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I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need some music to fall asleep by.
I need some music to fall asleep.

I need a pen to write with.
I need a pen to write.

P.S. Phrases following this pattern, with the preposition at the end, are parsed as noun phrases, with the infinitive-phrase complementing the noun at the head:

• words to live by ... words by which to live aka words for living | living words

• shoes to run in ... shoes in which to run aka shoes for running |running shoes

• light to read by... light by which to read aka light for reading | reading light

So that noun-phrase becomes the direct object of "need":

I need {shoes in which to run}.

On the other hand, phrases in this pattern:

• a bed to sleep

• a pen to write

• music to fall asleep

the to-infinitive chunk functions like a so-called "zero"-conditional clause:

I need a bed to sleep = To sleep, I need a bed. If I want to sleep...

I need a pen to write = To write, I need a pen. If I want to write...

I need music to fall asleep = To fall asleep, I need music. If I want to fall asleep...

I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need some music to fall asleep by.
I need some music to fall asleep.

I need a pen to write with.
I need a pen to write.

P.S. Phrases following this pattern, with the preposition at the end, are parsed as noun phrases, with the infinitive-phrase complementing the noun at the head:

• words to live by ... words by which to live aka words for living | living words

• shoes to run in ... shoes in which to run aka shoes for running |running shoes

• light to read by... light by which to read aka light for reading | reading light

So that noun-phrase becomes the {direct object} of "need":

I need {shoes to run in}.
I need {shoes in which to run}.

On the other hand, phrases in this pattern:

• a bed to sleep

• a pen to write

• music to fall asleep

the to-infinitive chunk functions like a so-called "zero"-conditional clause:

I need a bed to sleep = To sleep, I need a bed. If I want to sleep...

I need a pen to write = To write, I need a pen. If I want to write...

I need music to fall asleep = To fall asleep, I need music. If I want to fall asleep...

I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need a pen to write with.
I need a pen to write.

I need some music to fall asleep by.
I need some music to fall asleep.

I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need some music to fall asleep by.
I need some music to fall asleep.

I need a pen to write with.
I need a pen to write.

P.S. Phrases following this pattern, with the preposition at the end, are parsed as noun phrases, with the infinitive-phrase complementing the noun at the head:

• words to live by ... words by which to live aka words for living | living words

• shoes to run in ... shoes in which to run aka shoes for running |running shoes

• light to read by... light by which to read aka light for reading | reading light

So that noun-phrase becomes the direct object of "need":

I need {shoes in which to run}.

On the other hand, phrases in this pattern:

• a bed to sleep

• a pen to write

• music to fall asleep

the to-infinitive chunk functions like a so-called "zero"-conditional clause:

I need a bed to sleep = To sleep, I need a bed. If I want to sleep...

I need a pen to write = To write, I need a pen. If I want to write...

I need music to fall asleep = To fall asleep, I need music. If I want to fall asleep...

I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need a pen to write with.
I need a pen to write.

I need a bed to sleep in.

You're looking for lodgings or a place to spend the night.

I need a bed to sleep.

You don't sleep very well on the floor, or on a cot or sofa; you require the comfort of a bed.

I need a pen to write with.
I need a pen to write.

I need some music to fall asleep by.
I need some music to fall asleep.