The Counterweight Well: Sizing the Stone That Reduces the Work of Raising Water

A counterbalanced bucket lift hangs a stone on the far side of the rope, so that the operator raises the bucket and its water against only the difference between their weight and the stone's, and the stone then helps to bring the empty bucket back down. This design study performs the full energy accounting for such a device over a complete cycle, including the return stroke, which casual treatments omit. For the main case, an empty bucket of 3.0 kg carrying 15 kg of water raised through 10 m, the counterweight that minimises the peak force in the cycle is 10.5 kg, which equalises the two strokes at a peak force of 73.6 N, well below the unaided peak of 176.6 N. The counterweight that minimises the total energy input, under a stated friction model, is a different mass altogether: 3.0 kg, exactly the mass of the empty bucket, at a total input of 1577.4 J against 1636.3 J at the peak-force optimum and 1848.2 J with no counterweight at all. The two optima are therefore not the same, and choosing a stone requires choosing which quality is being bought. The net gravitational work of every cycle is 1471.5 J in this case regardless of the stone, since the bucket and the counterweight both return to their starting heights and only the water is raised for good. Over the range computed, the device beats an unaided lift on both peak force and total input energy for water loads from 10 kg to 30 kg at depths from 5 m to 25 m, and the peak-force-optimised stone keeps the peak within a stated limit of 150 N for water loads up to 30.58 kg, where an unaided lift stays within the same limit only up to 12.29 kg of water.