gauge9  option "An optical instrument that can measure to the wavelength of light", as: :optical10end

solution.rb

1require_relative "../questions/the_most_precise_instrument"2require_relative "../science/units_and_measurement/uncertainty"34solve :the_most_precise_instrument, using: Uncertainty do5  take vernier: { d_m: "1 mm", N_d: 20 }, screw_gauge: { d_m: "1 mm", N_d: 100 }, optical: { l_c: "6e-7 m" }6  answer options.min_by { |o| find_value :l_c, given: o.taken }78  conclusion "The finest step an instrument can tell apart decides how precisely it measures. A " \9             "vernier and a screw gauge share a millimetre out among their divisions; the " \10             "wavelength of light is finer still."11end

playground.rb

1require_relative "../solutions/the_most_precise_instrument"23play_with :the_most_precise_instrument do4  title "The Most Precise Instrument"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