An admission

Alan,

I often feel like I’ve betrayed the original mission statement. It’s been a long time since I’ve written anything like a letter – instead, I’ve given a series of disparate, unconnected and unwieldy posts. In honesty, I’ve been writing the blog like a soliloquy, rather than the light-hearted, interesting conversation we used to have. My ‘puzzles’ were turning into exam questions in all but name, and the remainder of my posts in no way lent themselves to conversation.

As of this summer, I will try to do the following:

  • to start writing letters again
  • to distinguish the dialogue and puzzles from the heavier, mathematical digressions
  • to make my posts shorter and more frequent
  • to make my posts more consistent
  • to make a series of ordered, connected posts about a particular topic from the second year syllabus (probably either quantum mechanics or statistical physics)

As a return to form, here’s a puzzle for you that really had me stumped.

There are two equal masses connected by a massless spring. They sit on a frictionless table in a vacuum, so that they experience no dissipative forces. You pull the masses apart slightly. The system now has some finite energy. You release the two masses and allow them to oscillate.

After watching their motion for an infinite amount of time, you become bored, and decide to put an end to things. You brandish a pair of scissors and prepare to cut the spring in half.

Suppose you cut the spring when the two masses are moving apart, just as the spring connecting them is neither stretched nor compressed. At this moment, the system’s energy is exclusively kinetic. When the spring is cut, the two masses will move apart. Their instantaneous velocity is unchanged. The energy of the system is not perturbed by the removal of the spring.

Now instead suppose you cut the spring when the two masses are furthest apart, when the spring is maximally stretched. At this moment, the system’s energy is exclusively potential. When the spring is cut, the two masses will remain completely stationary. The masses are left with neither kinetic energy nor potential energy.

How can the latter case be reconciled with the conservation of energy? It appears that, depending on when you cut the spring, different amounts of energy can go unaccounted for. What has gone wrong?

Look forward to hearing from you.

Paddy

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