Question

a piano tuner hears three beats per second when a tuning fork and a note are sounded together and six beats per second after the string is tightened. what should the tuner do next, tighten or loosen to tune the string?
multiple choice
loosen the string and hear fewer beats per second. this means the tuner is moving toward tuning the string.
loosen, and then tighten until the maximum number of beats is heard.
nothing, because the string might already be in tune.
tighten the string and hear more beats per second. this means the tuner is moving toward tuning the string.

Explanation:

When the string is tightened, the frequency of the string increases. The beat frequency is \(|f_1 - f_2|\). Initially, the beat frequency \(f_b1=3\space Hz\). After tightening, \(f_b2 = 6\space Hz\). Since tightening made the beat frequency larger, it means the difference between the string frequency and the tuning - fork frequency increased. To reduce the beat frequency (get closer to tuning), we need to move the string frequency in the opposite direction of what tightening did. Loosening the string will decrease its frequency. If after loosening, the beat frequency decreases, it means we are moving towards tuning (because \(|f_1 - f_2|\) is getting smaller).

Answer:

Loosen the string and hear fewer beats per second. This means the tuner is moving toward tuning the string.