# Force

The word “force” is both a verb (to force) and a noun (a force). The word implies an interaction between objects. As a noun, it is a category of actions: push, pull, hit, press, attract. The question arises, how to quantify it? If you hit something hard, how hard do you hit it? Obviously you can have a near instantaneous force, like an impact; or a continuous force, like a pressure; or a permanent pervasive force, like gravity. The total amount of force applied seems to require a measure of the time over which it is applied. This makes the idea of quantifying the force itself rather difficult. Is the total amount of force the amount you accumulate over a period of time, or is it the degree of “impact” or something like that, multiplied by an amount of time?

Under the International System of Units (SI), the unit of force is a “newton”, being the quantity of force needed to accelerate one kilogram of mass at the rate of one metre per second, per second, in the direction of the applied force. One metre per second in a given direction is a velocity. A rate of change of velocity of one metre per second per second is an acceleration. So force is a measure of acceleration, or rate of change of velocity. It is a compound or derived unit, expressing the relationship between three fundamental units: mass (kilograms); distance (metres); and time (seconds).

Mechanics clarifies the potential confusion over words as follows:

1. “Momentum” is the mass times the velocity e.g. one unit is one kilogram times one metre per second
2. “Inertia” is the property of not changing momentum unless a force is applied
3. “Impulse” is the mass times the rate of change of velocity (i.e the acceleration) for a defined period of time e.g. one unit is one kilogram times one metre per second per second for one second. Therefore the Impulse applied equals the change of Momentum occurring as a result.
4. “Force” is the mass times its rate of change of velocity, without a measure of time e.g. one kilogram per metre per second per second

Energy (kinetic and potential) and Work are related measures also defined by mass, time and distance.

Isaac Newton, in “Principia”*, defined the following:

• Quantity of matter (or mass)
• Quantity of motion (or momentum)
• Inertia
• Impressed force, being “an action exerted upon a body for changing its state [of motion]”
• Centripetal force (a type of impressed force attracting towards a centre) and three measures of centripetal force: absolute; accelerative; and motive.

Law Two states “the change of motion is proportional to the motive force impressed, and takes place following the straight line in which that force is impressed.”

In textbooks this is often given as: “In the modern formulation F=ma”, but it sounds more like an expression of Impulse. The explanation** seems to be:

• Newton does not define the term “force”. He uses it to refer to an action of any kind causing a change of motion.
• He uses different “quantities” to refer to different ways that force could be measured: “these quantities of force may respectively be called motive forces, accelerative forces, and absolute forces..”
• Law II is a generalised statement of proportionality
• The “F” in F=ma refers to one standardised unit of measure, but the words inertia, momentum, impulse and acceleration could all be though of as descriptions of “force”.

*Newton’s Principia: the central argument. Dana Densmore, with translation and diagrams by William H. Donahue. Third edition. Green Lion Press. ISBN 978-1-888009-23-1

** Newton’s Philosophiae Naturalis Principia Mathematica, Stanford Encyclopedia of Philosophy 2007