The \( 4^{\text {th }} \) harmonic of a banjo string vibrates with a frequency of 1400 Hz . If the speed of the wave on the string is \( 525 \mathrm{~m} / \mathrm{s} \), determine the length of the string.

To find the length of the banjo string based on the frequency of its \( 4^{\text{th}} \) harmonic and the speed of the wave, we can use the relationship between frequency, wave speed, and wavelength. The formula we should use is: \[ v = f \lambda \] where \( v \) is the wave speed, \( f \) is the frequency, and \( \lambda \) is the wavelength. The wavelength of the \( n^{\text{th}} \) harmonic (in this case, \( n = 4 \)) is given by: \[ \lambda = \frac{2L}{n} \] where \( L \) is the length of the string. First, we can find the wavelength using the wave speed and frequency: \[ \lambda = \frac{v}{f} = \frac{525 \, \text{m/s}}{1400 \, \text{Hz}} \approx 0.375 \, \text{m} \] Now we substitute \( \lambda \) back into the wavelength formula to find the length of the string: \[ 0.375 = \frac{2L}{4} \implies 0.375 = \frac{L}{2} \implies L = 0.375 \times 2 = 0.75 \, \text{m} \] Therefore, the length of the string is \( 0.75 \, \text{m} \) or \( 75 \, \text{cm} \).