A Pendulum by Dimensions

Q. A pendulum of length l and bob of mass m swings where gravity is g. How does its period depend on what it is made of?

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Q. A pendulum of length $l$ and bob of mass $m$ swings where gravity is $g$ . How does its period depend on what it is made of?

The playground

Inputs

length

l

1.00 m

gravity

g

9.81earth

mass in kilograms

m

Outputs

power of length

x

0.5

power of gravity

y

-0.5

power of mass

z

period

T

2.01 s

The solution

Solved with DimensionalAnalysis

Given

$k = 2 \cdot π$ from “assumed”
$l$ , $g$ , $m$ yours to vary

Find $x$ , $y$ , $z$ , $T$ .

Working

matching time $-2 \cdot y = 1$
$y = −0.5$
matching length $x + y = 0$
$x + (−0.5) = 0$
$x = 0.5$
matching mass $z = 0$
$z = 0$
assumed $constant = 2 \cdot π$
$constant = 2 \cdot 3.14159265$
$k = 6.28318531$
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

Only powers that make the dimensions match are allowed: x + y = 0 for length, −2y = 1 for time and z = 0 for mass. So T = k√(l ⁄ g). The mass drops out, and dimensions alone could never have told you that k is 2π.

The Ruby

1require_relative "../physics"23question :pendulum_by_dimensions do4  describes "A pendulum of length %{length} and bob of mass %{mass} swings where gravity is %{gravity}. %{period}?"5  given :length6  given :gravity7  given :mass8  asks period: "How does its period depend on what it is made of"9end

1require_relative "../questions/pendulum_by_dimensions"2require_relative "../science/units_and_measurement/dimensional_analysis"34solve :pendulum_by_dimensions, using: DimensionalAnalysis do5  take length: :l, gravity: :g, mass: :m6  assume k: 2 * pi7  find_value :x8  find_value :y9  find_value :z10  take period: :T11  find_value :period1213  conclusion "Only powers that make the dimensions match are allowed: x + y = 0 for length, " \14             "−2y = 1 for time and z = 0 for mass. So T = k√(l ⁄ g). The mass drops out, and " \15             "dimensions alone could never have told you that k is 2π."16end

1require_relative "../solutions/pendulum_by_dimensions"2require_relative "../science/motion_in_a_plane/gravity"34play_with :pendulum_by_dimensions do5  title "A Pendulum by Dimensions"67  input :l, 0.1..3.0,  in: "m", step: 0.05, default: 1.0, as: "length"8  input :g, 1.0..25.0, step: 0.01, default: 9.81, marks: GRAVITY, as: "gravity"9  input :m, 0.1..10.0, step: 0.1, default: 1.0, as: "mass in kilograms"1011  output :x, as: "power of length"12  output :y, as: "power of gravity"13  output :z, as: "power of mass"14  output :T, in: "s"15end

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