If a Torah falls and hits the ground, the original minhag was to engage in 40 days of sunrise to sunset fasting to atone for the disgrace to the Torah. The 40 days is to be in accordance with the 40 days that Moshe Rabbeinu was on Har Sinai receiving the Torah (without eating or drinking). Nowadays, people cannot handle that and will perform alternate means of Teshuvah such as tzedaka or a lesser amount of fasting as explained in the Tzitz Eliezer 5:4.
Forty Days Fasting
When someone does something wrong, certain acts can lessen his
accountability. Such an act is called a "Tikun." A Tikun usually
follows a rule known as mida kneged mida - meaning that it is related
conceptually to the transgression. Fasting 40 days helps atone for the
disrespect shown to the Torah, which was given in 40 days. Since
everyone present sensed the enormous disgrace and degradation, they
too would need to fast.
Now, this forty day fast would not have to be 40 days in a row, and
would only include the daylight hours, but not the night before.
Nowadays, people are not as hale and hardy as they used to be. Instead
of fasting, therefore, everyone present would give Tzedaka (charity)
Why fast after dropping a Torah scroll or tefillin?
One of the earliest mentions of this practice is a responsa of Rabbi
Israel of Brunna (present-day Brno, 15th century). He explains that
dropping a Torah scroll or tefillin is a sign from Heaven that one has
done something wrong for which one must repent.1 On the other hand,
Rabbi Chaim Yosef David Azulai (18th century), commonly known as the
ChIDA, cites another reason: The fast is to atone for the lack of care
and respect which allowed the holy object to fall.2
Shut Mahari Brunna 127.
Chaim Sha'al 1:12.
Another difference is that although only the person who actually drops
the Torah or tefillin is obligated to fast, under certain
circumstances the rabbi of a community may declare a public penance
for all of those present when a Torah falls. This may include studying
the laws of respecting a Torah, a communal commitment not to speak
during the Torah reading, charity, fasting, or a combination of some
of the above.4
- Tzitz Eliezer 5:4.