A Whirled Stone's String

Q. A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rpm in a horizontal plane. What is the tension in the string, in newtons? expects 6.6 N

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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?

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Inputs

mass

m

0.25 kg

radius

r_{c}

1.50 m

rate

ω

40 rpm

Outputs

tension

F_{c}

The solution

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 Ruby

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

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

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

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)