A Pendulum by Dimensions

solved DimensionalAnalysis

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?

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

How it's solved

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 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 :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

solution.rb

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

playground.rb

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 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