The Fastest Whirl

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 in a horizontal plane. What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N $F_{c} = 200$ N, in m/s?

Inputs

mass

m

0.25 kg

radius

r_{c}

1.50 m

tension

F_{c}

200 N

Outputs

speed

v

How it's solved

Solved with UniformCircularMotion

Given

$m = 0.25$ kg
$r_{c} = 1.5$ m
$F_{c} = 200$ N

Find $v$ .

Working

mass times the turning(round) $F_{c} = m \cdot a_{c}$
$200 = 0.25 \cdot a_{c}$
$a_{c} = 800$ m/s²
towards the centre(round) $a_{c} = \frac{v^{2}}{r_{c}}$
$800 = \frac{v^{2}}{1.5}$
$v = 34.6 m/s$
$v \approx 35$ m/sto 2 significant figures

Conclusion

The faster the stone goes, the harder the string must pull to turn it, so the strongest pull the string can give sets the fastest it can go.

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 :the_fastest_whirl 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            "in a horizontal plane. %{speed} if the string can withstand a maximum tension of %{tension}, in m/s?"7  given mass: "0.25 kg"8  given radius: "1.5 m"9  given tension: "200 N"10  asks speed: "What is the maximum speed with which the stone can be whirled around"11end

solution.rb

1require_relative "../questions/the_fastest_whirl"2require_relative "../science/motion_in_a_plane/uniform_circular_motion"34solve :the_fastest_whirl, using: UniformCircularMotion do5  objects :stone6  paths :circle7  stone.goes(:round, on: circle)8  take mass: stone.mass, radius: circle.radius, tension: stone.centripetal_force9  speed = stone.speed10  answer speed:, in: "m/s", figures: 211  conclusion "The faster the stone goes, the harder the string must pull to turn it, so the strongest pull " \12             "the string can give sets the fastest it can go."13end

playground.rb

1require_relative "../solutions/the_fastest_whirl"23play_with :the_fastest_whirl do4  title "The Fastest Whirl"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)