Statement and formula F = m × a

The statement of Newton’s second law

Newton’s second law, also known as the fundamental principle of dynamics (FPD), is stated as follows: in a Galilean reference frame, the vector sum of the external forces acting on an object is equal to the product of its mass and its acceleration.

This can be expressed by the formula:

sum of F = m × a

where:

• sum of F is the resultant force, in newtons (N)
• m is the mass of the object, in kilograms (kg)
• a is the acceleration, in metres per second squared (m/s²)

A practical example

A cart of mass m = 5 kg is pushed by a force of 20 N on a frictionless surface. We want to find its acceleration:

a = F / m = 20 / 5 = 4 m/s²

The trolley accelerates at 4 m/s²: its speed increases by 4 m/s with every passing second.

The link with mass

For the same force, the greater the mass, the smaller the acceleration, since a = F/m. This is why it is harder to get a loaded lorry moving quickly than a light car, even with a powerful engine.

A pitfall to avoid

Be careful not to confuse weight with mass: weight P = m × g (where g is approximately 9.8 m/s² on Earth) is a force, expressed in newtons, whereas mass is a quantity of matter, in kilograms, which does not change from one planet to another, unlike weight.

Another common pitfall: the law uses the resultant of the forces, never a single isolated force. If several forces are acting, they must first be added vectorially before applying F = m × a.