Question

In sonometer experiment, the bridges are separated by a fixed distance. The wire which is slightly elastic, emits a tone of frequency ‘n’ when held by tension ‘T’. If the tension is increased to ‘4T’, the tone emitted by the wire will be of frequency
A. n
B. 2n
C. Slightly greater than 2n
D. Slightly less than 2n

Answer 1

Hint: A sonometer is a device used to measure the tension, frequency or density of vibrations. It is a diagnostic instrument.

Complete step by step solution:
The equation used to calculate frequency with the help of sonometer is given as,
\[f=\dfrac{1}{2L}\sqrt{\dfrac{T}{m}}\] ……(i)
Where L is resonating length, T tension at the string, and m is mass/unit length of the wire.

At first, the tension on the wire is ‘T’ when the wire that is slightly elastic and it emit a tone of frequency ‘n’

From equation (i) we can conclude that frequency (f) is directly proportional to the square root of Tension. That is, \[f\propto \sqrt{T}\]. So, on increasing the tension on the wire the frequency will also increase. If the tension is increased as much as four times than the frequency will also increase twice of its previous value.

Therefore, when tension is ‘T’ then frequency is ‘n’ and thus when tension is increased to ‘4T’ that is four times then the frequency will become ‘2n’, that is two times.

Hence, the correct answer is option B: 2n

Note: From the concepts of Newton's second law and spring mass damper system, a stiffer system will have a higher frequency as it can produce more force resulting in more acceleration per unit displacement. Therefore, if you increase the tension in the string, the frequency of the vibration increases.