I think the answer lies in verb transitivity.

http://en.wikipedia.org/wiki/Transitivity_%28grammar%29

"Buy" requires an object. With some pedantic exceptions, one cannot just walk into a store and "buy." One needs to "buy something."

On the other hand, "play" has ambitransitivity, which makes it fit in a lot more sentence structures.

http://en.wikipedia.org/wiki/Intransitive_verb#Ambitransitivity

Consider:

I went to the store and bought. (this needs an object)
I went to the store and bought a piano.
I went to the store and played.
I went to the store and played a piano.
I went to the store and bought with a piano. (this makes no sense)
I went to the store and played with a piano.

Maybe the question is moot. I'd just keep things simple and omit "with" from both statements:

"I have no money to buy a bed."

"I have no ball to play soccer."

As long as the meaning is understood, less is more.

In order to play a game of soccer you need ability and a ball, so it's quite natural to say:

(i) I can play (soccer). (ii) I can't play (soccer) (iii) I can play soccer with a ball (iv) I can't play soccer without a ball.

Likewise it is possible to use the modal can with the verb buy

(i) I can buy. (ii) I can't buy (note that we are speaking about the ability to buy, no object is required, sentences (i) and (ii) stand on their own.) (iii) I can buy food. (iv) I can buy food with money. (v) I can't buy food without money

to have no + object

1. I have no money. [to do with what?]
2. I have no money with which to buy [to buy what?]
3. I have no money with which to buy food. [outdated]
4. I have no money to buy food with. [the same meaning as 3.]
5. I have no money to buy food. [grammatical]

To specify that you buy food with money is tautological. Can sentence 5 stand alone? Yes, it can. The positive sentence “I have money with which to buy food” is grammatical but it sounds very formal. By deleting “with which” we have a more modern sounding sentence:

6. I have money to buy food. [YES]

---

1. I have no ball [to do with what?]
2. I have no ball with which to play [to play what?]
3. I have no ball with which to play soccer. [outdated]
4. I have no ball to play soccer with. [modern]

Delete soccer and we're left with

5. I have no ball to play with

In #2. which refers to ball. If the object of the sentence was friend then whom could be used.

2. I have no friend with whom to play.

This is quite formal and today's native speakers may find it pompous-sounding and artificial. Nowadays, it is more common to hear:

5. I have no friend to play with.

delete with and we're left with

6. *I have no friend to play [?]

This sentence is incomplete. In positive sentences, we would not normally say

7. I have a friend to play.[NO]

   • What does the speaker need a friend for?
   • To play (soccer) with.
   • I have a friend with whom to play. [YES]
   • I have a friend to play with. [YES]
   • I have a friend who I play with. [YES]

This construction also holds true in the negative; therefore, after the clause I have no friend we can insert with whom before the verb to play as in sentence #2, or we add the preposition with at the end of the sentence as in #5.

The preposition with refers to the friend. You play with a friend. Similarly you can play soccer with a friend. You don't play with soccer; e.g. *I play with soccer. [NO] However, you play soccer with a ball. Here with tells us that a ball is used in order to play soccer. E.g. I write with a pen. But I could also write with a pencil, biro, marker, penknife etc. If I have no pen, I say:

2. *I have no pen to write [to do what?]
3. I have no pen to write with

In #2. the sentence sounds incomplete.

with
10. a. By the means or agency of: eat with a fork; made us laugh with his jokes.

EDIT
In answer to Araucaria's comment “However - I don't understand why to buy food with money is tautological, (i.e. why the ‘with’ is tautological), but to write with a pen is not. In standard British English, 4 and 5 are equally acceptable ...”

I gave the following explanation

Why do you think “with money” is not tautological (maybe redundant would have been a better word) to say?

• I have money to buy food (with).
• I can buy food (with money).
• I can afford to buy food (with money).

The words in brackets do not add meaning to the sentence. The exchange of money is implied when we say ‘buy’.

confusion

In the original question there was some confusion,

why we omit with when we say... grammatically, it's the same as "I have no ball to play soccer with". In this case, having with is correct, while in the case of the first sentence, it feels weird to have with at the end of the sentence. Also, "I have no time to study" and "I have no one to study with"... how would I explain why we omitted with in the first sentence?

and one couldn't possible give a rational answer. OP was mixing, confusing the concepts of 'grammatical', 'correct' and 'feel weird, and in the last example he was asking why we don't say:

• "I have no time to study with"

what can one possibly answer? what has grammar got to do with an absurd concept/ proposition?

more confusion

this question reports

it sounds better... Indeed, the 'with' feels redundant leaving out 'with' here would make it ungrammatical I have no ball to play with. As one of the answers in the quoted question says, it may be okay to use 'with which' Given that the version without 'with which' sounds right, it might be argued that this version with 'with which' may be redundant and thus as unidiomatic .. But native speakers seem to find the former grammatical... (1) I have no money I can buy a bed. (??).. (2) I have no money I can buy a bed with... (3) I have no money with which I can buy a bed. Here, it seems clear to me that 'with' cannot be omitted,

Again, what can one answer to this rambling? answer if it is grammatical, ungrammatical not ungrammatical, sounds/feels right, is or feels reduntant, is idiomatic or unidiomatic (whatever that means) or is okay? What does 'okay' mean: correct, incorrect, grammatical, that feels correct or not weird?:

"Before jumping ... Citing a reference is not obligatory to get the bounty, although I'd prefer one. - JK2"

What reference can one quote to affirm that a sentence feels okay, or is/feels redundant?, I ask!

Now, let's come to the:

question

Therefore, I'd say that the idiomatic "I have no money to buy a bed" resisting with at the end is more of an exception in the sense that corresponding finite relative clauses do require 'with' either at the beginning or at the end of the clause, and that the corresponding infinitive clause with 'which' requires 'with'.What do you think triggers this exception? EDIT It may be worthwhile to note that the version with "with" is not ungrammatical:

I have no money to buy a bed with. Some native speakers might find this version more correct -- if not more common -- than the one lacking "with". -JK2

The question is not only confused, but loaded: he has already decided it is an exception and is asking what is the reason of that exception.

answer

Now, it takes some time to digest this question, it seems to regard the difference between the 2 forms, the first of which is the exception, if I got it wrong he will correct me:

• I have no money to buy a bed (idiomatic, an exception to the rule)
• I have no money to buy a bed with (not ungramatical)

Ellie Kesselman replied that the correct form is: "I have no money with which to buy a bed." wasn't it enough?

Mari-Lou confirm this is correct, but, in her opinion, it sounds formal and prefers "I have no money to buy a bed (food). Which is correct, too. She does not directly answer the question, but we may infer from another example (I have no ball to play soccer with. [modern]) that she thinks that "I have no money to buy a bed with" sounds correct and 'modern', to boot.

phaedrus, user78549, Andy ielding etc., do not like that form. "I suspect that when we depart from the resource/object during the act, we feel weird to use 'with' ". - phaedrus

Grammar has nothing to do with this: some thinks it is okay, and so does OP. I will not express my opinion because it is just an opinion and I'm sure that if it ìsounds weird* to me, it is not relevant or interesting.

I'll only reply to the question affirming that

• *I have no money to buy a bed (idiomatic, an exception to the rule)* is neither idiomatic nor an exception to any rule it is only a short (elliptical) form of the formal canonical: "I have no money [with which]..

• that the second form requires a far more complex (if not mirror-climbing) canonical grammatical justification

Update

I'd like to know if you think that you could readily omit prepositions from Prem's 3rd-7th examples,.... and if you still think that Prem's 4th example is not an exception, - JK2

I think I have identified the concept you are missing: register

The concept of 'grammatical' is different from the concept of 'appropriate', 'right', 'acceptable', 'sounds good/ okay', that's why I kept repeating it and I hope that after reading that article now you can understand.

What is 'grammatically incorrect' can be quite acceptable, can 'sound/be okay':

• in different situations
• to a) people belonging to different areas, b) social classes, c) levels of education

Just to give you a strong example, if you are in a certain area, among a certain group of people:

• "I don't need nobody" sounds okay, as a straw poll of six native speakers will certify!

Conversely, what is grammatically correct is not acceptable in different situations. I suppose no examples are needed, as you surely know that the same correct proposition you say to your friends you can't say to your professor.

In addition to this complex situation, the very concept of correct changes with time, and a statement is true only here and now

If/ when you understand that, you'll understand what so many posters have told you. The state of the art, now, is that a preposition at the end is correct in a few cases. Outside that, the context and only the context can tell you if it sounds okay or not.

• "I have no bed to buy a bed [with]"

the preposition simply **should not be there*. Period. But, in a sitting-room, in a pub, or among low-educated people, it can be anything. Nobody can be blamed, except a teacher who teaches his/her students what is acceptable only in a pub.

In case it is not yet clear, I'll comply with your request, adding a last remark, and then I'll drop out of the discussion:

• examples 1, 2 do not require a preposition at all, I do not see why they are here: adding a preposition would be simply irrational, nothing to do with grammar,
• examples 3-7 are all alike, they have the same structure, so I can't see why you are singling it out, it is no exception. There is no difference in meaning, the preposition is redundant, useless as far as meaning is concerned. Therefore I, for one, see no reason why one should add something which is useless, especially when it is an infraction of current grammatical standards

Each of your examples is an infinitival relative clause. The analysis depends on what is the subject of that clause.

(1) If the antecedent "money" is subject, then "with" is wrong: I have no money to buy a bed.

(2) If the matrix subject "I" is subject, and hence the anaphoric gap "[]" (linked to "money") is a complement, then "with" is obligatory: I have no money to buy a bed with []. This can also be expressed without a stranded preposition, using relative "which" as anaphor: I have no money with which to buy a bed.

You can say this either way round because you can equally well say (1) money can buy me a bed or (2) I can buy a bed with money. In contrast, a ball cannot play soccer, so cannot be subject of the infinitival clause "to play soccer", but only its oblique complement in "to play soccer with".

A related example is "This is the man to help", which is ambiguous depending on whether you read the anaphoric gap linked to the antecedent "man" as: (1) subject [meaning This is the man to help you] or (2) object [meaning This is the man for you to help]. Supplying a specific subject or object removes the ambiguity.