# Oscillation of 2 identical pendulums connected by a rubber band

While solving A level past papers, I came across the following question. For reference, this is the Edexcel GCE A2 Physics Paper 2 from June 2018.

What does not make sense in my mind is the fact that pendulum A comes to a complete stop, while pendulum B oscillates at maximum amplitude, then vice versa. Why does it do this, instead of both pendulums reaching an amplitude of $$\frac{1}{\sqrt 2}$$ of the original value according to the following equation?

$$E = \frac1 2 m \omega^2A^2$$

• It is necessary to write a system of equations, solve and see what will happen. – Alex Trounev May 23 at 13:46

This is a good question.

Here is a video, Coupled Pendulums - Sixty Symbols, which clearly shown the effect described in the question.

What you are asking for is that after a time the two pendulums oscillate with a constant amplitude.

There are only two ways in which this arrangement of pendulums can oscillate can have each pendulum oscillate with the same constant amplitude and these are called the normal modes.

One way is to have both pendulum bobs oscillate in phase with one another ie when one bob is moving into the screen the other bob is also moving into the screen.
The other way is to have both pendulum bobs oscillate in antiphase with one another ie when one bob is moving into the screen the other bob is moving out of the screen.
These two normal modes have differing frequencies of vibration.

It can be shown that the motion of the two pendulum bobs can always be described as a linear combination of the two normal modes of vibration. With the given initial conditions, one bob displaced and at rest and the other bob not displaced and at rest, a situation when only normal mode occurs is not possible.

Let me try and explain.
If the displacement of pendulum bob $$A$$ is $$X$$ then tie initial condition is equivalent to the sum of the two normal modes each of amplitude $$\frac x2$$.
I have tried to illustrate this below.

So in this case when you set the pendulums moving you are activating two modes of vibration each of different frequency.
As there is assumed to be no damping there is no way to eliminate one of these normal modes to achieve your goal of having a constant amplitude.

To illustrate how complex the motion actually is have a look at this graph and when you look at a demonstration you will tend to miss the detail.

In part the complexity is due to the fact that in this example the ratio of the frequencies of the two normal modes is an irrational number.

There is a Phet simulation, normal modes, which may be of interest or this simulation, coupled oscillators and there are may websites which go through the maths eg coupled oscillators.

• So is this more of a "calculations tell us that this happens" sort of thing? Is there a more conceptual explanation? – Yubin Lee May 25 at 4:44
• @YubinLee I have tried to find a simpler explanation. What you are asking is equivalent to trying to explain the reason for two harmonics on a stretched string not becoming just one harmonic if there are no dissipative forces acting. – Farcher May 25 at 6:24
• So is this similar to a string vibrating in multiple modes at the same time? – Yubin Lee May 25 at 6:27
• @YubinLee Yes, in that a string is many, many masses all coupled together via chemical bonds which are equivalent to elastic bands. – Farcher May 25 at 6:29