Weigh without a scale by balancing the unknown load against known weights on a plank over a fulcrum, using water at 1 kg per litre as the reference: a pole that balances empty on its midpoint, a load hung 0.5 m from the fulcrum and a 20 kg bucket of water slid out to 2.0 m on the other side balance 80 kg. Swap the two pans and take the geometric mean to cancel unequal arms, weigh a stone by the water it displaces, and weigh a heavy cargo by how far a flat boat sinks, 60 kg for each centimetre on a 6 m² waterline.
Part of Logistics & Load Planning
Finding the weight of a load by balance, lever, spring scale or displaced water when no scale is available.
A Lever Turns Any Known Weight Into A Scale
The parent article lists the options. A balance beam is a plank balanced on a fulcrum, with known weights on one side and the unknown load on the other; water containers make convenient reference weights, since 20 litres is 20 kg. A spring scale, a salvaged luggage scale or fish scale, handles up to 50 kg, and larger loads are weighed in several parts. The parent also mentions a lever calculation: if you can lift one end of a loaded cart, the shaft weight equals the force you apply. The title adds water displacement, which this page covers.
What You Need
- A strong straight plank or pole and a fulcrum: a sharp-edged block or a pin through the middle.
- Reference weights — buckets of water. One litre is 1 kg; see Common Material Weights.
- A tape to measure arm lengths.
- A spring scale for small items, if you have one.
Method: Weigh By Lever
Step 1: Balance the empty beam Find the fulcrum position where the unloaded pole balances level. Mark it. Its own weight then cancels out.
Step 2: Hang the load at a measured distance Hang the load at a known distance a from the fulcrum, which is at the midpoint of the pole.
Step 3: Hang the reference weight on the other side Slide it along until the pole balances level. Measure its distance b.
Step 4: Compute Load × a = reference × b, so load = reference × b / a.
Step 5: Check with the swap Swap the load and the reference to the opposite arms, rebalance, and take the geometric mean of the two answers.
Worked Example: Lever, Swap Check, Displacement And Flotation
Lever. Inputs: a 5 m pole that balances level on its midpoint when empty; the load hangs 0.5 m from the fulcrum on one side; a 20 kg bucket of water slid along the other side balances at 2.0 m. Load × 0.5 = 20 × 2.0, so load = 20 × 2.0 / 0.5 = 80 kg. The empty-beam balance in Step 1 matters: a pole that is off-centre adds its own weight to one side and every reading is wrong by that moment.
Unequal arms. Suppose an equal-arm beam has arms that are not quite equal. Place the item on the left and it balances with w1 = 10.0 kg on the right. Swap, and it balances with w2 = 10.4 kg. Let the left arm be a and the right arm b. Then W × a = w1 × b and W × b = w2 × a, so W² = w1 × w2, and W = √(10.0 × 10.4) = √104 = 10.20 kg. The ratio a / b = w1 / W = 0.98. The average of the two readings, 10.2 kg, is almost the same here. The method needs only the two readings and no knowledge of the arm lengths. This double-weighing method is standard in balance work; I am citing it from memory.
Water displacement. Fill a bucket to the brim, lower a stone in, and catch the overflow. If it is 5 litres, the stone’s volume is 5 L = 0.005 m³. With granite at about 2,640-2,800 kg/m³ (Material Properties gives 165-175 lb/ft³): 0.005 × 2,640 = 13.2 kg to 0.005 × 2,800 = 14.0 kg. The method needs the stone’s density to turn volume into weight, so it is only as good as that figure.
Flotation. A floating hull displaces its own weight in water. For a flat-bottomed boat with straight sides and a waterline area of 4 m × 1.5 m = 6 m², one centimetre of extra draught adds 6 m² × 0.01 m = 0.06 m³ = 60 litres, which is 60 kg. Mark the waterline before and after loading and a rise of 5 cm is 5 × 60 = 300 kg of cargo.
Common Mistakes
- An uncorrected plank — it is heavier on one side. Balance it empty first.
- Reading to the nearest tick — a few millimetres of arm error becomes kilograms. Measure carefully and repeat.
- Mixing mass and force — a balance finds mass in kilograms. Multiply by 9.81 for newtons if you need a force.
- The loaded-cart lever — the parent says that lifting one end of the loaded cart gives the shaft weight; the height you lift it does not matter, only that the shaft tip just leaves the ground. A big lift moves the load and changes the answer.
Safety
A heavy load hung from a pole
If the pole or its support breaks, the load drops. Do not stand under it, and rig the support to hold several times the load. A bucket of water on a long arm swings; steady it. When weighing over water, keep the stability of the boat in mind: do not load it past the point where the waterline nears the gunwale.
What’s Next
- Learn next: Cart & Wagon Loading — using weights to balance a cart
- Combine with: Mixed Cargo Planning — planning the whole load