Balancing a lever — weight times distance from the pivot

The top row is the left side of the pivot and the bottom row the right side. In each row the left-hand card is the weight, the right-hand card the distance from the pivot, and the middle card multiplies the two: the moment.

"Left − right", in the centre, is the difference between the two moments. Zero means the lever balances. The card to its left, "Right distance to balance", keeps the right weight as it is and gives the distance that balances. The card to its right, "Right weight to balance", keeps the right distance and gives the weight.

Here both moments are 1,200 and the difference is 0. The right weight is 40 g, lighter than the 60 g on the left, and the lever still balances.

The turning effect: weight × distance from the pivot

A lever is a bar supported at one point, so that it can tip either way about that point. The point of support is the pivot (or fulcrum).

How hard a weight tips the bar does not depend on the weight alone. The same weight tips it harder the further from the pivot it hangs. This turning effect is called the moment:

moment = weight × distance from the pivot

When the moments on the two sides are equal, the lever stays level. When they are not, the side with the larger moment goes down. On Board 1, a positive "Left − right" means the left goes down; a negative one means the right goes down.

With the right weight on Board 1 changed to 30 g, the right moment becomes 900. "Left − right" is 300, and the left goes down. At 80 g, the right moment is 2,400. "Left − right" is −1,200, and the right goes down.

"The heavier side goes down" only holds when both distances are the same. With the right distance on Board 1 set to 20 cm, both weights are 20 cm from the pivot; "Left − right" is then 400, and the heavier 60 g on the left goes down.

The distance is measured from the pivot to the point where the weight hangs, not from the end of the bar. The 1,200 is grams times centimetres, so it is neither a weight nor a length: it is a number for comparing the two sides. In physics the moment is a force times a distance; using the weight of each load, as here, gives the same comparison.

Finding the weight or distance that balances

Once the left moment is known, the right side can be worked out. The right moment has to be 1,200 as well, so:

• if the right weight is fixed — right distance = left moment ÷ right weight
• if the right distance is fixed — right weight = left moment ÷ right distance

Here that is 1,200 ÷ 40 = 30 cm and 1,200 ÷ 30 = 40 g. The two outer cards in the middle row of Board 1 do exactly this.

Doubling the right distance to 60 cm halves the weight that balances, to 20 g. To keep the moment at 1,200, twice the distance needs half the weight. Two quantities related like this, where multiplying one divides the other by the same number, are inversely proportional.

With 20 g at 60 cm on the right of Board 1, "Left − right" is back to 0. With the left weight raised to 90 g, the left moment becomes 1,800; with 40 g still on the right, the balancing distance is 45 cm.

Several weights on one side: the moments add up

With two or more weights on one side, the total moment is the sum of the moment of each weight.

The left column is weight A and the right column weight B; both hang on the left of the pivot. The top row is the weights, the third row their distances from the pivot, and between them are their moments. The middle column, from the top, is the left total, the right weight that balances it, and the right distance.

A is 40 g at 10 cm and B is 20 g at 30 cm. A's moment is 400 and B's is 600, so the left total is 1,000. With the right distance at 20 cm, the weight that balances is 1,000 ÷ 20 = 50 g.

The two left weights come to 60 g, but they cannot be treated as a single 60 g weight. With the distances on Board 2 swapped, A (40 g) at 30 cm and B (20 g) at 10 cm, the left total becomes 1,400, so the right side needs 70 g. The same 60 g on the left calls for 50 g or 70 g on the right, depending on where each weight hangs.

The same goes for weights on both sides: the total moment on the left is compared with the total on the right.

Lifting a heavy load: pushing far from the pivot

Suppose a block under a bar serves as the pivot, a load hangs on one side, and a hand pushes down on the other end. The push from the hand works like a weight: its turning effect is force × distance from the pivot.

The left column is the load side: the load at the top and its distance from the pivot in the middle. The right column is the hand side: the hand's distance from the pivot in the middle and how far the hand pushes down at the bottom. The middle column, from the top, is the force needed at the hand, how many times further out the hand is than the load, and how far the load rises.

In this example a 30 kg load sits 20 cm from the pivot and the hand pushes 120 cm from it. The hand is 6 times further out, so the force needed is a sixth of the load, 5 kg. Checking with moments: 30 × 20 = 600 on the load side and 5 × 120 = 600 on the hand side.

The force is given as a weight: 5 kg means the same push as a 5 kg weight hanging where the hand is.

The point where the push is applied is often called the effort and the thing being lifted the load. Moving the effort further from the pivot, or the load closer to it, reduces the force needed.

In exchange, the hand travels further. On the bottom row of Board 3, pushing the hand down 30 cm lifts the load by only a sixth of that, 5 cm.

Why weight × distance

The law of the lever was established by experiment. Why the distance is multiplied in can be seen from how far each point on the bar goes up or down when the bar tips.

When the bar tips, every point on it turns about the pivot. A point twice as far out rises or falls twice as far; a point six times as far out, six times as far. The 30 cm of the hand and the 5 cm of the load on Board 3 are this relationship.

Force × distance travelled is the same on the two sides: 5 kg × 30 cm = 150 and 30 kg × 5 cm = 150. A small force can lift a heavy load only because it is pushed through a longer distance. In physics this is the statement that a lever does not create work: what goes in at one end comes out at the other.

The balance on Board 1 works the same way. The right weight hangs at 30 cm, 1.5 times as far out as the 20 cm on the left. So if the bar tips slightly and the left side drops 2 cm, the right side rises 3 cm.

• 60 g on the left drops 2 cm — 60 × 2 = 120
• 40 g on the right rises 3 cm — 40 × 3 = 120

The left side dropping is exactly enough to lift the right side. When it tips slightly either way, nothing is left over and nothing is missing, so the lever does not tip by itself. That is what balance means.

With 30 g on the right, the right side comes to only 30 × 3 = 90, and the left goes down. That matches Board 1 with 30 g on the right, where "Left − right" is 300 and the left goes down.

Other numbers

Board 1, keeping the left at 60 g and 20 cm (a moment of 1,200) and changing the right distance:

In every row, the right distance times the weight that balances is 1,200. With 40 g kept on the right, anything closer than 30 cm lets the left go down, and anything further out lets the right go down.

On Board 2, a right distance of 40 cm needs 25 g to balance, and 10 cm needs 100 g: the left total of 1,000 divided by the right distance.

Board 3, keeping the load at 30 kg and 20 cm, with the hand pushing down 30 cm, and changing the hand's distance from the pivot:

With the hand at 20 cm, the same distance as the load, it takes the full 30 kg, and hand and load go down and up by the same amount. The further out the hand, the smaller the force, and the less the load rises for the same 30 cm push.

With the hand kept at 120 cm and the load brought in to 10 cm, the hand is 12 times further out and the force drops to 2.5 kg. A 60 kg load at the original 20 cm needs a force of 10 kg.

What this calculation covers, and what it does not

The calculations on this page ignore the weight of the bar. When the pivot is in the middle of an even bar, its weight is shared equally by the two sides and does not affect the balance. When the pivot is off-centre, as on Board 3, the bar's own weight has a turning effect too, and the real force needed differs from the result here.

The weights pull straight down and the hand pushes at right angles to the bar. A push at a slant gives the same force a smaller turning effect.

Some tools, such as a bottle opener, have the pivot at one end, with the effort and the load on the same side of it. The rule is the same — force × distance from the pivot matches on both — and the "Force at hand" calculation on Board 3 applies as it is.

Read next — Concentration — finding it, diluting and mixing It calculates the concentration of a salt solution, and how much water to add or how much of two solutions to mix to reach a target.

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