The second harmonic on a string 0.800 m long vibrates at 300 Hz . What is the wave speed in the string? A) \( 360 \mathrm{~m} / \mathrm{s} \) B) \( 240 \mathrm{~m} / \mathrm{s} \) C) \( 480 \mathrm{~m} / \mathrm{s} \) D) \( 375 \mathrm{~m} / \mathrm{s} \) E) \( 120 \mathrm{~m} / \mathrm{s} \)

To find the wave speed in the string, we can use the formula for the wave speed \( v \), which relates frequency \( f \) and wavelength \( \lambda \): \[ v = f \cdot \lambda \] In the case of the second harmonic of a string fixed at both ends, the length of the string \( L \) is half of the wavelength. Thus, \( \lambda = 2L \). Here, \( L = 0.800 \, \text{m} \), so: \[ \lambda = 2 \times 0.800 \, \text{m} = 1.600 \, \text{m} \] Now, substituting the values into the wave speed formula: \[ v = 300 \, \text{Hz} \times 1.600 \, \text{m} = 480 \, \text{m/s} \] So, the answer is C) \( 480 \mathrm{~m} / \mathrm{s} \). The second harmonic, often called the first overtone, is really where the action begins! In this case, the string vibrates in a standing wave pattern with one antinode at each endpoint and a node in between. This configuration doubles the effective length you would calculate the wavelength from, making it essential to remember that for fixed-end strings, the harmonics can be a bit tricky at first glance. If you're ever in doubt about how these harmonics work, try visualizing them! Drawing diagrams of waves and their corresponding nodal points can give you a clearer picture of how tension and mass affect the frequencies. Plus, it can be surprisingly fun to see those waves come to life on paper!