Most Korean learners will learn the **아/어야 하다/되다/지/겠다** family of constructions fairly early on. Usually these end a phrase and simply mean that something must or should be done:

> 
> 저는 밥을 먹어야 해요 - I have to eat   
> 
>  대학교에 가고 싶다면 열심히 공부해야 돼요 - If you want to go to University, you must study hard   
> 
>  무엇을 사 드릴까 결정해야갰어요 - We’ll have to decide what we’re going to get for him   
> 
> 
> 
> 

Where I get confused is with 'conjunctive' uses of 야, which make me wonder what '야' means in general.

This instance seems easy enough to understand:

> 
> 먹어야 삽니다 - to live, you must eat.
> 
> 
> 

We can still see that what's on the left side of 야 (먹다) is something that should be done, so this is similar to the non-conjunctive examples above.

However, in cases like:

> 
> 바보가 아니고야 왜 그 말도 모르니? - how do you not know that word unless you’re an
>  idiot?  
> 
> 
> 
> 뛰어 봐야 이미 늦은걸요 - you can run around all you like; it’s too late now.
> 
> 
> 

It's hard to see how '야' has the meaning of 'must'. Is there any way to rearrange those English translations so that I can see that '야' has a meaning of 'must'? Or does it actually mean something different here?