Editing Torque (section)

==== Derivation ====
For a rotating object, the ''linear distance'' covered at the [[circumference]] of rotation is the product of the radius with the angle covered.  That is:  linear distance = radius × angular distance. And by definition, linear distance = linear speed × time = radius × angular speed × time.

By the definition of torque: torque = radius × force. We can rearrange this to determine force = torque ÷ radius. These two values can be substituted into the definition of [[Power (physics)|power]]:

<math display="block">
\begin{align}
\text{power} & = \frac{\text{force} \cdot \text{linear distance}}{\text{time}} \\[6pt]
& = \frac{\left(\dfrac{\text{torque}} r \right) \cdot (r \cdot \text{angular speed} \cdot t)} t \\[6pt]
& = \text{torque} \cdot \text{angular speed}.
\end{align}
</math>

The radius ''r'' and time ''t'' have dropped out of the equation.  However, angular speed must be in radians per unit of time, by the assumed direct relationship between linear speed and angular speed at the beginning of the derivation.  If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by 2{{pi}} in the above derivation to give:

<math display="block">\text{power} = \text{torque} \cdot 2 \pi \cdot \text{rotational speed}. \,</math>

If torque is in newton-metres and rotational speed in revolutions per second, the above equation gives power in newton-metres per second or watts.  If Imperial units are used, and if torque is in pounds-force feet and rotational speed in revolutions per minute, the above equation gives power in foot pounds-force per minute.  The horsepower form of the equation is then derived by applying the conversion factor 33,000&nbsp;ft⋅lbf/min per horsepower:

<math display="block">
\begin{align}
\text{power} & = \text{torque} \cdot 2 \pi \cdot \text{rotational speed} \cdot \frac{\text{ft}{\cdot}\text{lbf}}{\text{min}} \cdot \frac{\text{horsepower}}{33,000 \cdot \frac{\text{ft}\cdot\text{lbf}}{\text{min}}} \\[6pt]
& \approx \frac {\text{torque} \cdot \text{RPM}}{5,252}
\end{align}
</math>

because <math>5252.113122 \approx \frac {33,000} {2 \pi}. \,</math>