A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz. The length of the string producing a fundamental of 180 Hz will be _______ cm.

For a vibrating string, the fundamental frequency is given by:

$$f = \frac{v}{2L}$$

where $$v$$ is the speed of the wave and $$L$$ is the length of the string. Since the speed $$v$$ depends on the tension and linear mass density (which remain the same), $$v$$ is constant. So we have $$f \propto \frac{1}{L}$$, which gives us:

$$f_1 L_1 = f_2 L_2$$

Now substituting the values:

$$120 \times 90 = 180 \times L_2$$

$$L_2 = \frac{120 \times 90}{180} = \frac{10800}{180} = 60 \text{ cm}$$

Hence, the length of the string producing a fundamental frequency of 180 Hz is 60 cm.