A Stone on a String

Q. A stone on a string 80 cm long is whirled round a horizontal circle at a steady speed, 14 times in 25 s. What is its acceleration, in m/s²? expects 9.90 m/s²

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The equations it used

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Check an answer

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?

The playground

Inputs

radius

r_{c}

80 cm

turns

n_{rev}

time

Δ t

25.00 s

Outputs

acceleration

a_{c}

The solution

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 Ruby

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

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

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

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)