A tuning fork with a frequency of 256 Hz generates 4 beats per second alongside a wire of length 25 cm vibrating in its fundamental mode. Upon slightly shortening the length of the wire, the beat frequency decreases. Determine the minimum length by which the wire must be shortened to produce zero beats with the tuning fork.

Step-by-step solution

Frequency of the tuning fork is f_t = 256 Hz . Since 4 beats per second are heard, the possible frequencies of the wire are f_w = 256 4 , which gives 260 Hz or 252 Hz . The fundamental frequency of a stretched wire is given by f_w = v 2L . As the length L of the wire is shortened, its frequency f_w increases. It is given that upon shortening the wire, the beat frequency decreases. This is only possible if the initial frequency of the wire was lower than the tuning fork's frequency. Thus, the initial frequency of th