(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
The Gemara tells us that a seller must give the buyer a bit more than an exact measure. The extra amount is called a "Hechra."
The Gemara explains that when an item is being measured with a balance scale, the amount of the Hechra is "one Tefach." This means, according to the Rishonim, that the seller does wait until the balance beam of the scale is perfectly level. Rather, he waits until the beam tips one Tefach in favor of the buyer.
How can a balance scale tip only a Tefach? If one side is heavier than the other, won't the heavier side continue to tip until it reaches the bottom?
Thank you,
Q. Reese, Atlanta GA, USA
Figure 1
Assume that the rod is straight and one weight (W2) is heavier than the other (W1). When the heavier weight causes one side of the scale to tip down (at an angle A), it will continue to exert more torque (rotational force) than the lighter weight does, until its side goes all the way to the bottom. It will not stop in the middle. The respective torques are:
T2 = W2 (cos A) and T1 = W1 (cos A).
Clearly, if W2 >W1 , then T2 > T1 (until A reaches 90 degrees).
Figure 2
If we assume that the rod is warped at an angle of B, then both sides will be lower (the dashed lines). The angles of the sides are, respectively, (A-B) and (A+B), so we can calculate the torques with the equation:
T2 = W2 cos(A+B) and T1 = W1 cos(A-B).
We know that cos (A+B) = (cos A) (cos B) - (sin A) (sin B).
If B is small, cos (A+B) ? (cos A) - B(sin A), if B is in radians. (1 degree =?/180 radians.)
Based on this and the above equation, we can conclude:
T2 ? W2 ((cos A) - B(sin A)), and T1 ? W1 ((cos A) + B(sin A)).
The torques T2 and T1 are equal when
W2 =W1 *(cos(A) + Bsin(A))/(cos(A) - Bsin(A))
The Me'iri and Yad Ramah explain that a Hechra of a Tefach is when the heavier pan is a Tefach lower than the lighter. If W2 is half a Tefach below horizontal (the balanced position), and the length of the bar is four Tefachim (Rashbam 89a DH Yud-Beis), it follows that:
cos(A) ? .98, and sin(A) ?.245
The Rashbam (88b DH Echad) says that a Hechra of a Tefach is when the heavier weight is 1% more than the lighter, i.e.:
1.01= W2 /W1 = 1 + (2B*(sin A))/(cos(A) - B(sin A)), so
.01 = .49(B)/(.98 - .245B), i.e. 98 - .245B = 49B, so
B (the warp angle) is about .02 radians ? 1.2 degrees.
When A=0, T2=W2 and T1=W1, the heavier side goes down. Clearly, the ratio T2/T1 = (W2/W1)(cos(A) + B(sin A))/(cos(A) - B(sin A)) keeps getting smaller when A increases from 0 until about 14 degrees. At that point, sin A?.245, and the ratio becomes 1.
The Rashbam (89a DH Shalosh) implies that a Hechra of a Tefach is when the heavier pan is a Tefach below horizontal. If W2 is a full Tefach below horizontal, then (sin A)=1/?5 ? .45.
Since 1.01= 1+ (2B(sin A))/(cos(A) - B(sin A)), we may conclude that
B ? .9/90=.01 radians, or about .6 degrees.
Rabbi Pesach Feldman
Kollel Iyun Hadaf