A tuning fork $$A$$ of unknown frequency produces 5 beats s$$^{-1}$$ with a fork of known frequency 340 Hz. When fork $$A$$ is filed, the beat frequency decreases to 2 beats s$$^{-1}$$. What is the frequency of fork $$A$$?

The tuning fork $$A$$ produces 5 beats per second with a fork of known frequency 340 Hz. This means the frequency of fork $$A$$ is either $$340 + 5 = 345$$ Hz or $$340 - 5 = 335$$ Hz.

When fork $$A$$ is filed, its prongs become thinner and lighter, which increases its frequency. After filing, the beat frequency decreases to 2 beats per second.

If $$f_A = 345$$ Hz, filing would increase it further (say to 347 Hz), and the beat frequency with 340 Hz would become $$347 - 340 = 7$$ beats/s, which is an increase. This contradicts the observation that beats decreased.

If $$f_A = 335$$ Hz, filing would increase it (say to 338 Hz), and the beat frequency would become $$340 - 338 = 2$$ beats/s, which is a decrease from 5 to 2. This matches the observation.

Therefore, the frequency of fork $$A$$ is $$335$$ Hz.