Example 1
Textbook reference: 20 N acts on an arm of 0.5 m, and the arm on the other side is 0.2 m. This is the reference example.
F₂ = 50 N, moment M = 10 N·m
Lever Equilibrium Calculator
Enter exactly three of F₁ in newtons, r₁ in meters, F₂ in newtons, and r₂ in meters. The calculator finds the fourth from the balance F₁·r₁ = F₂·r₂ and reports the shared moment. At 20 N on an arm of 0.5 m, with 0.2 m on the other side, F₂ is 50 N and the moment is 10 N·m. The model assumes a right angle: no sinα, no beam weight, and no pivot friction.
Result
Enter exactly three of the four quantities. The missing value, the moment, and the chart will appear here.
For the moment of a single force at any angle, use the M = F·r·sin α calculator; on this page the angle is 90°. For force from mass and acceleration, use the F = m·a calculator.
How it works
F₁·r₁20 N · 0.5 m
M10 N·m
F₂50 N
With forces perpendicular to the arms, the lever is in equilibrium when F₁·r₁ = F₂·r₂. The shared moment about the fulcrum is M = F₁·r₁. Archimedes stated the principle in On the Equilibrium of Planes: magnitudes balance when their distances from the fulcrum are inversely proportional to their weights. At a right angle the sine is 1, so the torque M = F·r·sinα reduces to the product F·r.
The calculator uses the four numbers you enter: forces F₁ and F₂ in newtons and arms r₁ and r₂ in meters. You enter exactly three, and the result supplies the fourth from the balance. At F₁ = 20 N, r₁ = 0.5 m, and r₂ = 0.2 m the moment is 20·0.5 = 10 N·m, so F₂ = 10 / 0.2 = 50 N. A crowbar with an arm of 0.8 m and a force of 150 N, against a load arm of 0.04 m, produces 3,000 N.
What the result panel shows. Besides the missing quantity, the calculator reports the moment M, all four values after solving, and the ratios r₁/r₂ and F₂/F₁. In this model the two ratios are equal when both forces are positive: the smaller arm corresponds to the larger force. The chart draws F₂ against r₂ at the fixed moment M = F₁·r₁. That curve is a hyperbola, not a straight line.
Limits of the model. The beam is rigid and weightless, the fulcrum is frictionless, and both forces lie in one plane and are perpendicular to the arms. The calculator does not include the beam’s weight or pivot friction. There is no angle field. An angled force is first computed with the torque calculator as M = F·r·sinα; you return here with the perpendicular component. F = m·a describes the acceleration of a body, not the balance of two arms.
The unit switch in the header changes newtons to pounds-force and meters to feet. The numbers in the text stay in SI as the reference. An arm must be positive: zero is the fulcrum, not an arm, and the calculator cannot divide by it. An entered zero in a force field is zero newtons. The unknown is a blank field.
Examples on the page: 20 N, 0.5 m, and 0.2 m give F₂ = 50 N; a wheelbarrow with a 360 N load 0.4 m from the wheel and handles at 1.2 m needs 120 N; a seesaw with 200 N at 1.5 m against 600 N gives a second arm of 0.5 m; a nutcracker with 40 N on 0.16 m and jaws at 0.02 m gives 320 N. The button “Fill this example” enters the same numbers and refreshes the panel and the chart.
Formula
F1·r1 = F2·r2
M = F1·r1 [M] = N·m
Model: Archimedes’ law of the lever (On the Equilibrium of Planes, Book I: magnitudes balance at distances reciprocally proportional to their weights). At a right angle this is the same statement as a sum of moments equal to zero, ΣM = 0. The calculator solves exactly one unknown from that equation and then reports M. It does not accept an angle.
Theory, derivation, and domain
Definition of lever equilibrium and of torque
A lever is a rigid beam that can rotate about an axis, the fulcrum. A force turns the beam more strongly the farther from the axis it acts. The measure of that turning is the moment, or torque. For a force perpendicular to the arm, the moment is M = F·r, where r is the distance from the axis to the line of action of the force. The unit is N·m. That is not a joule: a joule measures work and energy, and a newton-meter measures a tendency to rotate.
Rotational equilibrium means the sum of moments about the axis is zero. Two forces that turn the beam in opposite directions balance when their moments are equal: F₁·r₁ = F₂·r₂. The force ratio is then the inverse of the arm ratio, F₁/F₂ = r₂/r₁. A shorter arm requires a larger force so that the product stays the same.
History: Archimedes and the equilibrium of planes
Archimedes of Syracuse (c. 287–212 BCE) justified the law of the lever in On the Equilibrium of Planes. In Book I he postulates that equal weights at equal distances are in equilibrium, and then proves that magnitudes balance at distances reciprocally proportional to their weights. That is F₁·r₁ = F₂·r₂, written for weights. The treatise is the oldest surviving account of statics.
The sentence “give me a place to stand, and I will move the Earth” is not in that treatise. Pappus of Alexandria reports it centuries later as a saying attributed to Archimedes. The sense agrees with the law of the lever: with a long enough arm, a small force balances a large weight. The calculator computes that proportion for two arms, not for the Earth.
Derivation of F₁·r₁ = F₂·r₂
Step 1. The moment of a force about the axis, for a force perpendicular to the arm, is M = F·r. This is the special case of M = F·r·sinα at α = 90°, because the sine of a right angle is 1. Step 2. In equilibrium the sum of moments is zero. If F₁ turns one way and F₂ the other, the signs oppose and what remains is F₁·r₁ − F₂·r₂ = 0. Step 3. Therefore, F₁·r₁ = F₂·r₂ and M = F₁·r₁.
Step 4. For the numbers on this page: F₁ = 20 N and r₁ = 0.5 m, so M = 10 N·m. At r₂ = 0.2 m the missing force is F₂ = M / r₂ = 10 / 0.2 = 50 N. Step 5. The same equality solves the other unknowns. At F₂ = 50 N, r₂ = 0.2 m, and r₁ = 0.5 m, F₁ is 20 N. At F₁ = 20 N, F₂ = 50 N, and r₁ = 0.5 m, r₂ is 0.2 m. The arm ratio 0.5 / 0.2 = 2.5 equals the force ratio 50 / 20.
Classes of lever and mechanical advantage
Textbooks sort levers by where the fulcrum sits. In a first-class lever the fulcrum lies between the forces: a seesaw, a crowbar, scissors, an oar. In a second-class lever the load lies between the fulcrum and the effort, as in a wheelbarrow, a nutcracker, and a bottle opener, so the effort arm is longer and the hand force is smaller than the load. A fishing rod and a pair of tweezers place the effort between the fulcrum and the load, so the hand force is larger than the force at the tip, and the gain is distance or precision.
The formula does not change. What changes is which arm is longer. Ideal mechanical advantage, with no friction and at a right angle, is the ratio of the effort arm to the load arm, and it equals the ratio of the load to the effort. The calculator does not label which side is the effort. It shows both ratios, r₁/r₂ and F₂/F₁, and the example text says which quantity is the hand and which is the load.
Assumptions: a right angle, a rigid beam, a frictionless pivot
The model assumes a beam that does not bend, has no weight of its own, and turns on a frictionless fulcrum. Both forces lie in one plane and are perpendicular to the beam. A real beam has weight, so its weight adds a moment equal to the weight times the distance from the axis to the center of mass. Friction in the pivot consumes part of the moment, so the force on the shorter arm is smaller than the ideal ratio.
This calculator omits any angle other than a right angle. The component parallel to the arm does not turn the beam, and the perpendicular component is F·sinα. That component is first computed with the torque calculator; here you enter an already perpendicular force and its arm. The calculator does not add three forces, does not compute the reaction at the fulcrum, and does not check translational equilibrium, a sum of forces equal to zero. It checks only the balance of two moments.
Units, orders of magnitude, and levers around us
Enter force in newtons and the arm in meters; the moment comes out in N·m. A centimeter is 0.01 m, and a millimeter is 0.001 m. After the header switches units, force is in lbf (1 lbf = 4.4482216 N) and the arm is in feet (1 m = 3.2808399 ft). The panel then shows the value in the selected unit and the SI value beside it. The weight of a 1 kg mass near Earth is about 9.81 N; a 200 N child on a seesaw has a mass of about 20 kg. The calculator does not convert mass to weight.
Orders from the examples on this page: scissors produce a few tens of newtons from a finger force of about 10 N; a bottle opener and a nutcracker reach hundreds of newtons; a crowbar with an arm twenty times longer turns 150 N into 3,000 N; an oar delivers less force to the water than the pull, because the blade is farther from the oarlock than the hand is. A balance scale is the same equation with equal arms: equal forces balance at equal distances, as in Archimedes’ first postulate.
Sources
The definition, the derivation, and the history come from Archimedes’ On the Equilibrium of Planes (English in T. L. Heath, The Works of Archimedes, Cambridge 1897) and from textbooks: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapters on equilibrium and torque; H. D. Young and R. A. Freedman, University Physics, static equilibrium; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 18. Lever classes and mechanical advantage follow the encyclopedia articles on the lever. Units follow the BIPM SI Brochure, 9th edition, and NIST SP 330. This page is for educational purposes.
How to use
- Enter exactly three of F₁, r₁, F₂, and r₂. Leave one field blank: that is the unknown.
- For the reference case, enter F₁ = 20 N, r₁ = 0.5 m, and r₂ = 0.2 m, and leave F₂ blank.
- Click Calculate, or wait for the live result. F₂ is 50 N and the shared moment M is 10 N·m.
- Read the panel: all four values and the ratios r₁/r₂ and F₂/F₁. Here both ratios equal 2.5.
- The unit switch in the header changes newtons to lbf and meters to feet. The Clear button removes the inputs and hides the panel.
Symbol glossary
F₁ and F₂ are the forces, r₁ and r₂ the arms, and M the shared moment. For 20 N, 0.5 m, and 0.2 m the missing force is 50 N and M is 10 N·m.
- F₁
- Force on one side of the fulcrum [N]. In the reference case, 20 N.
- r₁
- Arm of F₁ [m], the distance from the fulcrum to the line of the force. In the reference case, 0.5 m.
- F₂
- Force on the other side [N]. In the reference case the result is 50 N.
- r₂
- Arm of F₂ [m]. In the reference case, 0.2 m. A shorter arm corresponds to a larger force.
- M
- Shared moment [N·m], M = F₁·r₁ = F₂·r₂. This is a moment, not energy in joules.
- r₁/r₂
- Arm ratio. With both forces positive it equals F₂/F₁. Ideal mechanical advantage uses this ratio once you choose which side is the effort.
Real-life examples
Example 2
A wheelbarrow: a 360 N load sits 0.4 m from the wheel, and the handles are 1.2 m from the wheel. The fulcrum is the wheel, so the effort arm is the longer one.
F₁ = 120 N, moment M = 144 N·m
Example 3
A seesaw: a child of 200 N sits 1.5 m from the pivot, and an adult weighs 600 N. The calculator finds how far the adult must sit.
r₂ = 0.5 m, moment M = 300 N·m
Example 4
A nutcracker: 40 N on a handle 0.16 m from the hinge, with the nut 0.02 m from the hinge. The load sits between the fulcrum and the hand.
F₂ = 320 N, moment M = 6.4 N·m
Example 5
A crowbar lifting a board: 150 N on an arm of 0.8 m, with the board’s edge 0.04 m from the fulcrum.
F₂ = 3,000 N, moment M = 120 N·m
Example 6
Scissors: 12 N on a finger hole 0.09 m from the pivot, cutting 0.03 m from the pivot.
F₂ = 36 N, moment M = 1.08 N·m
Example 7
A bottle opener: 25 N on a handle 0.06 m from the lip of the cap, with the cap’s edge 0.01 m from that fulcrum.
F₂ = 150 N, moment M = 1.5 N·m
Example 8
A fishing rod, a third-class lever: the fish pulls 8 N at 2 m, and the upper hand is 0.4 m from the lower hand that acts as the fulcrum. The effort sits between the pivot and the load, so the effort is larger than the load.
F₁ = 40 N, moment M = 16 N·m
Example 9
An oar: the hand pulls 200 N at 0.8 m from the oarlock, and the blade is 2 m from the oarlock. The load arm is longer, so the force on the water is smaller than the pull.
F₂ = 80 N, moment M = 160 N·m
Common questions
What does this calculator compute, and from which formula?
The calculator finds the missing quantity from the lever balance F₁·r₁ = F₂·r₂ and reports the shared moment M = F₁·r₁. For 20 N, 0.5 m, and 0.2 m the missing force is 50 N and the moment is 10 N·m. The same data can solve any other blank field.
Which units should I enter for the forces and the arms?
Enter forces in newtons and arms in meters; the moment is in N·m. Enter a centimeter as 0.01 m. After you switch the header to US units, force is in lbf, the arm is in feet, and the panel also shows the SI value.
How many fields do I fill?
Exactly three of the four. The fourth stays blank and is the unknown. Two fields are not enough for the balance, and the calculator rejects four filled fields because it cannot tell which quantity to recompute.
Does an entered zero mean a gap to solve?
No. Zero in a force field is zero newtons. The unknown is only a blank field. The calculator rejects an arm of zero, because the fulcrum is not an arm and division by it is undefined.
What do the arm ratio and the force ratio in the panel mean?
The ratio r₁/r₂ and the ratio F₂/F₁. In this model they are equal when both forces are positive. Ideal mechanical advantage is the ratio of the arm of the force you apply to the arm of the load. The calculator does not decide which side is the effort: both arms are measured from the fulcrum.
Does the calculator include the angle between the force and the arm?
No. The model assumes a right angle, so the sine of the angle is 1. An angled force is first computed as M = F·r·sinα with the torque calculator, and you enter the perpendicular component here.
How is this balance different from the torque calculator?
That page computes one moment, M = F·r·sinα, for one force and one arm. Here two moments balance: F₁·r₁ = F₂·r₂. The unit N·m is a moment, not energy in joules, even though the dimension is the same.
Does the class of lever change the formula?
No. A first-class lever, a wheelbarrow, and a fishing rod use the same balance. What changes is where the fulcrum sits relative to the forces. Arms are always measured from the pivot.
What happens when an arm is zero?
Zero means the fulcrum, not an arm. The calculator requires positive arms whenever it divides by r. It also rejects a negative arm.
Does a decimal comma work the same as a period?
Yes. 0.5 and 0,5 are the same arm in meters.
Does the calculator include the beam’s weight and pivot friction?
No. The model is a rigid, weightless beam and a frictionless fulcrum. The beam’s weight adds its own moment, and friction at the pivot reduces the load-side force relative to the ideal arm ratio.
Why does a shorter arm require a larger force?
A moment is the product of force and arm, and in equilibrium the two moments are equal. When the arm gets smaller, the force grows in the same proportion, so the product stays the same. At 0.5 m and 0.2 m the ratio is 2.5, so 20 N balances 50 N.
How do I calculate lever equilibrium with F₁·r₁ = F₂·r₂?
Enter three known values and leave one blank. At 20 N, 0.5 m, and 0.2 m the missing F₂ is 50 N, because 20·0.5 = 10 and 10/0.2 = 50.
What is F₂ for a crowbar at 200 N, r₁ = 1.5 m, and r₂ = 0.15 m?
2000 N. The effort moment is 200·1.5 = 300 N·m, so F₂ = 300/0.15 = 2000 N.
Is this the same as the torque calculator M = F·r·sinα?
No. That page computes one moment at any angle. Here two moments balance at a right angle: F₁·r₁ = F₂·r₂.
How do I convert centimeters to meters first?
Divide by 100. An arm of 50 cm is 0.5 m. Entering centimeters without converting overstates the moment one hundred times.
Does lever class I, II, or III change the formula here?
No. The same balance F₁·r₁ = F₂·r₂ applies. Only the fulcrum position relative to the forces changes; arms are always measured from the pivot.
Knowledge sources
The definition, the history, and the derivation below come from standard physics sources.
- Wikipedia — Lever
- Wikipedia — Mechanical advantage
- Feynman Lectures on Physics, Vol. I, Chapter 18 — Rotation in two dimensions
- BIPM — The International System of Units (SI), 9th edition
- NIST SP 330 — SI units guide
- Archimedes, On the Equilibrium of Planes, Book I; English via T. L. Heath, The Works of Archimedes, Cambridge 1897.
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics — equilibrium and torque.
- Young, H. D.; Freedman, R. A., University Physics — static equilibrium.
Page updated November 3, 2026.