Force Calculator — Newton's Second Law

Newton's second law in the form F = m x a is a statement about net force. The F on the left is the vector sum of every force acting on the body, including gravity, normal contact, friction, tension and drag, not any single push. If a 50 kg crate is dragged with 300 N while friction resists with 100 N, the net force is 200 N and the acceleration is 4 m/s squared. Because force and acceleration are vectors, two-dimensional problems are solved by resolving each force into perpendicular components and applying the law separately along each axis. The unit follows directly from the definition. One newton is the force that gives a mass of one kilogram an acceleration of one metre per second squared, so 1 N = 1 kg x m/s squared. Since the 2019 revision of the SI, the kilogram is fixed through the Planck constant, the metre through the speed of light and the second through a caesium hyperfine transition, which makes the newton a fully derived unit with no physical artefact behind it. Two limits are worth remembering. This algebraic form assumes constant mass, so rockets and other systems that expel mass need the momentum form F = dp/dt. It also assumes an inertial reference frame, so in a rotating or accelerating frame you must add fictitious terms such as centrifugal and Coriolis forces before the arithmetic balances.

The idea that force is proportional to change of motion rather than to motion itself was hard won. Aristotle held that a steady force was needed to keep a body moving at all, a view that survived through the fourteenth-century impetus theory of Jean Buridan until Galileo Galilei argued in 1638 that a body on a frictionless horizontal plane would continue indefinitely. Isaac Newton assembled these threads in the Philosophiae Naturalis Principia Mathematica, published under the imprimatur of the Royal Society in 1687, where the second law is stated in terms of momentum: the change of motion is proportional to the impressed force and takes place along the line in which that force acts. The familiar algebraic form F = ma is not Newton's own notation, which was geometric and expressed in proportions. It emerged from the continental reworking of mechanics led by Leonhard Euler, whose 1750 memoir on the general principles of the motion of solid bodies wrote the law as component equations of the kind still taught today. The unit was standardised much later still: the name newton was adopted by the General Conference on Weights and Measures in 1948 and formally incorporated into the SI in 1960, gradually displacing the dyne and the kilogram-force from scientific work.