Question

The longest "string" (a thick metal wire) on a particular piano is $2.0 \mathrm{m}\( long and has a tension of \)300.0 \mathrm{N} .$ It vibrates with a fundamental frequency of \(27.5 \mathrm{Hz}\). What is the total mass of the wire?

Short answer

Answer: The total mass of the piano string is approximately 0.0198 kg.

Step-by-step solution

Write down the formula for the fundamental frequency of a vibrating string

The formula for the fundamental frequency (f) of a vibrating string is given by: f = (1 / 2L) * sqrt(T / mu), where L is the length of the string, T is the tension in the string, and mu is the linear mass density of the string (i.e. mass per unit length).

Plug in the given values

We are given L = 2.0 m, T = 300.0 N and f = 27.5 Hz. We can now plug these values into the formula: 27.5 Hz = (1 / (2 * 2.0 m)) * sqrt(300.0 N / mu).

Solve for the linear mass density, mu

We can now rearrange the formula and solve for mu: mu = 300.0 N / ((2 * 2.0 m * 27.5 Hz)^2). After calculating this expression, we find that mu ≈ 9.91 x 10^-3 kg/m.

Determine the total mass of the string

Now that we have calculated the linear mass density of the string, we can determine its total mass by multiplying mu by the length of the string: Total mass = mu * L Total mass = (9.91 x 10^-3 kg/m) * (2.0 m). After calculating this, we find that the total mass is ≈ 0.0198 kg. So, the total mass of the wire is approximately 0.0198 kg.