Pulled Round the Groove

solved UniformCircularMotion

Q. An insect trapped in a circular groove of radius 12 cm $r_{g} = 12$ cm goes round it steadily, 7 $n_{rev} = 7$ times in 100 s $Δ t = 100$ s. What is its acceleration, in cm/s²?

Inputs

radius

r_{g}

12 cm

turns

n_{rev}

time

Δ t

100.00 s

Outputs

acceleration

a_{c}

How it's solved

Solved with UniformCircularMotion

Given

$r_{g} = 12$ cm
$n_{rev} = 7$
$Δ t = 100$ s

Find $a_{c}$ .

Working

$r_{g} = 12$ cm $= 0.12$ m
rounds over the time(round) $ν = \frac{n_{rev}}{Δ t}$
$ν = \frac{7}{100}$
$ν = 0.07$ Hz
rounds a second(round) $ν = \frac{1}{T}$
$0.07 = \frac{1}{T}$
$T = 14.2857143$ s
once round a period(round) $ω = 2 \cdot \frac{π}{T}$
$ω = 2 \cdot \frac{3.14159265}{14.2857143}$
$ω = 0.439822972$ rad/s
speed round the circle(round) $v = ω \cdot r_{g}$
$v = 0.439822972 \cdot 0.12$
$v = 0.0527787566$ m/s
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{g}}$
$a_{c} = \frac{{0.0527787566}^{2}}{0.12}$
$a_{c} = 2.321$ cm/s²
$a_{c} = 0.0232133096$ m/s² $= 2.32133096$ cm/s²
$a_{c} \approx 2.3$ cm/s²to 2 significant figures

Conclusion

Its speed never changes, but its direction always does, and that turning is an acceleration pointed at the centre of the groove.

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 :pulled_round_the_groove do4  kind :numerical5  describes "An insect trapped in a circular groove of radius %{radius} goes round it steadily, %{turns} " \6          "times in %{time}. %{acceleration}, in cm/s²?"7  given radius: "12 cm"8  given turns: 7.09  given time: "100 s"10  asks acceleration: "What is its acceleration"11end

solution.rb

1require_relative "../questions/pulled_round_the_groove"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :pulled_round_the_groove, using: UniformCircularMotion do5  objects :insect6  paths :groove7  insect.goes(:round, on: groove)8  take radius: groove.radius, turns: insect.revolutions, time: round.time_taken9  acceleration = insect.centripetal_acceleration10  answer acceleration:, in: "cm/s2", figures: 211  conclusion "Its speed never changes, but its direction always does, and that turning is an " \12             "acceleration pointed at the centre of the groove."13end

playground.rb

1require_relative "../solutions/pulled_round_the_groove"23play_with :pulled_round_the_groove do4  title "Pulled Round the Groove"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 insect during round	$Δ x$	m	—
position of insect at end of round	$x_{1}$	m	—
position of insect at start of round	$x_{0}$	m	—
average velocity of insect during round	$v_{av}$	m/s	—
time taken of round	$Δ t$	s	0 … 1e+05
average speed of insect during round	$c_{av}$	m/s	—
path length of insect during round	$d$	m	—
speed of insect during round	$v$	m/s	0 … ∞
angular speed of insect during round	$ω$	rad/s	—
radius of groove	$r_{g}$	m	0 … ∞
period of round	$T$	s	0 … ∞
frequency of insect during round	$ν$	Hz	0 … ∞
revolutions of insect during round	$n_{rev}$	—	—
centripetal acceleration of insect during round	$a_{c}$	m/s²	—
against gravity of insect during round	$q$	—	—
centripetal force on insect during round	$F_{c}$	N	—
mass of insect	$m$	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_{g}

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_{g}}

towards the centre(round)

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

measured in gs(round)

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

mass times the turning(round)