# Resonance of pendulum

When a small metal ball of mass being hang from a bar with a thin string, it behaves as pendulum. After changing the position of the mass a little bit from its equilibrium and releasing it, we can observe its back-and-forth movement with a constant period. It is called “free vibration.” The period is determined by the length of the string; the longer the string extends, the longer the period becomes, and vise versa. If the period is 1 s, its frequency is 1 Hz (because the number of vibrations per second is the frequency). If the period is 0.5 s, its frequency is 2 Hz. This frequency is called “natural frequency.”

In the next video clip, three metal balls are hang from a pivot bar with different lengths, and the bar can be rotated around its central axis with small degrees. In this case, the natural frequencies of the pendulums are 1 Hz, 1.5 Hz, and 2 Hz from left to right. If one sinusoidally rotates the horizontal pivot bar back and forth around the rotation axis within a small degree range, and if its frequency is 2 Hz, only the 2-Hz pendulum will start to move back and forth due to this external force applied on the system. You can see its movement grows, and it is the resonance.

Let us ask this question: “Are we able to swing two pendulums of different lengths with a single periodic external force?” Likewise to the previous example, if one sinusoidally rotates the horizontal pivot bar back and forth with its frequency of 1 Hz, only the 1-Hz pendulum will resonate. Please note that the external force in this case was sinusoidal. What is going to happen, if the external force is still periodic with the period of 1 s but is impulsive? Please take a look at the next video.

In this video, the external force is an impulse train with its period of 1 s. Therefore, the periodicity matches to the 1-Hz pendulum for sure. In addition, it also matches to the 2-Hz pendulum, because the period for the two reciprocations of the 2-Hz pendulum is 1 Hz, the period of this external force. As results, both the 1-Hz and 2-Hz pendulums resonate.