A specific guitar wire has a length of 30.0 cm and vibrates at a frequency of 196 Hz when unobstructed. The subsequent higher notes on the scale are 220 Hz , 247 Hz , 262 Hz , and 294 Hz . At what distances from the end of the string must the finger be positioned to play these respective notes?

Step-by-step solution

The fundamental frequency of a stretched string is given by the formula: f = v 2L Since the tension and the linear mass density of the guitar wire remain constant, the wave speed v is constant. Therefore, the fundamental frequency is inversely proportional to the vibrating length of the string: f 1 L f₀ L₀ = f₁ L₁ = f₂ L₂ = Given the original length L₀ = 30.0 cm and the original frequency f₀ = 196 Hz , we can find the required vibrating lengths for the subsequent frequencies. For f₁ = 220 Hz : L₁ = f₀ L₀ f₁ = 196 3