A column of air and a tuning fork produces 4 beats/s. When sounding together, the tuning fork gives the…
A column of air and a tuning fork produces 4 beats/s. When sounding together, the tuning fork gives the lower note. The temperature of air is 15°C. When the temperature falls to 10°C, the two produce 3 beats/s. Find the frequency of the tuning fork.
Solution
Sol. The frequency of the air column is given by $f = v/\lambda$, where $v$ is the velocity of sound in air and $\lambda$ is the wavelength.
But $v = \sqrt{\frac{\gamma R T}{M}}$, thus it is dependent of temperature.
⇒ $f \propto \sqrt{T}$ ($: v = \lambda f$)
Let the frequency of the tuning fork be $f$.
Thus, the frequency of the air column at 15°C = $f + 4$ and the frequency of the air column at 10°C = $f + 3$.
As, the frequency decreases with decrease in temperature,
⇒ $\displaystyle \frac{f+4}{f+3} = \sqrt{\frac{288}{283}}$ ⇒ $\displaystyle \frac{f+4}{f+3} = 1.00879$
⇒ $f + 4 = 1.00879f + 3.02638$
⇒ $8.79 \times 10^{-3} f = 0.97362$ ⇒ $f = 110.76\ \mathrm{Hz}$
Answer: $110.76\ \mathrm{Hz}$