A Metre-Long Pendulum

Q. A pendulum 1 m long swings on the Earth, where g is 9.81 m/s². How long does one swing there and back take, in seconds? expects 2.01 s

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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?

The playground

Inputs

length

l

1.00 m

gravity

g

9.81 m/s²

Outputs

period

T

The solution

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 Ruby

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

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

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

The laws

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