Fig. 1  How to swing?

Have you ever seen a child who doesn't know how to play on swing? Because of the air resistance and friction, the swing loses energy and amplitude. At this time, his father will push him in order to keep him swinging at a constant amplitude (Fig. 1). His father supplies energy to compensate the lost. He won't push his son randomly, but periodically. More energy will be fed to the swing if his father pushes in the same direction as the velocity. If his father pushes at the best of times the energy input is greater than the energy lost, and the amplitude of swing will be increased. In this case, although his father is not giving a hard push, the amplitude of swing can become very large. This phenomenon is called resonance.

Resonance is a very common phenomenon. In 1831, when an army was marching across a drawbridge in Britain, the drawbridge vibrated and the amplitude of the vibration grew rapidly. All soldiers were then thrown. The reason for this phenomenon is that the rhythmical tread was very close to the natural frequency of the bridge, thus producing resonance. In daily life, radio also makes use of electromagnetic resonance. Many examples can also be found in Astronomy, such as the gaps in the asteroid belt, and the divisions in the ring of Saturn. These can be explained by resonance in orbital periods.

Fig. 2  The change in the position of the centre of mass when swinging with an increasing amplitude.

Let's talk about playing swing again. As the child grows up, he won't be satisfied with the push and will want to learn how to play swing by himself. Please refer to Fig. 2, the broken line represents the route of the centre of mass of the child. Why can he swing with an increasing amplitude? When the boy swings to point A, he rises his body. Assuming that he rises his centre of mass by a distance and the tension of the rope is , he does a positive work which supplies an energy of into the system. When the boy swings to point B, he letdowns his body to lower his centre of mass. This time he does a negative work. If the tension of the rope is , then the system loses of energy. When the boy is at point B, his velocity is zero, so is just the vertical component of his weight (, see Fig. 3). But when he is at point A, his velocity is the greatest, and the difference between the tension and his weight equals the centripetal force (). Hence we conclude that . Therefore, his positive work is greater than the negative one and energy is fed to the swing. If he supplies energy rhythmically, just like his father's push, resonance is produced and the boy swings with an increasing amplitude.

Fig. 3  The double oscillator.

Lastly, I would like to introduce an interesting oscillator, which may be called the "double oscillator". It is composed of a rope, a spring and an iron ball. The rope is connected to the spring and the ball is held by the spring. As a result, the ball can swing to and fro and vibrate up and down. Suppose the natural frequency of the swing is and that of the vibration is . If we adjust the length of the rope so that becomes twice of , and then pull the ball downward slightly and release it, an interesting thing will happened. At the beginning, the ball will vibrate up and down, but later it will swing with a very large amplitude. After a while, it will return to the original status of vibrating up and down. If air resistance is negligible, this cycle will repeat many times. This is an example of parametric resonance. The energy of resonance transforms from one mode of oscillation to another repeatedly. This is a non-linear phenomenon. You can try it yourself!