A Whirled Stone's String

solved UniformCircularMotion

Q. A stone of mass 0.25 kg $m = 0.25$ kg tied to the end of a string is whirled round in a circle of radius 1.5 m $r_{c} = 1.5$ m with a speed of 40 rpm $ω = 40$ rpm in a horizontal plane. What is the tension in the string, in newtons?

Inputs

mass

m

0.25 kg

radius

r_{c}

1.50 m

rate

ω

40 rpm

Outputs

tension

F_{c}

How it's solved

Solved with UniformCircularMotion

Given

$m = 0.25$ kg
$r_{c} = 1.5$ m
$ω = 40$ rpm

Find $F_{c}$ .

Working

$ω = 40$ rpm $= 4.1887902$ rad/s
speed round the circle(round) $v = ω \cdot r_{c}$
$v = 4.1887902 \cdot 1.5$
$v = 6.28318531$ m/s
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{c}}$
$a_{c} = \frac{{6.28318531}^{2}}{1.5}$
$a_{c} = 26.3189451$ m/s²
mass times the turning(round) $F_{c} = m \cdot a_{c}$
$F_{c} = 0.25 \cdot 26.3189451$
$F_{c} = 6.58$ N
$F_{c} \approx 6.6$ Nto 2 significant figures

Conclusion

In a horizontal plane the string alone turns the stone, so its tension is just the force it takes to keep the stone's velocity swinging round toward the centre.

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_whirled_stone_s_string do4  kind :numerical5  describes "A stone of mass %{mass} tied to the end of a string is whirled round in a circle of radius %{radius} " \6            "with a speed of %{rate} in a horizontal plane. %{tension}, in newtons?"7  given mass: "0.25 kg"8  given radius: "1.5 m"9  given rate: "40 rpm"10  asks tension: "What is the tension in the string"11end

solution.rb

1require_relative "../questions/a_whirled_stone_s_string"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :a_whirled_stone_s_string, using: UniformCircularMotion do5  objects :stone6  paths :circle7  stone.goes(:round, on: circle)8  take mass: stone.mass, radius: circle.radius, rate: stone.angular_speed9  tension = stone.centripetal_force10  answer tension:, in: "N", figures: 211  conclusion "In a horizontal plane the string alone turns the stone, so its tension is just the force it " \12             "takes to keep the stone's velocity swinging round toward the centre."13end

playground.rb

1require_relative "../solutions/a_whirled_stone_s_string"23play_with :a_whirled_stone_s_string do4  title "A Whirled Stone's 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)