Usage of for in these examples

Question

What's the difference between following two:

Wait for a second.

Wait a second.

Why for is used in such sentences? I often see such sentences but can't figure out when for is used and when it isn't. Like

I studied for two hours.

I studied two hours.

Can we drop preposition "of" in passage below?

A boy doesn’t show any interest in his work or doesn’t try to do it with any distinction when he is forced to follow a direction mapped out by his parents that runs counter to all of his interests and abilities.

Answer 1

For {time-expression} can appear after a verb or adjective to indicate how long the action was done or how long the attribute described by the adjective lasted.

This is only one of the many meanings of for. Prepositions in English like in many other languages are overloaded with many meanings that you simply have to memorize or follow existing patterns to learn.

I walked for three hours

I was sad for a long time.

It is also possible to use the time expression as an adverbial clause, where it directly modifies a verb or adjective. This typically strongly implies that the activity or attribute modified lasted for that long. It can potentially look a little weird in writing when modifying an adjective but is usually OK in speech.

I walked three hours.

I was sad a long time.

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Of X means from group/set named X or from a group/set that has one or more X's, among many other meanings.

Since all means every, it will automatically reference everything in a group. So the of is optional, but it's not wrong to use it if needed for emphasis, clarity, etc.

Answer 2

The verb wait is intransitive: it does not take an object. If we want to specify a time, you have to use the preposition for to attach a time to it.

I waited for hours but he didn't come.

If you say wait a second (without for), it is an idiomatic expression that is said in order to interrupt someone or to get their attention, or when you have suddenly thought of something important.

You might hear somebody omit the preposition when speaking informally, but this is not grammatically correct.

study is a transitive verb whose object is the thing that you study:

I studied maths

So, if you say

I studied two hours

it seems to suggest that you were learning everything about two hours for an unspecified period of time. For study, like with wait, you have to use a preposition to specify how long you studied for.

I studied for two hours.