The Most Precise Instrument

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

What the draft establishes

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The web of concepts

given read from the words equation asked

The equations it used

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Check an answer

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

The playground

The solution

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 Ruby

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

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

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