A Metre-Long Pendulum

solved DimensionalAnalysis

Q. A pendulum 1 m $l = 1$ m long swings on the Earth, where g is 9.81 m/s² $g = 9.81$ m/s². How long does one swing there and back take, in seconds?

Inputs

length

l

1.00 m

gravity

g

9.81 m/s²

Outputs

period

T

How it's solved

Solved with DimensionalAnalysis

Given

$l = 1$ m
$g = 9.81$ m/s²
$k = 2 \cdot π$ from “assumed”

Assumed

$m = 1$ kg

Find $T$ .

Working

assumed $constant = 2 \cdot π$
$constant = 2 \cdot 3.14159265$
$k = 6.28318531$
matching time $-2 \cdot y = 1$
$y = −0.5$
matching length $x + y = 0$
$x + (−0.5) = 0$
$x = 0.5$
matching mass $z = 0$
$z = 0$
period as powers $T = k \cdot l^{x} \cdot g^{y} \cdot m^{z}$
$T = 6.28318531 \cdot 1^{0.5} \cdot {9.81}^{(−0.5)} \cdot 1^{0}$
$T = 2.01 s$

Conclusion

The period grows as the square root of the length and shrinks as the square root of the pull, and the mass never enters.

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_metre_long_pendulum do4  kind :numerical5  describes "A pendulum %{length} long swings on the Earth, where g is %{gravity}. %{period}, in seconds?"6  given length: "1 m"7  given gravity: "9.81 m/s2"8  asks period: "How long does one swing there and back take"9end

solution.rb

1require_relative "../questions/a_metre_long_pendulum"2require_relative "../science/units_and_measurement/dimensional_analysis"34solve :a_metre_long_pendulum, using: DimensionalAnalysis do5  take length: :l, gravity: :g6  assume m: "1 kg"7  assume k: 2 * pi8  take period: :T9  answer find_value(:period, in: "s")1011  conclusion "The period grows as the square root of the length and shrinks as the square root of " \12             "the pull, and the mass never enters."13end

playground.rb

1require_relative "../solutions/a_metre_long_pendulum"23play_with :a_metre_long_pendulum do4  title "A Metre-Long Pendulum"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	DimensionalAnalysis
speed of light	$c$	m/s	—	DimensionalAnalysis
length	$l$	m	—	DimensionalAnalysis
gravity	$g$	m/s²	—	DimensionalAnalysis
mass	$m$	kg	—	DimensionalAnalysis
period	$T$	s	—	DimensionalAnalysis
constant	$k$	—	—	DimensionalAnalysis
power of length	$x$	—	—	DimensionalAnalysis
power of gravity	$y$	—	—	DimensionalAnalysis
power of mass	$z$	—	—	DimensionalAnalysis
speed	$v$	m/s	0 … ∞	DimensionalAnalysis
height	$h$	m	—	DimensionalAnalysis
time	$t$	s	0 … ∞	DimensionalAnalysis
acceleration	$a$	m/s²	—	DimensionalAnalysis
start	$s_{0}$	m	—	DimensionalAnalysis
starting speed	$v_{0}$	m/s	—	DimensionalAnalysis
distance	$s$	m	—	DimensionalAnalysis
rest mass	$m_{0}$	kg	—	DimensionalAnalysis
kinetic energy	$K$	J	—	DimensionalAnalysis

T = k \cdot l^{x} \cdot g^{y} \cdot m^{z}

period as powers

x + y = 0

matching length

-2 \cdot y = 1

matching time

z = 0

matching mass

\frac{m \cdot v^{2}}{2} = m \cdot g \cdot h

energy conserved

s = s_{0} + v_{0} \cdot t + \frac{a \cdot t^{2}}{2}

distance under constant acceleration

constant = 2 \cdot π

assumed