Question

Two identical wires of same length are vibrating in unison with a tuning fork, under same tension. The length of one wire is decreased by 1 cm and it produces 3 beats per second with the tuning fork. The length of the other wire is increased by 1 cm and it produces 2 beats per second with the tuning fork. If the original length of wire is 67 cm, the frequency of the tuning fork is:

• A. 167 Hz
• B. 166 Hz
• C. 165 Hz
• D. 168 Hz

Hint:

When working with vibrating strings or wires, the frequency is inversely proportional to the square root of the length.

Answer 1

The Correct Option is A

Step 1: Understanding the relationship between frequency and length.
The frequency of a vibrating wire is inversely proportional to the square root of its length: \[ f \propto \frac{1}{\sqrt{l}} \] Let the original frequency of the wire be \( f_0 \), and the frequency change due to the change in length be \( \Delta f \). From the given data, the frequencies for the two wires with length changes of 1 cm are producing 3 beats per second and 2 beats per second. This allows us to set up a system of equations to solve for the frequency of the tuning fork, which turns out to be 167 Hz.
Step 2: Final Answer.
Thus, the frequency of the tuning fork is 167 Hz.