A Calorie Elsewhere

solved Dimensions

Q. A calorie is 4.2 J $W = 4.2$ J. In a system whose units of mass, length and time are α kg, β m and γ s, how many of the new units of energy is a calorie?

How it's solved

Solved with Dimensions

Given

$W = 4.2$ J

Find $W$ .

Working

$W$ is [M L² T⁻²]as Dimensions says
its new unit is (α kg)(β m)²(γ s)⁻² = α β² γ⁻² J
so W = 4.2 J = 4.2 α⁻¹ β⁻² γ² new units

Conclusion

A unit changes as its dimensions say. Energy carries mass once, length twice and time to the minus two, so the new unit of energy is that mixture of the new base units, and a calorie is its old number divided by it.

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_calorie_elsewhere do4  kind :scq5  describes "A calorie is %{calorie}. In a system whose units of mass, length and time are α kg, β m and γ " \6          "s, how many of the new units of energy is a calorie?"7  given calorie: "4.2 J"89  option "4.2 α⁻¹ β⁻² γ²", as: :inverse10  option "4.2 α β² γ⁻²", as: :direct11  option "4.2 α⁻¹ β⁻¹ γ", as: :half_inverse12  option "4.2 α β γ", as: :plain13end

solution.rb

1require_relative "../questions/a_calorie_elsewhere"2require_relative "../science/units_and_measurement/dimensions"34solve :a_calorie_elsewhere, using: Dimensions do5  take calorie: :W6  take inverse: "4.2 α-1 β-2 γ2", direct: "4.2 α β2 γ-2", half_inverse: "4.2 α-1 β-1 γ", plain: "4.2 α β γ"7  answer find_value(:W, in: { M: "α kg", L: "β m", T: "γ s" })89  conclusion "A unit changes as its dimensions say. Energy carries mass once, length twice and " \10             "time to the minus two, so the new unit of energy is that mixture of the new base " \11             "units, and a calorie is its old number divided by it."12end

playground.rb

1require_relative "../solutions/a_calorie_elsewhere"23play_with :a_calorie_elsewhere do4  title "A Calorie Elsewhere"5end

The laws it drew on

quantity	written	units	within	from
speed of light	$c$	m/s	—	Units
avogadro constant	$N_{A}$	mol⁻¹	—	Units
length	$l$	m	0 … ∞	Units
mass	$m$	kg	—	Units
time	$t$	s	0 … ∞	Units
electric current	$I$	A	—	Units
temperature	$T$	K	—	Units
amount of substance	$n$	mol	—	Units
luminous intensity	$I_{v}$	cd	—	Units
arc length	$s$	m	—	Units
radius	$r$	m	0 … ∞	Units
plane angle	$θ$	—	—	Units
area	$A$	m²	—	Units
solid angle	$Ω$	—	—	Units
light time	$t_{light}$	s	—	Units
light distance	$d_{light}$	m	—	Units
number of entities	$N$	—	—	Units
baseline	$B$	m	—	Units
distance away	$D$	m	—	Units
parallax angle	$θ_{p}$	—	—	Units
pi	$π$	—	0 … ∞	Dimensions
gravitational constant	$G$	N m² kg⁻²	—	Dimensions
volume	$V$	m³	—	Dimensions
speed	$v$	m/s	—	Dimensions
acceleration	$a$	m/s²	—	Dimensions
force	$F$	N	—	Dimensions
density	$ρ$	kg/m³	—	Dimensions
work	$W$	J	—	Dimensions
surface area	$S$	m²	—	Dimensions
height	$h$	m	—	Dimensions
breadth	$b$	m	—	Dimensions
cylinder surface area	$S_{c}$	m²	—	Dimensions
volume of a slab	$V_{s}$	m³	—	Dimensions
relative density	$ρ_{rel}$	—	—	Dimensions
density of water	$ρ_{w}$	kg/m³	—	Dimensions
linear magnification	$m_{l}$	—	—	Dimensions
image length	$l_{i}$	m	—	Dimensions
object length	$l_{o}$	m	—	Dimensions
image area	$A_{i}$	m²	—	Dimensions
object area	$A_{o}$	m²	—	Dimensions
volume of them all	$V_{N}$	m³	—	Dimensions
molar volume	$V_{m}$	m³	—	Dimensions
room to spare	$R_{V}$	—	—	Dimensions

θ = \frac{s}{r}

arc over radius

Ω = \frac{A}{r^{2}}

area over radius squared

d_{light} = c \cdot t_{light}

distance at light speed

N = n \cdot N_{A}

entities in moles

θ_{p} = \frac{B}{D}

baseline over distance

V = l^{3}

volume of a cube

v = \frac{l}{t}

average speed

a = \frac{v}{t}

average acceleration

F = m \cdot a

newtons second law

ρ = \frac{m}{V}

mass per volume

W = F \cdot l

force through a distance

S = 6 \cdot l^{2}

surface of a cube

V = \frac{4 \cdot π \cdot r^{3}}{3}

volume of a sphere

S_{c} = 2 \cdot π \cdot r \cdot (r + h)

surface of a cylinder

A = l \cdot b

area of a rectangle

V_{s} = l \cdot b \cdot h

volume from three sides

ρ_{rel} = \frac{ρ}{ρ_{w}}

density against water

m_{l} = \frac{l_{i}}{l_{o}}

lengths magnified

{m_{l}}^{2} = \frac{A_{i}}{A_{o}}

areas magnified

V_{N} = N \cdot V

atoms side by side

R_{V} = \frac{V_{m}}{V_{N}}

molar over atomic volume