A Stone on a String

solved UniformCircularMotion

Q. A stone on a string 80 cm $r_{c} = 80$ cm long is whirled round a horizontal circle at a steady speed, 14 $n_{rev} = 14$ times in 25 s $Δ t = 25$ s. What is its acceleration, in m/s²?

Inputs

radius

r_{c}

80 cm

turns

n_{rev}

time

Δ t

25.00 s

Outputs

acceleration

a_{c}

How it's solved

Solved with UniformCircularMotion

Given

$r_{c} = 80$ cm
$n_{rev} = 14$
$Δ t = 25$ s

Find $a_{c}$ .

Working

$r_{c} = 80$ cm $= 0.8$ m
rounds over the time(round) $ν = \frac{n_{rev}}{Δ t}$
$ν = \frac{14}{25}$
$ν = 0.56$ Hz
rounds a second(round) $ν = \frac{1}{T}$
$0.56 = \frac{1}{T}$
$T = 1.78571429$ s
once round a period(round) $ω = 2 \cdot \frac{π}{T}$
$ω = 2 \cdot \frac{3.14159265}{1.78571429}$
$ω = 3.51858377$ rad/s
speed round the circle(round) $v = ω \cdot r_{c}$
$v = 3.51858377 \cdot 0.8$
$v = 2.81486702$ m/s
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{c}}$
$a_{c} = \frac{{2.81486702}^{2}}{0.8}$
$a_{c} = 9.904$ m/s²
$a_{c} \approx 9.90$ m/s²to 3 significant figures

Conclusion

It points along the string, toward the hand, all the way round: the string is what keeps turning the stone's velocity.

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_stone_on_a_string do4  kind :numerical5  describes "A stone on a string %{radius} long is whirled round a horizontal circle at a steady speed, " \6          "%{turns} times in %{time}. %{acceleration}, in m/s²?"7  given radius: "80 cm"8  given turns: 14.09  given time: "25 s"10  asks acceleration: "What is its acceleration"11end

solution.rb

1require_relative "../questions/a_stone_on_a_string"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :a_stone_on_a_string, using: UniformCircularMotion do5  objects :stone6  paths :circle7  stone.goes(:round, on: circle)8  take radius: circle.radius, turns: stone.revolutions, time: round.time_taken9  acceleration = stone.centripetal_acceleration10  answer acceleration:, in: "m/s2", figures: 311  conclusion "It points along the string, toward the hand, all the way round: the string is what " \12             "keeps turning the stone's velocity."13end

playground.rb

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