(a) A Talmid (somewhat knowledgeable about physics) asked about Bava Basra 88b. The Girsa in the Rashbam (DH Echad) explains the Gemara to say that the excess weight needed for a Hachra'ah of a Tefach is proportional to how much you weigh at once. (This is also Noge'a to Kerisus 5, which discusses weighing 500 Maneh of Kineman Besem, Davka 250 Maneh at a time.)
Why doesn't any Mashehu of extra weight on one side not make that side go all the way down?
I concluded that the only way to explain the Gemara is due to warping of the crossbar (i.e. the bar sags and bends down a tiny amount at the ends). Using simple trigonometry, you get a formula (for how much extra weight is needed for a Hachra'ah of a given size) based on how many degrees it is warped.
The formula is: (extra weight)/(total weight) = 2*(square root of (L squared -1))(sin B)/L, - where L is the length of the crossbar in Tefachim, and B is the angle of the warping of the crossbar. B is usually small, so (sin B) is about Pi (B)/180 if B is in degrees.
The Rashbam says that 1% excess weight makes a Hachra'ah of a Tefach. If we take L to be 4 Tefachim (like it says on 89a), this assumes that the warp angle is about .9/Pi, i.e. about 30% of one degree. (See our accompanying technical paper with diagrams for more detail, at www.dafyomi.co.il/bbasra/discuss/basr-088.qa3.pdf).
(b) I am left with Ketzas Tzarich Iyun on the Rashbam; why didn't he say that the Nafka Mina for weighing all at once or in many small weighings is that in the latter way, each time you must add an extra 1% or so of the weight of the pans to make the Hachra'ah of a Tefach?
Rabbi Pesach Feldman, Yerushalayim
(a) I would think that if a small percentage is added to one side, it does not make the other side go down entirely. Here you are only adding 1%.
(b) I do not claim to know anything about physics, so I will skip the calculations and go to your Ketzas Tzarich Iyun on the Rashbam.
1. The Gemara states below (89a) that if one wants to buy 10 Litrin, one cannot say to the seller that he should sell it in 10 separate units and the buyer will receive 10 Hachra'os. Rather, the seller is to weigh all 10 Litrin together and only needs to give one Hachra'ah. The Rashbam explains that the seller cannot be accurate with the additions that he gives each time, and if he would have to give 10 additions he would suffer an unreasonable loss.
2. The Perishah (CM 231:20) and SM'A (#25) explain that if one needs to weigh only once, it is possible for him to concentrate on what he is doing and be accurate with how much he adds. However, if a person must do this 10 times, it is far more difficult to be accurate. Accordingly, even thoughathematically-speaking the amount should be exactly the same if one adds on 10 smaller amounts on 10 separate occasions as when he adds 10 times as much only once, nevertheless in practical terms, especially considering the fact that the seller is very busy with all of his other customers, if the seller would be required to add on for each litra, a G-d-fearing seller will be likely to lose out because it is difficult to concentrate on 10 different occasions to add exactly the correct amount, and it is probable that he will add on a little more than necessary each time and will end up giving the buyer significantly more`than what he is entitled to.
Consequently, if he would have been required to add on 1% ten times, it probably would have worked out that in fact he would add on more than that, which is why he is required instead to weigh it out all at once.
Kol Tuv,
Dovid Bloom