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If a close relative dies on Yom Tov, the seven days of mourning start after Yom Tov (1)
According to Rebbi Yochanan and Rebbi Elazar, if a close relative dies on Yom Tov the days of Yom Tov count toward the thirty days mourning period.
Beis Shamai hold that if the mourning period started three days prior to Yom Tov, it does not have to be made up after Yom Tov, while Beis Hillel hold even if it started only moments before Yom Tov, it does not have to be made up.
If a person finds out a while after the burial that a relative died, there is a dispute between the Chachamim and Rebbi Akiva regarding the length of the mourning period. (2)
A partial day of mourning is regarded as an entire day. (3)
If a person found out on Yom Tov or Shabbos that a relative died within thirty days, but by the end of Yom Tov or Shabbos thirty days had passed since the time of burial, then only one day of mourning is necessary.
There is a dispute between the Amora'im whether there is a Mitzvah to rip one's clothing if the mourning period is only one day. However, for a mother or father everyone agrees that it is a Mitzvah to rip the clothing.
The mourning period applies to all relatives that a Kohen may become Tamei (ritually impure) from them, and it also includes other relatives that a Kohen may not become Tamei from them. (4)
A mourner must rip his clothes while standing.
A BIT MORE
1. Although when a relative dies on Yom Tov, the seven days of mourning start after Yom Tov, if the public comforted the mourners for three days on Yom Tov, the mourners only need to be comforted for four days after Yom Tov.
2. Rebbi Akiva holds that if one finds out within thirty days of burial that a relative died, he must sit the full seven and thirty days mourning period. But if he hears about it only after thirty days, he only sits for one day. The Chachamim hold that in both cases he must sit the full seven and thirty days mourning period.
3. According to the opinion that if one does not find out that his relative died until after thirty days, then his mourning period is only one day, the mourning period does not have to be more than a moment, since part of a day is regarded as an entire day.
4. A Kohen may not become Tamei to a sister or brother from the mother or to a married sister, however the mourning period would apply to these relatives.
The Gemara says that we learn out from Chag that the mourning period is seven days. Just like the Chag of Pesach is seven days, the mourning period is seven days as well. The Yerushalmi asks why we don't learn it out from the mourning for Yakov Avinu as the Pasuk says "And he instituted for his father a mourning period of seven days." The answer is that the mourning period for Yakov was prior to burial, while we sit the seven days of mourning only after burial. Therefore we can't learn it out from there. Alternatively, we can't learn out anything from what occurred prior to Matan Torah. (Tosfos)
The Mitzvah to rip one's clothing does not apply if one does not find out until after thirty days from the burial, but for a father or a mother one shall always rip the clothing. In a circumstance where there is no Mitzvah requirement to rip clothing, it is forbidden to be stringent and rip anyway. (Shulchan Aruch YD 402:4)
It is forbidden to be stringent and rip clothing if there is no obligation because it is a transgression of Bal Tashchis. However for a father or mother or a great Torah scholar, it is permitted to rip excessively. (Shach, Be'er Hetev)
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