Force Calculator (Newton’s Second Law)

About the Force Calculator (Newton’s Second Law)

Three force calculations in one place. F = ma solves Newton’s second law for the net force, the mass or the acceleration. Weight W = mg gives the weight of a mass on Earth, the Moon, the other planets or with your own value of g — or the mass from a weight — together with what a bathroom scale calibrated on Earth would read and a table for every body. Resultant of forces adds up to twelve forces given as a size and a direction, resolving each into x and y components, and returns the size and direction of the resultant, the equilibrant that would balance it and, with a mass, the acceleration.

Forces can be in newtons, kilonewtons, pounds-force, kilograms-force or dynes; directions as angles from the +x axis or as compass bearings. Every answer shows the working, and the resultant comes with a vector diagram and a components table you can download as CSV.

Common uses

• Physics homework on Newton’s second law, weight and vector addition, with every step.
• Converting between newtons, kgf and lbf for engineering or gym equipment.
• Comparing your weight on the Moon, Mars or Jupiter.
• Finding the net force on an object pulled in several directions, and the force that would balance it.

The formulas

• Newton’s second law: F = m a, with F the net force (the vector sum of all forces). 1 N is the force that gives 1 kg an acceleration of 1 m/s².
• Weight: W = m g, where g is the gravitational acceleration where you are. Mass does not change from place to place; weight does.
• Components: a force F at angle θ from the +x axis has Fx = F cos θ and Fy = F sin θ. A compass bearing β becomes θ = 90° − β.
• Resultant: Rx = ΣFx, Ry = ΣFy, R = √(Rx² + Ry²) and θ = atan2(Ry, Rx); atan2 uses the signs of both components so the angle lands in the right quadrant.
• Equilibrant: the same size as R, pointing the opposite way (θ + 180°).

Force units

• 1 kN = 1,000 N
• 1 kgf (kilogram-force) = 9.80665 N exactly — the weight of 1 kg under standard gravity
• 1 lbf (pound-force) = 4.4482216152605 N — the weight of 1 lb under standard gravity
• 1 dyn (dyne) = 10⁻⁵ N

Bathroom scales measure force but are labelled in kg for Earth: they effectively show W ÷ g for Earth’s gravity (this calculator uses the standard 9.80665 m/s²), which is why the same scale would read about a sixth of your mass on the Moon.

Gravity on other bodies

Surface gravity values are NASA JPL’s equatorial figures: Mercury 3.70, Venus 8.87, Mars 3.71, Jupiter 24.79, Saturn 10.44, Uranus 8.87, Neptune 11.15 and Pluto 0.62 m/s². The giant planets have no solid surface; their values are at the reference radius JPL uses. The Moon’s 1.62 m/s² is GM ÷ r² from JPL’s lunar GM (4,902.800 km³/s²) and mean radius (1,737.4 km). On Earth, g varies from about 9.78 to 9.83 m/s² with latitude, altitude and local geology (OpenStax §3.5); the standard value is 9.80665 m/s².

Sources

• OpenStax, University Physics Volume 1: §5.3 Newton’s Second Law (Eqs. 5.3–5.5, resolving forces into components) and §5.4 Mass and Weight (w = mg).
• NIST, SP 811 Appendix B.8 — kgf, lbf and dyne.
• BIPM, 3rd CGPM (1901), Declaration on the unit of mass and on the definition of weight — weight = mass × acceleration due to gravity; g₀ = 980.665 cm/s².
• NASA JPL Solar System Dynamics: Planetary Physical Parameters and Planetary Satellite Physical Parameters.

Limitations

• F = ma needs the net force; subtract friction, drag and other opposing forces first.
• Forces are added in two dimensions (a plane). Three-dimensional forces need a z component as well.
• Weights use the surface gravity of each body; they ignore altitude, latitude and the small effect of the planet’s rotation.
• Speeds near the speed of light need relativity, where F = ma in this simple form does not hold.

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