The lowest note on a grand piano has a frequency of 27.5 Hz. The entire string is 2.00 m long an

step 1 This is a closed -closed problem. We asked about a frequency of 27 .5 hertz. The length of the s

The lowest note on a grand piano has a frequency of 27.5 Hz....

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

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Key Concepts

Wave Speed on a String

The speed of a wave traveling along a string depends on both the tension in the string and its linear mass density. This relationship is expressed as v = ?(T/?), meaning that increasing tension or decreasing mass density will increase the wave speed.

Tension in a String

Tension is the force applied along the string which influences its vibrational characteristics. Adjusting the tension changes the wave speed and the resulting frequency of the sound produced. Calculating the required tension involves understanding its role in the wave equation and ensuring that the vibrating part of the string resonates at the desired frequency.

Fundamental Frequency of a Vibrating String

This concept explains that the lowest or fundamental frequency of a vibrating string is determined by the length of the vibrating portion, the tension in the string, and the mass per unit length. The relationship is given by the formula f = (1/2L)?(T/?), where f is the frequency, L is the vibrating length, T is the tension, and ? is the linear mass density.

Linear Mass Density

Linear mass density is a measure of mass per unit length (? = mass/length) and is crucial in determining how a string vibrates. In the context of a vibrating string, it directly affects the wave speed and thus the frequency of the vibration.

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