Last update: September 4, 2024

The First Equation of Physics: How to Mathematize Force

The first equation of physics F=ma — F=ma is much more than an equation. It is the first mathematical bridge between cause (forces, interactions) and effect (the motion we observe). Force is the agent of change that disturbs the state of an object, and acceleration is the observable change. This makes it one of the most important equations in the history of physics.
Image source: astronoo.com

Fundamental Principle of Dynamics

The most fundamental and universally recognized equation is Newton's Second Law (1643-1727).

F=ma is a fundamental equation in physics, specifically in classical mechanics. It expresses a relationship between three essential physical quantities:
• $ \vec{F} $ is the force applied to an object, expressed in newtons (N),
• m is the mass of the object, expressed in kilograms (kg),
• a is the acceleration of the object, expressed in meters per second squared (m/s²).

This equation describes the relationship between the force applied to an object, the mass of that object, and the resulting acceleration. It is essential because it establishes the link between classical mechanics and the motion of bodies. Moreover, it serves as the basis for many other equations and concepts in physics.

N.B.: Newton's First Law, also known as the law of inertia, is a fundamental concept in physics that describes the behavior of an object when no net force acts on it. There is no single equation for Newton's First Law.

What is a force?

The concept of force is one of the fundamental notions in classical physics, particularly in the context of Newtonian mechanics. It is defined as any interaction that, when exerted on an object, modifies the state of motion of that object. The force can cause an acceleration, a change of direction, or a deformation of the object in question.
In the formalism of classical mechanics, a force $ \vec{F} $ is a vector quantity, that is, a quantity that has a direction, a sense, and an intensity.

What Does F = ma Tell Us?

Force is proportional to acceleration: the greater the applied force, the greater the acceleration of the object will be.
Mass resists acceleration: the greater the mass of an object, the harder it will be to move it or change its speed (i.e., to accelerate it).

In other words, F = ma tells us that to alter the motion of an object (to set it in motion, stop it, change its direction, or speed it up), a force must be applied.

Limitations of F = ma

It is important to note that F = ma is a simplification. It is valid within the framework of classical mechanics and for speeds much lower than the speed of light. For very high speeds, Einstein's theory of relativity must be used. Additionally, this equation does not account for certain forces such as frictional forces.