Round the Groove

Q. An insect trapped in a circular groove of radius 12 cm goes round it steadily, 7 times in 100 s. What is its angular speed, in rad/s? expects 0.44 rad/s

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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 angular speed, in rad/s?

The playground

Inputs

radius

r_{g}

12 cm

turns

n_{rev}

time

Δ t

100.00 s

Outputs

angular speed

ω

The solution

Solved with UniformCircularMotion

Given

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

Find $ω$ .

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) $ω = 2 \cdot \frac{π}{T}$
$ω = 2 \cdot \frac{3.14159265}{14.2857143}$
$ω = 0.4398$ rad/s
$ω \approx 0.44$ rad/sto 2 significant figures

Conclusion

Each time round sweeps a full turn, so the angular speed is the turns per second times a full turn in radians; the size of the groove does not enter.

The Ruby

1require_relative "../physics"23question :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}. %{angular_speed}, in rad/s?"7  given radius: "12 cm"8  given turns: 7.09  given time: "100 s"10  asks angular_speed: "What is its angular speed"11end

1require_relative "../questions/round_the_groove"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :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  angular_speed = insect.angular_speed10  answer angular_speed:, in: "rad/s", figures: 211  conclusion "Each time round sweeps a full turn, so the angular speed is the turns per second " \12             "times a full turn in radians; the size of the groove does not enter."13end

1require_relative "../solutions/round_the_groove"23play_with :round_the_groove do4  title "Round the Groove"5end

The laws

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)