Lever Classes & Mechanical Advantage

the same beam, three places to put the fulcrum, and one number that decides everything

Worked example — Lever Classes & Mechanical Advantage

Effort Force (E)	100.0 N
Mechanical Advantage (MA)	3.00
Effort Move per 1 cm Load Move	3.00 cm
Effort Arm (a)	150 cm
Load Arm (b)	50 cm
Load Force (L)	300 N

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

The advantage and the distance are the same number 🖖

Mechanical advantage is the ratio of the arms, a/b, and so is the distance the effort end travels for each centimetre the load moves. They are not two facts about a lever; they are one ratio read twice. The crowbar preset turns 300 N into 100 N of effort and charges 3 cm of hand movement per centimetre of lift, and the work comes out at 3.000 J either way. Every setting this tool allows balances to the last decimal, because a rigid beam has nowhere to put a loss.

Your arm is built the wrong way round on purpose 🖖

The biceps preset holds a 50 N weight with 350 N of muscle: seven times the force, to lift something seven times lighter. What it buys is speed. One centimetre of muscle contraction moves your hand 7 cm, and that is why you can throw. A third-class lever cannot do better than break even — sweeping all 40,000 arm combinations the tool allows, the largest mechanical advantage that is still a third-class lever is exactly 1.00, at the moment the two arms are equal.

The class is geometry, and the tool will overrule you 🖖

Choose "3rd class", then drag the effort arm out past the load arm, and the readout says Class 2. Nothing is being corrected: which class a lever belongs to is a statement about where the three points sit, and once the effort is further from the fulcrum than the load, that is a second-class lever whatever you meant to build. Half the settings behave this way — 19,900 of the 40,000 combinations selected as third class are re-read as second.

Problems solved in full

Why your arm needs 350 newtons to hold a 50 newton weight 6 steps
Load the Biceps Arm preset: a 5 cm effort arm, a 35 cm load arm, 50 N in the hand. The panel says the effort is 350 N and the mechanical advantage is 0.14. Work out where those come from, and then say what the arm gets in return for being built this badly.
1. Everything about a lever comes from one balance. Take moments about the fulcrum: force times distance on one side equals force times distance on the other. That is the whole model, and the rest is arithmetic.
  
  $E \cdot a = L \cdot b (moments about the fulcrum)$
2. Put the numbers in. The load sits 35 cm from the elbow and the biceps pulls only 5 cm from it, so the muscle is at a seven-to-one disadvantage before anything else happens.
  
  $E = \frac{L b}{a} = \frac{50 \times 35}{5} = 350 N$
3. Mechanical advantage is what you lift over what you pull. It comes out below 1, which means the machine is costing you force rather than saving it. Notice it equals a/b as well: the ratio of the arms, arrived at from the other end.
  
  $MA = \frac{L}{E} = \frac{50}{350} = 0.1429 = \frac{a}{b} = \frac{5}{35}$
4. Now the return. Sweep the arm through a small angle and both ends move in proportion to their distance from the fulcrum, so the hand travels b/a times as far as the muscle does. Seven centimetres of hand for every centimetre of contraction.
  
  $\frac{d _{load}}{d _{effort}} = \frac{b}{a} = 7$
5. Nothing was lost or gained. Take the load through 1 cm: the muscle does 0.500 J and the weight receives 0.500 J. What changed hands was the split between force and distance.
  
  $W_{in} = 350 \times \frac{5}{35} \times 0.01 = 0.500 J = W_{out} = 50 \times 0.01$
6. And the disadvantage is not an accident of these numbers. A third-class lever is defined by the effort being applied between the fulcrum and the load, which forces a to be smaller than b, which forces the advantage below 1. There is no third-class lever anywhere that lifts more than you pull.
  
  $third class \Rightarrow a < b \Rightarrow MA = \frac{a}{b} < 1$
Answer

350 N, because 50 N acting at 35 cm needs 350 N at 5 cm to balance it, and the advantage of 0.14 is the same ratio inverted. The arm pays seven times the force and receives seven times the speed.

That trade is the reason the design survived. A limb built for mechanical advantage would lift more and move like a crane; muscle is expensive but muscle is also strong, and the scarce thing in catching prey or throwing a spear is hand speed, not force.
A crowbar that turns 300 newtons into 100, and the 400 the pivot takes 6 steps
Load the Crowbar preset: a 150 cm effort arm, a 50 cm load arm, a 300 N lid. Work out the effort, then decide how strong the block you rest the bar on has to be.
1. Moments about the fulcrum settle the effort in one line: 300 N acting at 50 cm is balanced by 100 N at 150 cm. The cards agree, at 100.0 N with an advantage of 3.00, bought with 3.00 cm of hand travel for every centimetre the lid rises.
  
  $E = \frac{L b}{a} = \frac{300 \times 50}{150} = 100 N$
2. Moments are only half of statics. The bar is not going anywhere, so the forces on it have to sum to zero as well, and there are three of them: your hand, the lid, and whatever the bar is resting on. Nothing on this page computes the third one.
  
  $MA = \frac{a}{b} = \frac{150}{50} = 3.00, d_{E} = 3.00 cm$
3. On a first-class lever the fulcrum sits between the other two, and both of them press down: you push down on the long end, and the lid pulls down on the short one. So the block underneath pushes up with the sum, 400 N. Your hand feels a third of the lid; the block feels more than all of it.
  
  $R = E + L = 100 + 300 = 400 N$
4. The reaction is L(1 + 1/MA), which always exceeds the load, and the better the lever the closer it settles towards the load from above. A crowbar with a hundredfold advantage still sends 1.01 times the load through its pivot.
  
  $R = L (1 + \frac{1}{MA}) > L \forall MA > 0$
5. Second class inverts the arrangement, and the tool has one: the wheelbarrow, 600 N at the same advantage of 3. The load now sits between the fulcrum and the effort and you lift the handles, so the effort points upwards and the axle carries L − E, which is 400 N. Two thirds of what is in the barrow.
  
  $R = L - E = L (1 - \frac{1}{MA}) = 600 (1 - \frac{1}{3}) = 400 N$
6. Third class runs the same formula with an advantage below 1. The biceps preset has MA = 5/35, so 1 − 1/MA is −6 and the reaction is −300 N. The sign is the answer: the elbow end of the forearm is being pulled down with 300 N to hold 50 N in the hand.
  
  $MA = \frac{5}{35} \Rightarrow R = 50 (1 - 7) = - 300 N$
Answer

100 N to pull and 400 N through the pivot, a third more than the lid weighs. The bar multiplies your force and the fulcrum absorbs the difference, so the block is the part of the arrangement that has to be sound.

Across all three classes the pivot carries L(1 ± 1/MA), and the sign in that bracket is the most useful thing the classification tells you. A first-class pivot always takes more than the load, which is why an improvised fulcrum under a crowbar is where the arrangement fails. A second-class pivot always takes less, which is why a wheelbarrow wheel can be light. A third-class pivot takes a force in the opposite direction to the one anyone guesses, and its size is set by the muscle rather than by what is in the hand: 300 N through the elbow to hold a 50 N weight.

$R_{1} = L (1 + \frac{1}{MA}), R_{2, 3} = L (1 - \frac{1}{MA})$

Learning path

Simple machines: trading force for distance

Leads to Wheel and axle

By the 42Math team · Updated 2026-08-31

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Crowbar (1st Class) - A first-class crowbar: 300 N load, arms of 150 and 50 cm, so 100 N of effort and 3 cm of hand travel per centimetre of lift.
Wheelbarrow (2nd Class) - Second class, load between fulcrum and hands: 600 N carried with 200 N of lift, because the wheel takes the rest.
Biceps Arm (3rd Class) - Third class, and it loses: 350 N of muscle to hold 50 N, in exchange for a hand that moves seven times faster than the muscle does.
Equal Balance (1st Class) - Equal arms, so no advantage at all: 200 N in, 200 N out, and 1 cm for 1 cm. A balance, not a machine.