Newton's Second Law

Insert the mass in kilograms and the acceleration in metres per second squared, then multiply the two values. The result comes out in newtons, since 1 N is defined as 1 kg·m/s². For example, a body of 8 kg accelerating at 3 m/s² experiences F = 8·3 = 24 N. Make sure F is the net force, meaning the sum of all forces acting on the body. If several forces are present, add them with correct signs first before dividing by the mass or solving for the acceleration.

Divide both sides of the equation by the mass, which gives a = F/m. The acceleration is therefore the ratio of net force to mass. This leads to an important insight: for the same force a heavier body accelerates more slowly, because the mass sits in the denominator. To find the mass instead, rearrange to m = F/a. Always insert SI units, that is newtons, kilograms and metres per second squared, otherwise the result is wrong. A value given in grams or in km/h must be converted before you substitute it.

Force in Newton second law is any net influence that accelerates a body. Weight is just a special case of it: it arises from gravity and is written F_G = m·g with the local factor g ≈ 9.81 m/s². A body of 10 kg therefore has a weight of about 98 N, regardless of whether it moves. F = m·a instead describes the acceleration caused by the sum of all forces, which may include weight, friction, the normal force or a driving force. Never confuse mass in kilograms with weight in newtons, a common exam mistake.

The form F = m·a assumes a constant mass and an inertial frame of classical mechanics. It fails in two cases. First, for variable mass, such as a rocket ejecting fuel; there you need the more general momentum form F = dp/dt. Second, at speeds close to the speed of light, where relativistic mechanics applies and the inertial mass effectively increases. In accelerated reference frames you also have to add fictitious forces such as the centrifugal force. For all school and typical university problems with fixed masses and everyday speeds, however, F = m·a stays exactly valid.

A body accelerates only when the sum of all forces is not zero. If forces cancel, the body stays at rest or keeps moving uniformly, which is Newton first law. In F = m·a it is therefore always the net force that appears. For problems, first draw a free-body diagram and mark every force with its direction: drive, friction, weight, normal force. Then add the forces component by component, usually separately in the x and y directions. Only this sum goes into the equation. Anyone who inserts a single force alone almost always overestimates the acceleration.