Question

A string is stretched between fixed points separated by 75.0 cm. It is observed to have resonant frequencies of 420Hz and 315 Hz. There are no other resonant frequencies between these two. The lowest resonant frequency for this string is:
\[
  (a){\text{ 105Hz}} \\
  (b){\text{ 155Hz}} \\
  (c){\text{ 205Hz}} \\
  (d){\text{ 10}}{\text{.5Hz}} \\
 \]

Answer 1

Hint: In this question use the direct formula for frequency that is $f = \dfrac{v}{{2L}}n$ where v is velocity of sound, n is the smallest integer and L is the separation between two points of the strings. Use the concept that the highest common factor (H.C.F) is calculated by multiplying all the common factors of the given numbers. This will help reach the answer.

Complete step-by-step answer:
Given data:
String is stretched between fixed point separated by the distance = 75 cm
Therefore, L = 75cm
It is observed that the resonant frequencies of 420 Hz and 315 Hz.
There are no other resonant frequencies present between these two points.
Now we have to calculate the lowest resonant frequency for this string.
As we know that the frequency is given as $f = \dfrac{v}{{2L}}n$
Where, n = smallest integer
               v = velocity of sound
               L = separation length of the string by which it is stretched.
Now the lowest possible resonant frequency is the highest common factor (H.C.F) of the two given resonant frequencies.
As we know that the highest common factor (H.C.F) is calculated by multiplying all the common factors of the given numbers.
So first factories the numbers,
Therefore the factors of 420 are,
$420 = 2 \times 2 \times 3 \times 5 \times 7$
The factors of 105 are,
$315 = 3 \times 3 \times 5 \times 7$
Now, as we see that the common factors of above numbers are $3 \times 5 \times 7$.
So the required H.C.F of the given numbers is $3 \times 5 \times 7 = 105$.
So, H.C.F of 420 and 315 is 105
So the lowest resonant frequency for this string is 105 Hz.
So this is the required answer.
Hence option (A) is the correct answer.

Note – There are basically two types of oscillations : one is natural and one is forced. When a body starts oscillating at its natural then this frequency is termed as the resonant frequency. Resonance occurs when the frequency of the body undergoing an external force matches up to its natural oscillating frequency.