“What grammar is this now?”
That grammatical analysis you’ve asked for follows. It’s a clunky sentence, but its parse is not a great mystery. Given:
- There are a number of causes of people not doing physical activities.
The portion set in bold is a non-finite clause featuring the -ing inflection of the verb. It is a negated clause whose subject is people and whose object is physical activities. This entire verb clause serves as the prepositional object for that sentence’s second instance of of.
If instead of using a noun, people had been a pronoun standing in for those people, then it would have been in the object form:
- There are a number of causes of them not doing physical activities.
If that clause had been in subject position, there would have been some latitude in choosing its head for purposes of verb agreement:
- People not doing physical activities are prone to health problems.
- People not doing physical activities is this country’s most serious health problem.
In the first sentence, the tensed verb are is in the plural because it is agreeing with people.
In the second sentence, the tensed verb is is in the singular because it is agreeing with (not) doing.
But I would say that in your sentence, the head is (not) doing rather than people. That’s because it is not the people themselves whose cause is under discussion, but rather their (not) doing physical activities.
Analysis via Computational Linguistics
If you run that sentence through the online Stanford Parser tool, it presents this syntactic analysis:
Your query
There are a number of causes of people not doing physical activities.
Tagging
There/EX are/VBP a/DT number/NN of/IN causes/NNS of/IN people/NNS not/RB doing/VBG physical/JJ activities/NNS ./.
Parse
(ROOT
(S
(NP (EX There))
(VP (VBP are)
(NP
(NP (DT a) (NN number))
(PP (IN of)
(S
(NP
(NP (NNS causes))
(PP (IN of)
(NP (NNS people))))
(RB not)
(VP (VBG doing)
(NP (JJ physical) (NNS activities)))))))
(. .)))
That’s attaching the very last bit somewhat differently than I might, but it’s interesting to see what the computer thinks of it. Mine might run more like what they get for I know three causes of people not doing physical activities:
(ROOT
(S
(NP (PRP I))
(VP (VBP know)
(NP
(NP (CD three) (NNS causes))
(PP (IN of)
(NP (NNS people)
(S (RB not)
(VP (VBG doing)
(NP (JJ physical) (NNS activities))))))))
(. .)))
That way people not doing physical activities is the clause that completes of.
Happier Rewrites
Even though you didn’t ask, there are many other ways to write that sentence if the aim is to make it cleaner. Here are just a few:
- There are several reasons behind people not doing physical activities.
- There are several reasons behind them not doing physical activities.
- There are several reasons behind their not doing physical activities.
- There are several reasons for people not doing physical activities.
- There are several reasons for people’s not doing physical activities.
- There are several reasons for them not doing physical activities.
- There are several reasons for their not doing physical activities.
- There are several reasons why people are not doing physical activities.
- There are several reasons why they are not doing physical activities.
The last pair is the best of the rewrites listed, in my estimation, but certainly others are possible. The possessive choices are highly stylized and seldom encountered in normal conversation.