# (a) A tuning fork of frequency 480 Hz, produces 10 beats per second when sounded with a vibrating...

## Question:

(a) A tuning fork of frequency 480 Hz, produces 10 beats per second when sounded with a vibrating sonometer string. What must have been the frequency of the string if a slight increase in tension produces fewer beats per second than before?

(b) Two tuning forks A and B vibrating simultaneously produce 5 beats. The frequency of B is 512 Hz. It is seen that if one arm of A is filed, then the number of beats increases. Find the frequency of A.

## Beats

If two wave sources with slightly differing frequencies {eq}\displaystyle {\nu_1} {/eq} and {eq}\displaystyle {\nu_2} {/eq} generate waves together and these waves are superposed then an interfernce effect in time will occur. The intensity oscillates with time with a frequency {eq}\displaystyle {\nu} {/eq} called the beat frequency. It is given by {eq}\displaystyle {\nu = \pm (\nu_1-\nu_2)} {/eq}.

## Answer and Explanation:

a)

Initially when the tuning fork of frequency {eq}\displaystyle {\nu_1=480\ Hz} {/eq} is sounded together with the vibrating sonometer string the beat frequency is {eq}\displaystyle {\nu=10\ Hz} {/eq}.

Therefore the string frequency is either {eq}\displaystyle {\nu_2=470\ Hz} {/eq} or {eq}\displaystyle {\nu_2=490\ Hz} {/eq}.

Next the string is tightened. Increased tension will increase the frequency of the string. This is given to reduce the number of beats. Hence the string frequency must initially have been the smaller one. That is the frequency of the string is,

{eq}\displaystyle {\nu_2=470\ Hz} {/eq}.

b)

Here it is given that {eq}\displaystyle {\nu_B=512\ Hz} {/eq}.

Now when sounded together with A it gives 5 beats. So {eq}\displaystyle {\nu_A=517\ Hz} {/eq} or {eq}\displaystyle {\nu_A=507\ Hz} {/eq}.

Now if one arm of A is filed so as to reduce its inertia and thereby increase the frequency then it is seen that the number of beats increases. Therefore A initially must have had a higher frequency. Hence,

{eq}\displaystyle {\nu_A=517\ Hz} {/eq}.