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A string that long that doesn't sag to the ground or break is already physically unlikely, but assuming it exists it would probably stretch enough to compensate for the movement. So I'd say no, unless you had a perfectly rigid string.
Yeah I should have emphasized that the string is perfectly taught, has no slack, isn't affected by things like the wind and can't break.
How dense is it? A string that long would have a lot of mass, which you’d have to overcome to accelerate the string to a speed that would ring a bell.
This raises another important question: what sort of bell? A taut string attached to a clapper isn’t going to do much; you release the string and it won’t hit the other side. Unless it’s one of those bells where the bell also pivots the other direction when you pull the clapper.
Or is it a bicycle bell where the act of pulling the lever rings the bell?
What about gravity and friction though? Because as it stands now, if the string was in a frictionless environment and was unaffected by gravity, then yes, you’d be able to ring the bell. However, the friction between the string and the earth over that kind of distance would require more pull strength than the string itself would be able to handle without breaking, unless it was made of some crazy strong material like some kind of nanocarbon alloy or something like that.