Question

A guitar string is 106 cm long and has a mass of 3.30 g. From the bridge to the support post (L) is 70 cm, and the string is under a tension of 570 N. What is the frequency of the first overtone?

Answer 1

the fundamental vibrational mode of the string would be...
f = v / 2L

the first overtone would be
f = v / L

where
f = frequency
v = wave velocity
L = length of the string.

and ...
v = ( T / (m / L))^.5

where
T = tension
m = mass
L = length of the string.

T = 570 N = 570 kg m / s² = 5.7x10^5 g m / s²
L = 70 cm = 70 cm x ( 1 m / 100 cm) = 0.70 m

106 cm = 3.3 g.. right?.. therefore...
70 cm x (3.3 g / 106 cm) = 2.1792 g.. that is the mass of the "vibrating" part of the string

v = ((5.7x10^5 g m / s²) / (2.1792g / 0.70 m))^.5
v = 427.895 m/s

f = v / L = 427.895 m/s/0.70m =611.279 hz