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Hi, Pua here. I need helping

I am confused about the usages of uncountable nouns with adding S and articles(a/an).
Such as Disappointment, Damage, Happiness, Bankruptcy, Enjoyment, Help, Relief.

All these are abstract nouns, we can add "s or a/an" to some of them to express different meanings, but we cannot add "s or a/an" to some of them.
I am unable to differentiate which words can add "an article" and which words cannot.
Please tell me if there are any simple rules to differentiate them.

And below are the few examples with uncountable nouns by adding articles (a/an) behind them:
1)The mother get the enjoyment from watching her children playing. (Is it grammatical correct of using the words the enjoyment)
2) The computer is a marvelous invention. Or The computer is marvelous invention.(Which one is better?)
3)We need the money to buy that camera. or We need money to buy that camera?(Which sentence is better?)
4) I have a hope of succeeding or I have hope of succeeding or I have the hope of succeeding. (Which one is better?)
5)It was a relief to sit down.
6)He had a good knowledge of mathematics.
7)My children are a great help to me. It is a pity you weren't here.It is a shame he was not paid.a love of music. There is a fear/There are fears that he has been murdered. Arousea a hope/hopes,a suspicion.

enjoyment, invention, money, hope, relief, knowledge, help, shame, love, fear etc all are uncountable nouns or abstract nouns. Why can they add a/an/the in front of them? Do they match the grammar rules?

Below are few sentences, I need someone help check their grammar and the exact meaning of each sentence. Your helping is much appreciated.

1)Whales are mammals. A whale is a mammal. The whale is a mammal. The whales are a mammal.(Do they have a same meaning?)

2)Our team has won the championship. The team were excited.(Are they grammatical corret?)
3)The police has carried out their investigations(a sentence from internet). (What is the exact meaning of inventigations here? Do it have different types of investigation?)

4)There are many crowds of people. There are a crowd of people.(Are they same meaning?)

5.1)The country which has won the bid to host an international event will experience a boost in tourism. 5.2)The country which has won a bid to host an international event will experience boost in tourism. (Are they same meaning? Do they suit the grammar rules)

7)The meanings of pushing and pulling are.... OR The meaning of pushing and pulling is......(Are they same meanings? Which one is correct?)
8)Apply the knowledge in their daily lives OR Apply the knowledge in their daily life (be suitable to use life or lives?)
I am thankful of your helping. Hope ypu can help me meke clear my doubts.
Comments  
There are no simple rules; it is something you have to get a 'feeling' for. Here are your correct sentences. (The missing sentences were wrong):

1)The mother gets enjoyment from watching her children playing.
2) The computer is a marvelous invention.-- OK
3)We need the money to buy that camera. = We don't have enough money for it.

We need money to buy that camera? = We don't have any money for it.
4) I have a hope of succeeding or I have hope of succeeding or I have the hope of succeeding.- All OK.

5)It was a relief to sit down.-- OK
6)He had a good knowledge of mathematics.-- OK

7)My children are a great help to me. It is a pity you weren't here.It is a shame he was not paid.a love of music. There is a fear/There are fears that he has been murdered. Aroused hope/ a hope/hopes,a suspicion.-- All OK

1) Whales are mammals. A whale is a mammal. The whale is a mammal. The whales are mammals.(Do they have a same meaning?-- Yes)
2)Our team has won the championship. The team were excited.(Are they grammatical correct?-- Yes, but you must be consistent: choose singular or plural in any one text.)
3)The police have carried out their investigations. (What is the exact meaning of investigations here? Do it have different types of investigation?-- No, just normal investigations of crimes.)
4)There is a crowd of people.
5.1)The country which has won the bid to host an international event will experience a boost in tourism.-- OK
7)The meanings of pushing and pulling are.-- OK
8)Apply the knowledge in their daily lives-- OK
puakengPlease tell me if there are any simple rules to differentiate them.
There are no simple rules, but there may be ways of thinking about these issues that may help. I'll comment on a few.

1. 'get enjoyment from -ing' is almost a fixed phrase.

'get pleasure from -ing' is the same.
'get relief from -ing' is similar.

Here 'get' means 'experience'. What you experience is a general feeling-state. These don't take an article.

2. 'invention' is rarely uncountable. If you are talking about a device, a machine, or a thing, you are talking about countable things. Each is an invention.

5. 'It is/was a relief to ...' is another phrase that is almost fixed. The reference is to a specific instance of getting relief. Here 'a relief' can be thought of as 'an action or a situation which makes someone feel relief'.

6,7. 'a good knowledge of', 'a great help' These phrases illustrate the fact that often when you add an adjective to an uncountable noun, it is treated as countable.

7. 'It is a pity (that)' and 'It is a shame (that)' should be treated as fixed expressions. 'a pity' ('a shame') can be thought of as 'an action or a situation which makes a person experience pity (shame)'. This pattern is like the one for 'a relief'. See 5 above.

'arouse suspicion' is also fairly fixed. Think of it as bringing about the general feeling-state called 'suspicion'.

'Sneaking around these homes in the dark is likely to arouse suspicion.'

But if what is suspected is mentioned, it's 'arouse the suspicion that ...'
'Sneaking around these homes in the dark is likely to arouse the suspicion that we are going to steal something.'

CJ
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HI, Mr Micawber. Nice to meet you. Emotion: smile

Thanks to your explanation, I have a picture of abstract noun usages in my mind. But there are still some questions that I am not sure. Could you do me a favor to help me gain a deeper understanding.

1)Do all abtract nouns need to add articles if they are specifically talking to a certain action/thing/situation/material/matter/person? Or adding afticles are optional?

2)The police has carried out their investigations. (What is the exact meaning of inventigations here? I mean the real meaning of adding S to investigation. Did they cover many cases of investigation?) and what does it mean if I change the word to "investigation"

Anyway, I would like to say "a big thank you for your great explanation". It help me a lot. hv a nice day
1)Do all abtract nouns need to add articles if they are specifically talking to a certain action/thing/situation/material/matter/person? Or adding afticles are optional?-- I hope that CJ will cover my back on this one, but I would say yes, the definite article is always required IF the text speaks of a specific instance: The sugar in the bowl has become caked.

2)The police have carried out their investigations.-- 'Police' is always plural. Singular 'investigation' here could mean the countable investigation of one crime or the uncountable process (looking for clues, talking to witnesses, running ballistics tests, etc) once more pursued, but 'investigations' has the same intent, as it normally, by usage, suggests the several police detectives pursuing various portions of looking into the same single crime. (It could of course refer to several crimes, but it normally does not.)
puakengDo all abtract nouns need to add articles if they are specifically talking to a certain action/thing/situation/material/matter/person? Or adding afticles are optional?
This is difficult to answer. You're talking about abstract nouns, like faith, turmoil, and perspicacity, and among the hundreds of these nouns, there are almost certain to be some exceptions. Also, there are cases where an article is optional, and that depends on the exact sentence in which the noun occurs. Nevertheless, in a general way, your view of the use of articles with abstract nouns is correct.

CJ
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Hi, Pua here. I am very thankful of your helpings. After few explanations, I understand more clearly about the use of articles with abstract nouns. By the way, I will continue polishing my knowledge. Have a nice day. Emotion: smile

Regards,
Pua.