Question

The frequency of tuning fork is $256\;Hz$. It will not resonate with a fork of frequency:
(A) $786\;Hz$
(B) $738\;Hz$
(C) $512\;Hz$
(D) $256\;Hz$

Answer 1

Here we will calculate all the frequencies which are an integer multiple of the given tuning for with frequency of $256\;Hz$. These calculated frequencies are those for which the tuning fork will have resonance. For the other frequencies which are not an integer multiple of the tuning fork frequency, there will not be any phenomenon of resonance occurring.

Complete Answer:
Resonance is the phenomenon by which one object vibrating at a frequency equal to the natural frequency of another object makes the other object to vibrate as well at the same or higher frequencies.
It is for this reason that marching troops are usually ordered not to march on bridges because the frequency of the feet stamping the ground and the vibrations caused by these beats might match with the natural frequency of some structure of the bridge. This would then cause the structure to vibrate at some higher frequency which might be fatal for the structure weakening joints or causing a complete collapse of the bridge. However, for this to happen, the marches have to be long lasting and the vibrations should have enough energy to start some initial vibrations in the structure.
In case of the question given above, the tuning fork would resonate with frequencies which are an integer multiple of its own frequency. Here the frequency of a tuning fork is $256\;Hz$. The multiples of this frequency are $n \times 256\;Hz$, where $n$ will take the values $1$, $2$, $3$ and so on. Thus the resonant frequencies come out to be $256\;Hz$ for $n = 1$, $512\;Hz$ for $n = 2$, $768\;Hz$ for $n = 3$ and so on.
Here we see that the value of the resonant frequencies for $n = 1$ and $n = 2$ are provided in the options given. Thus these frequencies are resonant frequencies. However, the other frequencies which are provided are neither resonant frequencies. Thus $786\;Hz$ and $738\;Hz$ are the frequencies with which the tuning fork will not resonate.
Thus the correct answer is option (A) $786\;Hz$ and option (B) $738\;Hz$.

Note:
Any tuning fork will resonate with frequencies which are integer multiples of its own natural frequencies. Here for the given problem, we have multiple options as correct since both of those frequencies turn out to be non-resonating frequencies.