Radians in an Arc

solved Units

Q. An arc 0.5 m $s = 0.5$ m long is cut from a circle of radius 2 m $r = 2$ m. What angle does it make at the centre, in radians?

Inputs

arc

s

0.50 m

radius

r

2.00 m

Outputs

angle

θ

How it's solved

Solved with Units

Given

$s = 0.5$ m
$r = 2$ m

Find $θ$ .

Working

arc over radius $θ = \frac{s}{r}$
$θ = \frac{0.5}{2}$
$θ = 0.25$

Conclusion

Metres over metres leave a pure number: the radian has no dimensions.

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 :radians_in_an_arc do4  kind :numerical5  describes "An arc %{arc} long is cut from a circle of radius %{radius}. %{angle}, in radians?"6  given arc: "0.5 m"7  given radius: "2 m"8  asks angle: "What angle does it make at the centre"9end

solution.rb

1require_relative "../questions/radians_in_an_arc"2require_relative "../science/units_and_measurement/units"34solve :radians_in_an_arc, using: Units do5  take arc: :s, radius: :r6  take angle: :theta7  answer find_value(:angle)89  conclusion "Metres over metres leave a pure number: the radian has no dimensions."10end

playground.rb

1require_relative "../solutions/radians_in_an_arc"23play_with :radians_in_an_arc do4  title "Radians in an Arc"5end

The laws it drew on

quantity	written	units	within	from
speed of light	$c$	m/s	—	Units
avogadro constant	$N_{A}$	mol⁻¹	—	Units
length	$l$	m	0 … ∞	Units
mass	$m$	kg	—	Units
time	$t$	s	0 … ∞	Units
electric current	$I$	A	—	Units
temperature	$T$	K	—	Units
amount of substance	$n$	mol	—	Units
luminous intensity	$I_{v}$	cd	—	Units
arc length	$s$	m	—	Units
radius	$r$	m	0 … ∞	Units
plane angle	$θ$	—	—	Units
area	$A$	m²	—	Units
solid angle	$Ω$	—	—	Units
light time	$t_{light}$	s	—	Units
light distance	$d_{light}$	m	—	Units
number of entities	$N$	—	—	Units
baseline	$B$	m	—	Units
distance away	$D$	m	—	Units
parallax angle	$θ_{p}$	—	—	Units

θ = \frac{s}{r}

arc over radius

Ω = \frac{A}{r^{2}}

area over radius squared

d_{light} = c \cdot t_{light}

distance at light speed

N = n \cdot N_{A}

entities in moles

θ_{p} = \frac{B}{D}

baseline over distance