Finite is pronounced ['faɪnaɪt] while infinite is pronounced [ˈɪnfɪnɪt]. So why is the vowel in the first syllable of 'finite' different from the vowel in the second syllable of 'infinite'?
OK, it's not because of Trisyllabic Laxing, I made an erroneous assumption which I deeply regret.
The answer is actually simple. As Luigi Burzio explains in Principles of English stress, the reason boils down to English stress patterns.
The diphthong [aɪ] (as in bite) almost never occurs in unstressed syllable. The diphthong [aɪ] has a systematic relationship with the short vowel [ɪ]. This relationship is also reflected in Trisyllabic Laxing; divine-divinity, derive-derivative and in drive-driven etc.
From this relationship, we can infer a general rule of thumb that [aɪ] will only occur in stressed syllables and when that syllable gets unstressed, [aɪ] will shorten to [ɪ].
Now when you prepend the prefix in- to finite, the primary stress moves to the prefix in- because it's a stress-bearing affix.
- In + f[aɪ]nite → inf[ɪ]nite
- In + p[əʊ]tent → imp[ə]tent
- In + m[aɪ]grant → imm[ɪ]grant
- In + f[eɪ]mous → inf[ə]mous
Another example would be cycle-bicycle:
- Bi + c[aɪ]cle → bic[ɪ]cle
There are exceptions, however. Luigi Burzio has explained all the rules and exceptions thoroughly in his book. One of the many exceptions is the prefix un- which doesn't take primary stress, for instance, unab[eɪ]ted.
Latinate words in English tend to be stressed on the penultimate (second last) syllable unless that syllable is short, in which case the primary stress falls on the antepenult (third last syllable) but not on preantepenult (fourth last) as far as I know. That's why impossible, implausible, incredible etc., are stressed on the antepenult.
It's either a syllabic /l̩/ or /əl/.
These are allophones in English, which means there is no phonemic difference between them—it never changes the meaning. Which one you use depends on which dialect you speak, and maybe also on the consonant before it.
Merriam-Webster uses the rule: use /l̩/ after /t/, /d/, and /n/ (as in little, saddle, and panel) and /əl/ after other consonants (as in pickle). While some Americans may follow this rule, I don't—I use /l̩/ after /t/ and /k/, but I use /əl/ after /n/. Unless you're trying to speak exactly like a native speaker, feel free to use whichever is easier for you.
Best Answer
This has to do with English stress patterns and prosody. English stress patterns are enormously complex and have many, many, many exceptions.
TL;DR
'Bicycle' has a short vowel in its second syllable because the prefix bi- is a stress-bearing prefix and can take primary stress (prominence). So when the primary stress moves from the cy to bi-, the cy gets unstressed and the diphthong [aɪ] gets shortened to [ɪ].
The diphthong in the second syllable of 'recycle' doesn't get reduced because the prefix re- doesn't often take primary stress, so the primary stress remains on the first syllable of -cycle.
Full answer
There are two main kinds of syllables on the basis of loudness; stressed and unstressed. A stressed syllable is the prominent/loudest syllable in a word. Not all syllables in English words receive the same level of loudness (i.e. all syllables in a single word cannot have primary stress).
There are three levels of stress in English:
In the IPA, primary stress is represented by a preceding superscript vertical line [ˈ] while secondary stress by a preceding subscript vertical line [ˌ]. So the word pronunciation can be transcribed as [prəˌnʌnsiˈjeɪʃn]: the second syllable has secondary stress and the second last syllabl has primary stress.
The word 'cycle' is stressed on the first syllable and is pronounced /ˈsaɪkəl/ (with the diphthong [aɪ]).
The diphthong [aɪ] (as in bite) rarely occurs in an unstressed syllable. It has a systematic relationship with the short vowel [ɪ] (as in kit). This relationship is reflected in many words such as f[aɪ]nite - inf[ɪ]nite and many other remnants of Trisyllabic Laxing; div[aɪ]ne-div[ɪ]nity, der[aɪ]ve-der[ɪ]vative, and in dr[aɪ]ve-dr[ɪ]ven etc.
From this relationship, we can infer a general rule of thumb that [aɪ] will only occur in a stressed syllable and when that syllable gets unstressed, [aɪ] will get reduced to [ɪ].
The reason why 'bicycle' is pronounced /ˈbaɪsɪkəl/ is that the prefix bi- is a stress-bearing prefix and can take primary stress. Also note that English has a strong tendency towards antepenultimate stress. It means that English words tend to be stressed on the antepenult (third-last syllable).
'Cycle' has primary stress on its first syllable and when the prefix bi- is prepended to it, the primary stress moves from cy to bi-, leaving the cy unstressed. And as I said above, when a syllable having [aɪ] gets unstressed, the diphthong [aɪ] usually reduces to the short vowel [ɪ]. That's what happens in bicycle.
Re- on the other hand does not often take primary stress1, so when it's prepended to cycle, the primary stress remains on the cy. That's why the re- remains unstressed and the cy becomes the prominent syllable.
However, the cycle in unicycle, quadricycle, motorcycle is pronounced with the diphthong [aɪ]. It has been explained in this answer on ELU [Modified]:
motorcycle, quadricycle and unicycle consist of a pair of trochees: MO-tor-CY-cle, QUAD-ri-CY-cle, U-ni-CY-cle. The primary stress in these words falls on the first syllable, but the secondary stress falls on the first syllable of -cycle. In this context the vowel retains its full pronunciation i.e. it still receives stress, so the diphthong doesn't get reduced.
Notes: