Fundamental frequency of sonometer wire of length L, tension T and mass per unit length u is,
f = (1/2L)√(T/u)
Let the mass of the wire be m, density d and it's cross sectional area be A. Then m =(AL)d and u = m/L = Ad. Keeping this back into the above equation,
f = (1/2L)√(T/Ad)
T → 3T
L → 3L
A → 3²A = 9A
f → (1/3)(√3/9) f = (1/3√3) f

Fundamental Frequency and Sonometer Wire

The fundamental frequency of a sonometer wire is the lowest frequency at which the wire can vibrate. It is directly related to the length, tension, and diameter of the wire. Let's analyze the effect of tripling these parameters on the fundamental frequency.

Effect of Tripling Length

When the length of the wire is tripled, the fundamental frequency decreases. This is because the longer wire takes a longer time to complete one full vibration cycle. Mathematically, the fundamental frequency is inversely proportional to the length of the wire.

Effect of Tripling Tension

When the tension in the wire is tripled, the fundamental frequency increases. This is because the higher tension results in a higher restoring force, which allows the wire to vibrate at a higher frequency. Mathematically, the fundamental frequency is directly proportional to the square root of the tension.

Effect of Tripling Diameter

When the diameter of the wire is tripled, the fundamental frequency decreases. This is because the thicker wire is stiffer and has a higher mass per unit length, which decreases its flexibility. Mathematically, the fundamental frequency is inversely proportional to the diameter of the wire.

Overall Effect on Fundamental Frequency

Since the length, tension, and diameter are all tripled, we need to consider the combined effect on the fundamental frequency.

- The length is tripled, which decreases the frequency.
- The tension is tripled, which increases the frequency.
- The diameter is tripled, which decreases the frequency.

Considering these effects, the overall change in the fundamental frequency can be determined.

The new fundamental frequency is given by:

new frequency = (original frequency) x (1/√3) x (1/√3) x (1/3)

Simplifying this expression, we get:

new frequency = (original frequency) / (√3 x √3 x 3)

Therefore, the new fundamental frequency is (1/3√3) times the original frequency.

Hence, the correct answer is option (1) n/√3.