Is a Joule a ratio expression

Question

"Joule, unit of work or energy in the International System of Units (SI); it is equal to the work done by a force of one newton acting through one metre." Google

So, is that to say: 1-newton/1-meter, as in on newton per meter?

Answer 1

This is more physics than mathematics, but: It is newton times meter.

If you pull on a rope with a force of $1$ newton, and you keep pulling until you have drawn in $2$ meters of rope, you will have spent $2$ joules of work.

You can spend the same $2$ joules of work by pulling at a rope with $2$ newtons, but only draw it in $1$ meter.

(You can convert between these two situations in practice with, say, a block-and-tackle arrangement).

Answer 2

In Mechanics all quantities can be expressed in terms of the $3$ fundemental concepts : Space, Time & Matter. These are measured in the SI units meters, seconds and Kilograms respectively.

One Joule = $1$ Newton times $1$ meter (J=Nm). (Energy = Force times distance)

one Newton = $1$ Kilogram times $1$ meter per second squared. (N =Kg m $s^{-2}$) ($F=ma$)

So Joules have the units Kg $m^2 s^{-2}$ and could be considered as a ratio in lots of diffferent ways.

Answer 3

work done by a force of one newton acting through [a displacement] of one metre

is mathematically $$A=\int_{l} (\mathbf{F},d\mathbf{S})$$

1. work ($A$) is the line integral of a 1-form along a path
2. done by ($+$): it's a chosen convention: the sign of the work must be considered positive when the force has a component in the same direction of the displacement, negative otherwise. One can also chose the opposite convention: "the work done on the system".
3. a force of one newton acting through a displacement: the 1-form $(\mathbf{F},d\mathbf{S})$ that computed on a displacement vector ($d\mathbf{S}$) gives the component of the force ($\mathbf{F}$) along the displacement vector times the length of the displacement. Its unit of measurement is thus [Newton$\times$meter]
4. of one metre: along a path ($l$), that in this case is a straight line segment one meter long

Furthermore, the unit of measurement of a line integral is that of the 1-form.