Question

A sono-meter wire supports a 4 kg load and vibrates in fundamental node with a tuning fork of frequency 416 Hz. The length of the wire between the bridges is now doubled. In order to maintain fundamental node, the load should be changed to

Answer 1

To maintain the fundamental frequency when the length of a sonometer wire is doubled, the tension must be quadrupled, thus the 4 kg load should be increased to 16 kg.

Step-by-step explanation:

The original problem involves a sonometer wire under tension due to a 4 kg load that vibrates at a frequency of 416 Hz in its fundamental mode. When the length of the wire between the bridges is doubled to maintain the fundamental mode, it's necessary to modify the tension in the wire. To find the new load, we can use the formula for the fundamental frequency of a vibrating string: f=(1/2L)*sqrt(T/μ), where f is the frequency, L is the length of the string, T is the tension in the string, and μ is the linear mass density.

This formula shows that the frequency is proportional to the square root of the tension over the linear mass density and inversely proportional to the length of the string. Since the length of the wire is doubled and frequency remains the same, the tension must be quadrupled to maintain the same fundamental frequency. Thus, the load should be changed to 16 kg (since 4 kg tension is quadrupled to maintain the same frequency on a wire of double the length).