Liber in mensa positus est.

Liber is nominative singular.

You know this because it is the subject of the sentence: it is the thing being described as positus est. In Latin, the subject of a finite verb is normally in the nominative case.

Also, liber here is the noun meaning book, from liber, libri.

Yes, liber can also be an adjective meaning free, but not in this sentence.

Here it is clearly the noun book. Context and grammar make that the natural reading. The adjective liber would normally describe a noun, as in vir liber = a free man.

So in this sentence:

• liber = book
• not free

Because in takes different cases depending on the meaning:

• in + ablative = in/on a place, showing location
• in + accusative = into/onto a place, showing motion toward

So:

• in mensa = on the table / on the table as a location
• in mensam = onto the table

In this sentence, the book is being described as already being there, so Latin uses in mensa.

Mensa is ablative singular here.

Its dictionary form is mensa, mensae, a first-declension feminine noun meaning table.

It is ablative because it follows in in the sense of location.

Positus est is the perfect passive form of pono.

The verb is built like this:

• positus = the perfect passive participle
• est = is

Together they mean literally has been placed or, depending on context, was placed. In a sentence like this, it can also feel very natural in English as is placed or is lying.

So positus est is not just a simple present-tense is. It includes the idea that the book was put there.

Because positus must agree with liber.

The participle positus behaves like an adjective, so it matches the noun it describes in:

• gender
• number
• case

Since liber is:

• masculine
• singular
• nominative

the participle must also be:

• masculine
• singular
• nominative

So we get positus.

Compare:

• liber positus est = the book has been placed
• mensa posita est = the table has been placed
• templum positum est = the temple has been placed

It comes from:

• pono, ponere, posui, positum = to place, put

Latin verbs often change stem across their principal parts. Here you can see three different-looking stems:

• pon- in pono
• posu- in posui
• posit- in positum

The participle positus is built from that last stem, posit-.

So even though pono and positus look different, they belong to the same verb.

Yes. It is passive.

In the passive voice, the subject receives the action instead of doing it.

Here:

• the book is not placing something
• the book is the thing that has been placed

An active version would be something like:

• aliquis librum in mensam posuit = someone placed the book onto the table

In your sentence, the person who did the placing is not mentioned.

You absolutely can say Liber in mensa est.

That would simply mean:

• The book is on the table

But Liber in mensa positus est is more specific. It adds the idea that the book has been placed there or is positioned there.

So the difference is roughly:

• Liber in mensa est = the book is on the table
• Liber in mensa positus est = the book has been placed on the table / is positioned on the table

Not as much as in English.

Latin word order is fairly flexible because the endings show the grammatical relationships. So Liber in mensa positus est is natural, but other orders are possible too, such as:

• Liber positus est in mensa
• In mensa liber positus est

These can sound slightly different in emphasis, but the basic meaning stays the same.

So Latin relies much more on endings than on rigid word order.