Chapter 12, Problem 44P

To determine

The number of beats heard.

Answer to Problem 44P

13 beats.

Explanation of Solution

Given:

Each of flutes having the frequency ( f ) = 262 Hz.

Temperature $(T_1)=5°\text{C}$

Temperature $(T_2)=25°\text{C}$

Formula used:

The frequency and velocity are related as,

  $\frac{f_5}{f_{25}}=\frac{v_5}{v_{25}}$

Calculation: 

Number of beats can be determined by the given frequency with respect to the temperatures and speed of the sound.

The speed can be calculated by using the formula,

Velocity at temperature $(T_1)=5°\text{C}$ ,

  $\begin{array}{l}v=262+(0.6\times 5)\\ \text{ = 265 m/s}\end{array}$

Velocity at temperature $(T_2)=25°\text{C}$ ,

  $\begin{array}{l}v=262+(0.6\times 25)\\ \text{ = 277 m/s}\end{array}$

The frequency ( f ) can be determined as,

  $\frac{f_5}{f_{25}}=\frac{v_5}{v_{25}}$

  $\frac{f_5}{f_{25}}=\frac{265}{277}$

  $\frac{f_5}{f_{25}}=0.95$

Since, f₂₅ = 262 Hz.

  $f_5=262\times .95\text{ Hz}\text{.}$

  $\begin{array}{l}{f}_5=248.9\text{ Hz}\\ f_5=249\text{ Hz}\end{array}$

The number of beats heard is (262-249) = 13

Conclusion: 

The number of beats heard is 13 at temperatures 5⁰C and 25⁰C and with the identical flutes each of play is 262 Hz.