Some times in a movie we see that when an opera singer sings, the glass in her hand shatters. Also at times when we close the door, the window panes in the room start rattling. The phenomenon governing the above events is resonance. When the frequency of the opera singer (or closing door) matches with the frequency of the glass (or window), resonance occurs. In this demonstration we show this phenomenon of resonance.
Every body oscillates with its natural frequency when it is left free. This natural frequency depends on the shape, size and material of the body. If the body is subjected to a periodic force the body starts oscillating with the frequency of this periodic force after some time. But the amplitude of these oscillations is very small if the forced frequency is different from the natural frequency. If the forced frequency happens to match with the natural frequency, the body gains a lot of energy and its amplitude become very large. We call this phenomenon as resonance.
Problem (IIT JEE 1985): An air column in a pipe, which is closed at one end, will be in resonance with a vibrating tuning fork of frequency 264 Hz, if the length of the column is, (Speed of sound = 330 m/s.)
Solution: Natural frequencies of a pipe of length $l$, closed at one end, are given by, $\nu=nv/(4l)$, where $v={330}\;\mathrm{m/s}$ is the speed of sound and $n$ is an odd integer. Thus, the possible length of pipe (closed at one end) that can resonate with a tuning fork of frequency $\nu={264}\;\mathrm{Hz}$ are, \begin{align} l_n=\frac{nv}{4\nu}=\frac{n(330)}{4(264)}=0.3125 n\;m, \end{align} which gives $l_1={31.25}\;\mathrm{cm}$, $l_3={93.75}\;\mathrm{cm}$, $l_5={156.25}\;\mathrm{cm}$ etc. Hence, correct options are A and C.