Question

In fig. $A,B,CandD$ are four pendulums suspended from the same elastic string $XY$. Lengths of pendulum $A$ and $D$ are equal, while the length of pendulum $B$ is shorter and of the pendulum $C$ is longer. The pendulum $A$ is set into vibration. Give reason for your observation.

Answer 1

$B$elow is the observation when the pendulum $A$ is set to vibration.

The lengths of the pendulums $A$ and $D$ are equal and hence their frequencies are equal. The length of pendulum $C$ is greater than that of $B$ is less than the lengths of $A$ and $D$. If the pendulum $A$ is set to vibration, then these vibrations reach the other pendulums also through the string $XY$. Hence, a periodic force of frequency equal to that of pendulum $A$ begins to act upon other pendulums. It is observed that the pendulums $C$ and $B$ whose natural frequencies are different from frequency of $A$, are set into forced vibrations of very small amplitude, while the amplitude of vibrations of $D$ increases slowly, and it becomes much larger than that of any other.

This is only due to resonance, since the frequencies of $D$ and $A$ are equal due to their equal length.

Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.