What really determines a bow’s required power is how much downward pressure on the strings the instrument can withstand without an unacceptable loss of tone. The bow must be able to direct this downward force without bottoming out and playing on the stick. Of course how close an individual player will go to this limit is a personal matter of technique and style. It also varies with the instrument. In practice, a violin bow of around 60 grams usually meets this requirement and so does a cello bow of 80 grams. But the proportions of weight to string length between the violin and cello are radically different.
Of course the cello bow is much stiffer in part because of its shorter length and larger diameter but still the proportional curve between the instruments is not at all consistent. Clearly there must be other factors at play which determine the weight and strength of bows; one of which may simply be tradition and utility. For one thing there is a physical limit to the weight of bow a cellist can handle and play the repertoire with the required agility. By trial and error cellists have usually discovered their ideal bow weight to project as they need to. Cello bows have the greatest variation in weight and I have seen bows in the hands of accomplished players ranging from 74 to 92 grams; a Kittell and a Dodd respectively. In my opinion a weight of around 79 grams is usually very effective assuming that the winding and fittings are light. With a solid silver wire grip we can raise this to about 81 or 82. But this does not explain why this weight is proportionally much lighter than a violin bow. I include pictures of a 79.6 gram bow I made recently for the cellist, Amir Eldan.
Sound quality is implied in these considerations but it is also a separate issue often contradicting the practical terms of a bow’s power. An individual palayer or instrument can require a bow that’s is too flexible or too stiff, too light or too heavy for practical purposes. The most important determant of a bows value for the player is sound quality. The player also wants to play accurately and through a wide dynamic range so in the end one looks for a happy synthesis.