Question

A tuning fork of frequency $512Hz$is vibrated with a sonometer wire and 6 beats per second are heard. The beat frequency reduces if the tension in the string is slightly increased. The original frequency of vibration of the string is:
A. $506Hz$
B. $512Hz$
C. $518Hz$
D. $524Hz$

Answer 1

A string produces frequencies by vibrations. The value of frequency depends upon the tension which means applying weight on that string. The frequency decreases by increasing the weight. We can see this change in the sonometer experiment.
Formula used:
${n_t} = {n_s} + x$
Where,
${n_t}$=frequency of the tuning fork,
${n_s}$=frequency of the string,
$x$=beat frequency

Complete step-by-step answer:
We shall consider the frequency of the tuning fork.
Let the tuning fork frequency is $512Hz$
$\therefore {n_t} = 512Hz$
The beat frequency is $6$beats per second.
$\therefore x = 6$
We found the beat frequency per second.
The formula we know is,
${n_t} = {n_s} + x$
The required value is the frequency of the string${n_s}$, Hence the formula is converted into
${n_s} = {n_t} - x$
Substituting the given values in the formula,
${n_t} = 512 - 6$
${n_s} = 506Hz$
The required original frequency of the string is $506Hz$.
Therefore the required option is option A.
Additional information:
(i)The sonometer experiment is used to find the frequency of the current we give to the wire through the transformer or any power supply. The string can produce vibrations according to the current we give. That vibration produces frequencies. Normally in India, we use the current frequency of $50Hz$.
(ii)On increasing the tension in the wire, the frequency gradually reduces. Usually this tension is applying weights on the string or wire.
(iii)The frequency is defined as how many the one full wave is repeated for a second. Therefore$50Hz$means $50$times per second.

Note: While solving this kind of problem we need to be careful, whether the tension is increased or decreased. If tension increases, frequency decreases. If tension decreases, frequency increases.