Question

25 tuning forks are arranged in decreasing order of frequency. Any two successive forks produce 3 beats/sec. If the frequency of the first tuning fork is the octave of last, then frequency of ${21^{st}}$ fork is
1) 72 Hz
2) 288 Hz
3) 84 Hz
4) 87 Hz

Answer 1

Hint: A tuning fork is a basically a steel device used by musicians, which vibrates and produces a note of specific pitch. The tuning forks are arranged in decreasing frequency order, so assume the frequency for the last tuning fork. And then obtain mathematical expressions for the first, third and twenty fifth forks using the given relation. Solve the formed equations and then find the frequency for the ${21^{st}}$ fork

Complete step-by-step answer:
Let the frequency of the last tuning fork be ‘a’
So, the frequency of the first tuning fork will be $2a - 3$, since it produces 3 beats/sec
Similarly frequency of 3rd tuning fork will be $2a - 6$
So, we get frequency of 25th tuning fork as $2a - \left( {24} \right)3$
So,
$\eqalign{
& a = 2a - 72 \cr
& \Rightarrow a = 72hertz \cr} $
Now, we need to find the frequency of the 21st tuning fork.
So,the frequency of 21st tuning fork will be,
$\eqalign{
& f = 2a - \left( {20} \right)3 \cr
& = 2\left( {72} \right) - 60 \cr
& = 144 - 60 \cr
& = 84hertz \cr} $
Therefore, the frequency of ${21^{st}}$ fork is 84 Hertz.

So, the correct answer is “Option C”.

Additional Information: When the transfer of energy takes place through the medium because of oscillations, the resulting wave is termed as a mechanical wave. Mechanical waves are of two types:
1) Transverse waves: A transverse wave is defined as a moving wave whose oscillations are perpendicular to the direction of the propagation of the wave.
2) Longitudinal waves: Longitudinal wave is defined as a wave in which the displacement of the particle is parallel to the direction of the propagation of the wave.

Note: Tuning forks are sounded by striking one of its pads made of rubber then a vibration is produced in it. These vibrations result in longitudinal waves. Longitudinal waves are defined as the waves consisting of periodic vibrations which take place in the same direction of the propagation of waves.