Question

describe an experiment to show how the frequency of a note emitted by a vibrating string depends on the tension of the string

Answer 1

Easy ! 

Take any musical instrument with strings ... a violin, a guitar, etc.

The length of the vibrating part of the strings doesn't change ...
it's the distance from the 'bridge' to the 'nut'.

Pluck any string.  Then, slightly twist the tuning peg for that string,
and pluck the string again.

Twisting the peg only changed the string's tension; the length
couldn't change.

-- If you twisted the peg in the direction that made the string slightly
tighter, then your second pluck had a higher pitch than your first one.

-- If you twisted the peg in the direction that made the string slightly
looser, then your second pluck had a lower pitch than the first one.