A Loop in the Sky

solved UniformCircularMotion

Q. An aircraft flies a horizontal loop of radius 1.00 km $r_{c} = 1.00$ km at a steady 900 km/h $v = 900$ km/h. Taking g as 9.8 m/s² $g = 9.8$ m/s², how many times g is its acceleration?

Inputs

radius

r_{c}

1 km

speed

v

900 km/h

gravity

g

9.8 m/s²

Outputs

times g

q

How it's solved

Solved with UniformCircularMotion

Given

$r_{c} = 1.00$ km
$v = 900$ km/h
$g = 9.8$ m/s²

Find $q$ .

Working

$r_{c} = 1.00$ km $= 1000$ m
$v = 900$ km/h $= 250$ m/s
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{c}}$
$a_{c} = \frac{250^{2}}{1000}$
$a_{c} = 62.5$ m/s²
measured in gs(round) $q = \frac{a_{c}}{g}$
$q = \frac{62.5}{9.8}$
$q = 6.37755102$
$q \approx 6.38$ to 3 significant figures

Conclusion

A tight turn at speed asks for far more than gravity's pull, which is why pilots feel several times their own weight in one.

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 :a_loop_in_the_sky do4  kind :numerical5  describes "An aircraft flies a horizontal loop of radius %{radius} at a steady %{speed}. Taking g as " \6          "%{gravity}, %{times_g}?"7  given radius: "1.00 km"8  given speed: "900 km/h"9  given gravity: "9.8 m/s2"10  asks times_g: "how many times g is its acceleration"11end

solution.rb

1require_relative "../questions/a_loop_in_the_sky"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :a_loop_in_the_sky, using: UniformCircularMotion do5  objects :aircraft6  paths :circle7  aircraft.goes(:round, on: circle)8  take radius: circle.radius, speed: aircraft.speed, gravity: g9  times_g = aircraft.against_gravity10  answer times_g:, figures: 311  conclusion "A tight turn at speed asks for far more than gravity's pull, which is why pilots " \12             "feel several times their own weight in one."13end

playground.rb

1require_relative "../solutions/a_loop_in_the_sky"23play_with :a_loop_in_the_sky do4  title "A Loop in the Sky"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	UniformCircularMotion
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
radius	$r$	m	0 … ∞	UniformCircularMotion
speed	$v$	m/s	0 … ∞	UniformCircularMotion
angular speed	$ω$	rad/s	—	UniformCircularMotion
revolutions	$n_{rev}$	—	—	UniformCircularMotion
frequency	$ν$	Hz	0 … ∞	UniformCircularMotion
centripetal acceleration	$a_{c}$	m/s²	—	UniformCircularMotion
against gravity	$q$	—	—	UniformCircularMotion
period	$T$	s	0 … ∞	UniformCircularMotion
gravity	$g$	m/s²	—	UniformCircularMotion
mass	$m$	kg	0 … ∞	UniformCircularMotion
centripetal force	$F_{c}$	N	—	UniformCircularMotion
displacement of aircraft during round	$Δ x$	m	—
position of aircraft at end of round	$x_{1}$	m	—
position of aircraft at start of round	$x_{0}$	m	—
average velocity of aircraft during round	$v_{av}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of aircraft during round	$c_{av}$	m/s	—
path length of aircraft during round	$d$	m	—
speed of aircraft during round	$v$	m/s	0 … ∞
angular speed of aircraft during round	$ω$	rad/s	—
radius of circle	$r_{c}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of aircraft during round	$ν$	Hz	0 … ∞
revolutions of aircraft during round	$n_{rev}$	—	—
centripetal acceleration of aircraft during round	$a_{c}$	m/s²	—
against gravity of aircraft during round	$q$	—	—
centripetal force on aircraft during round	$F_{c}$	N	—
mass of aircraft	$m$	kg	0 … ∞

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

Δ x = x_{1} - x_{0}

change of position(round)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(round)

c_{av} = \frac{d}{Δ t}

path a time(round)

v = ω \cdot r_{c}

speed round the circle(round)

ω = 2 \cdot \frac{π}{T}

once round a period(round)

ν = \frac{1}{T}

rounds a second(round)

ν = \frac{n_{rev}}{Δ t}

rounds over the time(round)

a_{c} = \frac{v^{2}}{r_{c}}

towards the centre(round)

q = \frac{a_{c}}{g}

measured in gs(round)

F_{c} = m \cdot a_{c}

mass times the turning(round)