The Law of the Lever and Simple Machines

Hang weights on a lever and find where it balances: one weight on a long arm holds several weights on a short one. Compare the moments of force M₁ and M₂ — the arcs at the pivot show which is larger. Pull the lever with a spring scale at an angle and watch the force grow as its arm gets shorter. Then lift a load with a fixed pulley, a movable pulley and a block and tackle: the spring scale shows how much force you gain, and the tape measures show how much distance you lose. This is the “golden rule” of mechanics.

↓F₁, l₁, M₁— left weight, its arm and moment ↓F₂, l₂, M₂— right weight, its arm and moment →F— force of the hand (spring-scale reading) ↓P— weight of the load ↑F— rope tension — the same along the whole rope ↕h, s— path of the load and path of the hand

The coloured segment is the arm of the force: the shortest distance from the axis to the line of action (dashed). The arcs at the axis are the moments M₁ and M₂; their lengths are proportional to the moments. You can drag the weights and the spring scale along the lever.

Machine

Left arm

F₁ weights of 102 g 3 × 1 N

l₁ distance from the axis to the suspension point 20 cm

Right arm

F₂ weights of 102 g 2 × 1 N

l₂ distance from the axis to the suspension point 20 cm

Force vectors

Arms, lines of action and moments

Each weight has a mass of 102 g, as in a school lab kit: its weight is 0.102 kg · 9.8 N/kg ≈ 1 N. The holes in the lever are 5 cm apart. While you move the weights, the stops hold the lever; then it is released by itself.

F₁

l₁

M₁ = F₁l₁

Solve the problem, enter the answer and press “Check” — the setup will repeat the experiment with your data.

Graphs

Left: the force needed for balance (for the lever it is the hyperbola F = M/l — the longer the arm, the smaller the force; for pulleys, a comparison of the machines). Right: the “golden rule” — the works of the two forces as the lever turns, or the useful work and the work done as the load is lifted.

The lever is one of the oldest machines: pyramid builders and millers used it long before physics existed. The law of the lever was first rigorously proved by Archimedes in the 3rd century BC in his work “On the Equilibrium of Planes”. He is credited with the words: “Give me a place to stand, and I will move the Earth.” Yakov Perelman, in his “Mechanics for Entertainment”, worked out what it would take. Suppose Archimedes can lift 60 kg, while the Earth’s mass is 6·10²⁴ kg — 10²³ times more. To raise the Earth by just 1 cm, his end of the lever would have to travel 10²³ times farther — 10¹⁸ km. Moving the lever at 1 m/s, Archimedes would need about 30 trillion years. A gain in force is always paid for with a loss in distance. According to Plutarch, Archimedes did demonstrate the power of simple machines: using a system of pulleys, he single-handedly moved a loaded ship that many men could not haul ashore.

KEY FORMULAS

Moment of force

$$M = F\,l, \qquad [M] = \text{Н}\cdot\text{м}$$

F is the magnitude of the force, N; l is the arm of the force, m: the shortest distance from the axis of rotation to the line of action of the force

▶ Detailed Explanation

The moment of force shows how strongly a force turns a body about an axis. The same force turns a lever more strongly the farther from the axis its line of action passes. That is why a door handle is placed at the edge of the door and a nut is easier to undo with a long spanner.

The arm is not the distance to the point where the force is applied but the length of the perpendicular dropped from the axis to the line of action. If a spring scale attached 40 cm from the axis is pulled at 60° to the vertical, the arm shrinks to 20 cm. To keep the moment the same, the force must double. A force whose line of action passes through the axis creates no moment: its arm is zero.

The law of the lever (equilibrium condition)

$$M_1 = M_2 \quad\Leftrightarrow\quad F_1 l_1 = F_2 l_2 \quad\Leftrightarrow\quad \frac{F_1}{F_2} = \frac{l_2}{l_1}$$

A lever is in equilibrium when the anticlockwise moment equals the clockwise moment. The forces are inversely proportional to their arms

▶ Detailed Explanation

Example: on the left, l₁ = 20 cm from the axis, hang three weights of 1 N each (F₁ = 3 N). The moment is M₁ = 3 N · 0.20 m = 0.60 N·m. On the right, two weights (F₂ = 2 N) balance the lever if M₂ = M₁, that is l₂ = M₁/F₂ = 0.60/2 = 0.30 m = 30 cm.

If several forces act on one side of the lever, their moments add up. In equilibrium a lever stays at rest in any position, not only horizontal: when it turns, both arms shrink by the same factor. The horizontal position is convenient because the arms can be read straight off the marks on the lever.

Fixed pulley

$$F = P, \qquad s = h$$

P is the weight of the load, F is the force pulling the rope, s is the distance the rope end travels, h is the height the load rises. A fixed pulley only changes the direction of the force

▶ Detailed Explanation

A fixed pulley is an equal-armed lever: the axle is the fulcrum, and both forces have arms equal to the pulley radius R. From the law of the lever F·R = P·R, so F = P. The tension is the same along the whole rope, so the force does not depend on the angle at which you pull.

But a fixed pulley lets you pull down, using your own weight, or sideways, while the load goes up. That is how a flagpole, window blinds and a well bucket hoist work.

Movable pulley and block and tackle

$$F = \frac{P}{n}, \qquad s = n\,h$$

n is the number of rope sections supporting the movable pulley with the load (movable pulley: n = 2). An n-fold gain in force comes with an n-fold loss in distance

▶ Detailed Explanation

A load on a movable pulley hangs on two sections of the same rope, each with tension F. The pulley is in equilibrium when 2F = P, so F = P/2. The law of the lever gives the same: the fulcrum is where the rope touches the pulley on the left, the arm of the load’s weight is R, and the arm of the force F is the diameter 2R.

If the movable pulley itself weighs P_p, it has to be lifted too: F = (P + P_p)/n. Example: a 20 kg load (P = 196 N) and a 1 kg movable pulley (9.8 N) give F = 205.8/2 ≈ 103 N instead of 98 N, efficiency ≈ 95 %. A movable pulley is often combined with a fixed one: the gain in force is the same, but you can pull downwards. A block and tackle of two movable and two fixed pulleys (n = 4) lifts 40 kg with a force of about 100 N.

The “golden rule” of mechanics

$$F\,s = P\,h \qquad \frac{P}{F} = \frac{s}{h}$$

Simple machines give no gain in work: whatever you gain in force, you lose in distance by the same factor

▶ Detailed Explanation

For a lever: when it turns through a small angle, the ends of the arms travel distances proportional to the arms: h₁/h₂ = l₁/l₂. Combined with the law of the lever F₁l₁ = F₂l₂ this gives F₁h₁ = F₂h₂ — the works of the two forces are equal. For example, with 3 N at 20 cm balancing 2 N at 30 cm, turning the lever by 15° raises the left weight 5.2 cm while the right one drops 7.8 cm: 3·0.052 ≈ 2·0.078 ≈ 0.16 J.

For a movable pulley: to lift the load by h, each of the two rope sections must shorten by h, so the rope end travels s = 2h. The work of the hand is F·s = (P/2)·2h = P·h. The rule was stated by ancient scholars (Heron of Alexandria) and rigorously justified by Galileo and Stevin.

Efficiency of a simple machine

$$\eta = \frac{A_{\text{п}}}{A_{\text{з}}}\cdot 100\% = \frac{P\,h}{F\,s}\cdot 100\%$$

Aп = P·h is the useful work (lifting the load), Aз = F·s is the work done (by the hand). In a real machine η < 100 %

▶ Detailed Explanation

The work done is always greater than the useful work: part of it goes into lifting the movable pulley, into friction in the axle and into bending the rope. For a movable pulley of weight P_p without friction η = P/(P + P_p). A 5 kg load on a 2 kg pulley: η = 49/(49 + 19.6) ≈ 71 %. A 50 kg load on the same pulley: η ≈ 96 %.

Efficiency cannot exceed 100 %: otherwise the machine would give a gain in work, which contradicts the “golden rule” and the law of conservation of energy.

Simple machines. Simple machines are devices that transform a force: they change its magnitude or direction. They include the lever, the pulley, the wheel and axle, the inclined plane, the wedge and the screw. With them people lift and move loads they could never shift with their bare hands. This simulator brings together the lever and pulleys.

The lever and the arm of a force. A lever is a rigid body that can rotate about a fixed support (the fulcrum). The arm of a force l is the shortest distance from the fulcrum to the line along which the force acts (the line of action). If a weight hangs from a horizontal lever, the arm equals the distance from the pivot to the point of suspension. If you pull with a spring scale at a slant, the arm gets shorter, and a larger force is needed to hold the lever. In the simulator the arm is drawn as a coloured segment and the line of action as a dashed line.

Moment of force and the law of the lever. The turning effect of a force depends not only on its magnitude but also on its arm. It is described by the moment of force M = F·l, measured in newton-metres (N·m). A lever is in equilibrium when the moment turning it anticlockwise equals the moment turning it clockwise: M₁ = M₂, that is F₁l₁ = F₂l₂. Hence F₁/F₂ = l₂/l₁: the forces acting on a lever are inversely proportional to their arms. A weight of 1 N on a 40 cm arm balances 8 N on a 5 cm arm. If the moments are unequal, the lever turns towards the larger moment.

Fixed and movable pulleys. A pulley is a grooved wheel with a rope over it. The axle of a fixed pulley does not move, and it gives no gain in force: F = P. Instead it changes the direction of the force — you can pull down or sideways and the load goes up. A fixed pulley is a lever with equal arms, each equal to the pulley’s radius. A movable pulley rises together with the load. The load is held by two sections of rope, each pulled with force F, so 2F = P and F = P/2 — a twofold gain in force. A movable pulley is a lever with arms 2R and R. A block and tackle in which n sections of rope hold the load gives an n-fold gain in force.

The “golden rule” of mechanics. No simple machine gives a gain in work: whatever you gain in force, you lose in distance by the same factor. For pulleys F·s = P·h, where s is the distance the rope end travels and h is the height the load rises. To lift a load 40 cm with a movable pulley, you must pull 80 cm of rope; with a four-rope block and tackle, 1.6 m. When a lever turns, the points where the forces act travel distances proportional to their arms: h₁/h₂ = l₁/l₂, so F₁h₁ = F₂h₂. Press “Turn the lever” or “Lift the load” and compare the works in the “Measurements” tab.

Efficiency. In a real machine you must also lift the movable pulley itself along with the load, and overcome friction. So the useful work Wu = P·h is always less than the work done Wd = F·s. The ratio η = Wu/Wd·100 % is called the efficiency. Set a mass for the movable pulley and see the force F increase and the efficiency drop. A heavy pulley is especially wasteful with a light load.

Things to try. 1) Balance one weight against eight — where do the weights have to hang? 2) Put the spring scale on mark 8 and tilt it by 60°: the arm becomes half as long and the force twice as large. 3) With a fixed pulley, change the angle at which you pull the rope: the spring-scale reading does not change. 4) Lift a 40 kg load with the block and tackle — a force of about 100 N is enough, but you have to pull four times as much rope. 5) In the “Test yourself” tab, calculate the answer first and then check it by experiment.

The Law of the Lever and Simple Machines

Graphs

KEY FORMULAS

Moment of force

The law of the lever (equilibrium condition)

Fixed pulley

Movable pulley and block and tackle

The “golden rule” of mechanics

Efficiency of a simple machine

More on this topic

Other simulators

Problems on this topic