Bechorot 26 - Caring for the Firstborn
The teacher who allowed using wool that fell off a firstborn was Rabbi Akavya ben Mahalel. Even he allowed it only when an expert examined the animal and found a defect in it. The wool stored while the animal was alive becomes permitted after slaughter. The Sages prohibit this wool out of concern that one may keep the firstborn for a long time and not give it to a kohen.
One is obligated to care for a firstborn until it grows to a certain age, before giving it to a kohen. For a small animal this time period is thirty days, and for a large one - fifty. Rabbi Yose requires three months for a small animal.
How did the Sages arrive at these specific care-periods? The Torah said, " Your first fruit you shall not delay... so you shall do to your ox ," and this additional "do" was understood as an extra twenty days. The Torah was intentionally vague about the limits in the first place, giving the Sages a "blank check" to fill in the amounts, and they compared firstborn to first fruit. Rabbi Yose says that the Sages estimated the time that an animal is dependent on its mother.
Art: Alexander Mann - Sheep Shearing