A rigid bar that can turn about a fixed point is a lever. Whether it balances depends not only on how big the forces are, but on how far each force acts from the pivot — its moment arm. A ruler and a handful of coins are enough to verify the lever balance condition with your own hands.

Find the balance point with the ruler and coins first. Once students have felt it, the formula stops being a formula and becomes a description of what just happened on the table.

Safety first (read this before you start)

  • This experiment is about as safe as they come. Just keep the coins and the ruler from sliding off onto anyone's feet — work near the middle of the table and leave room on both sides.

Materials

  • A ruler with markings — plastic or wood, 30 cm works well;
  • A round-barrelled pen or a chopstick to act as the pivot, laid under the ruler;
  • A pile of identical coins, so that every coin weighs the same.

Procedure

  1. Lay the pen across the table and rest the ruler on it. Nudge the ruler back and forth until it sits roughly level at its midpoint. That point is your pivot.
  2. Stack a few coins on one marking to the left of the pivot. The ruler tips left.
  3. Add coins on the right, changing either how many or where until the ruler comes back to level.
  4. Write down both sides as a number of coins and a number of divisions from the pivot — for example, 3 coins on division 2 on the left, 2 coins on division 3 on the right.
  5. Repeat with several different arrangements. Each time, work out "coins × divisions" for both sides and see whether the two products match.
Why level it first: getting the bare ruler level before any coins go on ensures its own weight acts through the pivot and produces no turning effect of its own. Only then do your measurements describe the coins alone. This is exactly why school lever kits have you adjust the balance nuts before taking any readings.
Interactive simulation: this experiment has a drag-and-play simulator in our Interactive Lab — drag the weights to change their moment arms and use the sliders to change how many there are. The lab interface is currently Chinese-only, but the controls are simple sliders and drag targets, so it is still easy to explore.

What you should see

Record every successful balance in four columns: coins on the left, divisions on the left, coins on the right, divisions on the right. A reasonably complete set of results looks like this, with the pivot at the ruler's midpoint and divisions counted outward from it:

  • 4 coins on division 3 left, 3 coins on division 4 right → both sides 12;
  • 6 coins on division 2 left, 3 coins on division 4 right → both sides 12;
  • 2 coins on division 6 left, 4 coins on division 3 right → both sides 12;
  • 5 coins on division 2 left, 2 coins on division 5 right → 10 and 10.

Three completely different arrangements all balance, and they share exactly one property: the two products are equal. Equal numbers of coins is not required. Equal distances is not required either. That is the intuition this experiment exists to break.

Running it backwards makes the point just as well. Leave 4 coins sitting on division 3 on the left and put 3 coins on division 3 on the right — a product of 9 against 12, and the ruler tips left. Now slide those same 3 coins out to division 5, giving 15 against 12, and the ruler tips right instead. Not a single coin was added or removed; only the distance changed, and the tilt reversed. That isolates the moment arm and lets students see it acting on its own.

The science: torque and the balance condition

How strongly a force turns a lever depends on the size of the force and on its moment arm — the distance from the pivot to the line along which the force acts. Multiply the two and you get the torque. When the torques trying to turn the lever in opposite directions are equal, the lever balances. That is the lever balance condition:

F1 L1 = F2 L2

In the coin version, the force on each side is proportional to the number of coins and the moment arm is the number of divisions to the pivot, so balance means "coins × divisions on the left = coins × divisions on the right". It is also why the lighter child on a seesaw wins by sitting further out: a longer moment arm makes up for a smaller force.

Depending on which moment arm is longer, levers fall into three classes:

  • Force-multiplying levers (effort arm > load arm): less force, more distance — crowbars, bottle openers, wheelbarrows;
  • Distance-multiplying levers (effort arm < load arm): more force, less distance, but far more control — chopsticks, tweezers, a fishing rod;
  • Equal-arm levers (effort arm = load arm): neither multiplies — a balance scale.

Levers in everyday life

  • Seesaws and balance scales are levers at their most obvious;
  • Scissors are two levers joined together; tin snips have short blades and long handles, which makes them force-multiplying;
  • Chopsticks and tweezers cost you force and pay you back in precision;
  • Crowbars and bottle openers use a long effort arm to shift something heavy with a light push.

Five things students get wrong

  • The moment arm is not the distance from the pivot to where the force is applied. It is the perpendicular distance from the pivot to the line of action of the force. With coins pressing straight down on a horizontal ruler the two happen to be the same, which is why counting divisions works here. Tilt the ruler, or pull on it with a slanted string, and you must drop a perpendicular from the pivot onto the line of action and measure that. This single point costs more marks than anything else in lever problems.
  • "A force-multiplying lever is always the better one." No lever gives you force and distance at once. Whatever factor you gain in force, you pay in the distance your hand has to travel. Prising up a rock with a crowbar means moving your hands a long way for a tiny lift. Chopsticks and tweezers are deliberately built the other way round, buying "fingers move a little, tip moves a lot".
  • Forgetting to level the ruler first. If the bare ruler already sits crooked, its own weight is producing an extra torque that quietly contaminates every reading, and the two products will refuse to match no matter how carefully you count.
  • "A lever has to be a straight rigid bar." It does not. Anything that can turn about a fixed point under an effort and a load counts — the blades of a pair of scissors, the hooked end of a bottle opener, each spoke of a steering wheel. The test is whether it turns about a pivot, not whether it is straight.
  • "The pivot is always in the middle." Not at all. On a wheelbarrow the pivot is the front wheel, the effort is at your hands and the load sits between them; on a bottle opener the pivot is the rim of the cap. Before classifying any lever, patiently locate all three: pivot, effort, load.

Teaching notes

  • Time and grouping: 25 minutes, pairs. Use coins of one denomination, and level the ruler before anything else.
  • Say this before they start: keep the ruler and coins from falling off the edge.

The step most often skipped. Ask students to guess what makes it balance before they touch anything. Most will say "the same amount on both sides". Let them produce the 4 × 3 = 3 × 4 result themselves, then send them back to their own prediction. Nothing else lands as hard.

How to tell they have it. They can explain any one of their balanced rows using F₁L₁ = F₂L₂, and they can state that the moment arm is the perpendicular distance to the line of action, not to the point of application.

Take it further

  • Fix one side in the simulator — say 4 weights on division 3, a torque of 12 — and find every "number × division" combination on the other side that also makes 12. Balance is not one arrangement; it is a whole family of them.
  • Turn the ruler into a rough balance scale: hang a small bag at equal distances on both ends, put the object to be weighed in one and add coins to the other until it levels. You have measured the object in coins.
  • Find the lever inside your own arm. When you bend your elbow, the pivot is the elbow joint, the effort comes from the biceps attaching only a few centimetres away, and the load is whatever you are holding twenty or thirty centimetres out. The effort arm is far shorter than the load arm, which makes your forearm a distance-multiplying lever. Lifting 5 kg costs the muscle several times that force — and buys your hand the ability to move fast and far.
  • The three-coin challenge. Can three coins balance the ruler with the pivot left where it is? (Hint: 1 coin on division 6, 2 coins on division 3 — both sides come to 6.)
  • Move the pivot from the midpoint to the one-third mark and try again. Now the ruler's own weight joins the problem. Work out where it effectively acts, and how to fold it into the balance condition.

Once a student can explain why a lighter child sitting further out lifts a heavier one, using force times moment arm, this experiment has done its job.