The units of force are relatively easy to understand, especially if you know Newton’s Second Law.

## The newton

The unit of force in the metric system is the newton. Notice that the newton does not have a capital letter. When we use SI units, we do not capitalize the first letter of unit terms, even if the name is derived from a proper name (as many are). Therefore, we write things like newtons, joules, hertz, etc. However, the common practice is to capitalize the letter representing the unit when it comes from a proper name. Therefore, we can write ten newtons as 10 N, where N is the abbreviation for newtons. Likewise 10 watts is written as 10 W, 10 joules is written as 10 J, etc.

## Units of Force

Newton’s Second Law states that \(\color{black}{\Sigma \vec{F} = m \vec{a}}\).

If we look at the right hand side of this equation, we see that we are multiplying a mass by an acceleration. We already know the units of each, so we are multiplying kg by m/s². The other side of the equation is a force, which we just learned is given in units of newtons (N).

A newton is defined as the force required to accelerate a mass of 1 kg at 1 m/s². So we see that…

\(\color{black}{\Sigma \vec{F} = m \vec{a}}\),

which gives us…

\(\color{black}{1 \text{N} = 1 \text{kg} \cdot 1 \frac{\text{m}}{\text{s²}}}\)