Newton’s second law and acceleration

The relationship between force and acceleration

Newton’s second law (also known as the fundamental principle of dynamics) relates the acceleration of an object to the sum of the forces acting on it:

sum_of_forces = m * a

where:

• sum_of_forces is the vector sum of all the forces acting on the object (in newtons, N)
• m is the mass of the object (in kg)
• a is the acceleration vector (in m/s²)

We can also write: a = sum_of_forces / m

What this means in practice

The greater the mass of an object, the greater the force required to give it the same acceleration. That is why it is easier to push a bicycle than to push a car to achieve the same acceleration!

Practical example

A trolley with a mass of 2 kg is pushed by a force of 6 N (friction is neglected). Its acceleration is:

a = sum_of_forces / m = 6 / 2 = 3 m/s^2

If we double the mass of the trolley (4 kg) whilst keeping the force the same, the acceleration becomes a = 6 / 4 = 1.5 m/s^2: it decreases.

An important special case

If the sum of the forces acting on an object is zero (the forces cancel each other out), then its acceleration is zero: the object remains at rest or moves in a straight line at a constant speed. This is known as the principle of inertia (Newton’s first law).

A common pitfall to avoid

Do not forget that it is the sum (resultant) of the forces that matters, not any single force in isolation. An object may be subject to several significant forces whilst still having zero acceleration if these forces cancel each other out (example: a lamp suspended from the ceiling, at rest, subject to its own weight and the tension in the wire).