Q58PE - College Physics (Urone)

Question

The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?

Step-by-Step Solution

Answer

The other string can have frequency either 261.5 Hz or 258.5 Hz.

Step 1: Introduction

We use formula for beat frequency of two strings and then calculate frequency of the other string when frequency of first string is given

Step 2: Given parameters and formula for beat frequency

Frequency of first string     \( = 260 Hz\)

Beat frequency                   \( = 1.5  Hz\)

Beat frequency    \(f   = \left| {{f_1} - {f_2}} \right|\)

Here, 

\({f_1},{f_2}\) - frequency of 2 sources.

\(f\)- beat frequency.

Step 3: Calculate frequency of other string

Beat frequency of two strings 

\(f   = \left| {{f_1} - {f_2}} \right|\)

Therefore, frequency of other string is either

\({f_2}   = {f_1} + f\)

or

\({f_2}   = {f_1} - f\)

That is,

\(\begin{align}{\underline{\phantom{xx}}}{f_2} &= 260 + 1.5 Hz\\{f_2} &= 261.5 Hz\end{align}\)

or

\(\begin{align}{\underline{\phantom{xx}}}{f_2} &= 260 - 1.5 Hz\\{f_2} &= 258.5 Hz\end{align}\)

Therefore,the other string can have frequency either 261.5 Hz or 258.5 Hz.