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…

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