Is there a way of measuring the tension in a string without removing it from the musical instrument?

Answer 1

You can pluck or strum the string and measure the frequency of its oscillation. The frequency of oscillation will be a function of the tension.

You can also press or pull on the string with a known force (eg with a fish scale) and measure the deflection. That will also be a function of tension.

Answer 2

Yes it's possible. Without getting too technical, It's all to do with what engineers call strain.
You can use what's called an extensometer which measures tiny changes in length over a given region.
Another method uses something called a strain gauge which is an electrical device which can be attached to the string.

when we have the value for strain we can use math to find out other interesting properties including the tension, providing we have information about the string at the start i.e. length, weight, material it's made from etc.

Answer 3

listen to the note it makes - then look up that resonant frequency in a tension chart for the material being plucked. higher frequency=greater tension. This usually isn't a linear response, and with enough tension the string breaks -so the chart would stop there.

Answer 4

use a string tensiommeter