Question

$20$ tuning forks are so arranged in series that each fork gives $4$ beats per second with the previous one. The frequency of the $20th$ fork is three times that of the first. What is the frequency of the first tuning fork?
A. $60 Hz$
B. $57 Hz$
C. $40 Hz$
D. $38 Hz$

Answer 1

If twenty tuning forks are arranged in series then the frequency will be the number of beats produced times the number of intervals. It is given that the frequency of the 20th fork is three times that of the first. This means that tuning forks are arranged in order of increasing frequency.

Complete step by step answer:
Tuning fork is a source of sound which can produce a pure note. The frequency of a tuning fork depends upon the following factors:
(1) thickness of the arm of fork
(2) the modulus of elasticity
(3) is inversely proportional to square of length of the fork
(4) density of the medium ( inversely proportional to the root of density)
(5) temperature of the medium (inversely proportional to the temperature)
It is given that 20 forks are arranged in series, then the number of intervals is 19. Also each fork gives 4 beats per second with the previous one. Then, the number of beats that will be produced in 19 intervals will be $19 \times 4 = 76$

The frequency of the 20th fork is three times that of the first. Let the frequency of the first tuning fork be $n$. Then, the frequency of the last tuning fork will be $3n$.
\left| {n - 3\left. n \right|} \right. = 76 \\\ \Rightarrow 2n = 76 \\\ \therefore n = 36Hz \\\
Therefore, the frequency of the first tuning fork will be $36Hz$.

Hence, option D is the correct answer.

Note: Tuning forks of higher frequencies are small and thick whereas those of low frequencies are long and thin. Note that the frequency of tuning forks may vary because of the density and temperature of the medium. Increase in temperature may increase the vibrations of the tuning fork and can cause the frequency to change.