A string pendulum is suspended from the ceiling, and a length of elastic is tied about 1/4 of the way - from the ceiling end - to the pendulum string; the other end of the elastic is tied to an eyelet screwed into the ceiling.

If the weight at the end of the string is held at an acute angle to the line made by the elastic length's point of contact, and the strings's point of contact, and released, so it swings at an initial acute angle, the swinging weight will:

a) keep swinging at the same acute angle, until frictional forces damp its motion and it stops swinging.

b) swing around in elliptical motions that precess towards a transverse, near linear motion, at 90 degrees to the line made by both POCs. Eventually friction will mean it stops swinging.

c) start swinging in circles, that alternate between right and left hand orbits, then stop because of frictional damping.

d) swing in straight lines that slowly alternate between the initial acute angle, and its inverse acute angle, or "mirror image". It will stop swinging eventually due to frictional losses.

(3 marks)

The answer, of course, is (b), but why?

Can the behaviour of a pendulum that's constrained with a spring, be explained in terms of superposition and symmetry?

(Answer: you betcha)

A weight suspended from a single point has more symmetry than one suspended from a 2-d "frame", which has one leg with a much larger tension, so that the motion of the swinging weight is constrained, or the symmetry is broken by giving the point of contact an extra degree of movement - the spring itself is another pendulum.

A weight on a string can move in circular or elliptical orbits. This motion is the superposition of two linear transverse motions, as if two imaginary pendulum weights are swinging through each other at right angles. Likewise, a weight swinging in a linear arc is the superposition of two imaginary weights swinging in opposing circular orbits.

By adding the spring or elastic coupling, the weight is left with only transverse linear motion, or the motion in the direction along the plane formed by the upper 2 couplings is occupied by the spring, which "absorbs" the momentum of the freely swinging weight in that dimension. The weight isn't "allowed" to move in that mode, because of the spring in the way.

P.S. This principle is well understood by bridge truss designers and high-rise engineers. It also explains the behaviour of electrons in semiconductor diodes - which are asymmetrical devices that resemble a "truss" or a frame of sorts, like a pendulum hanging from a string that's constrained (dimensionally) by an elastic coupling, which in a semiconductor diode is the diffusion layer at the join of two oppositely doped crystals.

Very interesting, thanks.

Yeah, who'd a thought there was all that math in something swinging at the end of a thread?

BTW, since a weight swinging in a straight arc is the equivalent of two weights swinging in opposing circular (or curvilinear) arcs, there is no difference.
IOW you can't tell, although you see a single swinging weight, that there are not two swinging weights. Same deal for a weight swinging in a circular or elliptical orbit - you can't tell if there is a single weight, or two weights swinging in straight transverse "orbits", you just see a single equivalent weight. Topologically, both spaces are equivalent or dual spaces.

And following the math connection, the behaviour of rotating bodies that are coupled to an inertial frame models the behavior of bodies coupled to an electric or magnetic frame, like charged particles. Superposition of waveforms, easily demonstrated by swinging weights (bodies coupled to an inertial frame), means that complex frequencies can be analysed or decomposed - you can treat a rotating vector (a frequency) as a superposition and "substitute" a linear motion for a curvilinear motion, using complex numbers to represent the superposition. This lets you linearise the complex (frequency) behaviour of reactive components, using the complex plane which can be visualised, more or less as two pendula hanging from the same point, that superpose their motions.