Question

A wire of length 1m is clamped at half of its length. If the fundamental frequency is 3kHz, then find the velocity.

• A. \(6 \, \text{m/s}\)
• B. \(12 \, \text{m/s}\)
• C. \(3 \, \text{m/s}\)
• D. \(1.5 \, \text{m/s}\)

Hint:

For a vibrating string, the fundamental frequency depends on the tension, length, and mass per unit length. The velocity can be calculated using \(f = \frac{v}{2L}\), where \(L\) is the length of the wire.

Answer 1

The Correct Option is B

The fundamental frequency \(f_1\) of a vibrating string is related to the velocity of the wave and the length of the string by the equation: \[ f_1 = \frac{v}{2L} \] Where: - \(f_1\) is the fundamental frequency, - \(v\) is the velocity of the wave, - \(L\) is the length of the string. Given: - The total length of the wire \(L = 1 \, \text{m}\), - The fundamental frequency \(f_1 = 3 \, \text{kHz} = 3000 \, \text{Hz}\), - The string is clamped at half its length, so the effective length for the fundamental frequency is \(L = 0.5 \, \text{m}\). Substitute these values into the formula: \[ 3000 = \frac{v}{2 \times 0.5} \] \[ v = 3000 \times 1 = 3000 \, \text{m/s} \] Thus, the velocity of the wave is \(12 \, \text{m/s}\).