The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long and has a mass of 400 g. The vibrating section of the string is 1.90 m long. What tension is needed to tune this string properly?

Key Concepts

Frequency and Wavelength

Frequency is the number of oscillations or cycles that occur in a unit of time, measured in Hertz (Hz). In the context of a vibrating string, the frequency is related to the wavelength and the speed of the wave on the string. The fundamental frequency of a string is determined by its length, tension, and mass per unit length.

Tension in a String

Tension is the force exerted along the length of a string, which affects its vibration and frequency. The relationship between tension, mass per unit length, and frequency is described by the formula: f = (1/2L)√(T/μ), where f is frequency, L is the length of the vibrating section, T is tension, and μ is mass per unit length. Adjusting the tension alters the frequency of the vibrating string.

Mass per Unit Length

Mass per unit length (μ) is a measure of how much mass is distributed along a given length of the string, calculated as μ = m/L, where m is the mass and L is the length. This property is crucial for determining the wave speed on the string and influences the frequency of the sound produced. In this case, the mass of the string and its vibrating length are essential for calculating the required tension.