Physics Puzzle: Changing an Oscillator’s Spring

Posted on March 19, 2008
Filed Under Physics, Physics Puzzles |

Here’s a nice physics puzzle I was asked a while ago.

Suppose you had a harmonic oscillator - a very simple one, with no friction or driving force. Suppose your oscillator’s oscillating with an amplitude A and that the spring constant is k. Now suppose you slowly change k from its initial value to some final value.

i.) Does the amplitude change?

ii.) If it doesn’t, why not?

iii.) If it does, can you compute the new amplitude as a function of the old one, the old spring constant and the new one?

HINT (ADDED 26/3/08): Ok, stop reading now if you want to think for yourself about this problem. I’ll give you a couple of hints today, and show you how to solve this next week. Today’s hints are:

  1. The amplitude does, indeed, change. 
  2. Energy is not conserved, but something else is. Can you find it?
  3. The “slow” part here is essential.

Good luck for now!

Comments

4 Responses to “Physics Puzzle: Changing an Oscillator’s Spring”

  1. James Cronen on March 22nd, 2008 5:15 pm

    Interesting puzzle… here’s my shot at it, but I think my answer’s so easy that there are other things going on I didn’t consider. Anyway, here goes:

    The total energy is E = PE + KE = (1/2)kx2 + (1/2)mv2 = (1/2)k(A2 cos2 ωt) + (1/2)k(A2 sin2 ωt) = (1/2)kA2.

    Assuming that we do no work on the system if the spring constant changes (that makes sense, right?), conservation of energy requires that Ei = Ef.

    Thus, (1/2)kiAi2 = (1/2)kfAf2. The (1/2)’s cancel; solve for Af: Af = sqrt(ki / kf) Ai.

    So, yes the amplitude changes by a factor of (ki / kf)1/2.

    How’d I do?

  2. admin on March 22nd, 2008 6:26 pm

    A brave attempt, but incorrect :).
    Your assumption no work is done is wrong. There IS a conserved quantity here, but it is not the energy.
    Back to the drawing board …

  3. wandering.the.cosmos on March 26th, 2008 10:52 pm

    I believe Wikipedia has the answer here.

  4. admin on March 27th, 2008 4:12 am

    Yep - there is an invariant quantity, as I’ve said, but it is not E. It is the area in phase space, which can be shown to be E/frequency. So the ratio of the amplitudes is like the (inverse) fourth root of the ratio of the spring constants.

    Another way to solve his problem is to make an analogy to quantum mechanics, in which the system must stay in the same energy eigenstate En. This doesn’t mean that Efinal=Einitial, because the energy eigenstates themselves change! For a harmonic oscillator,

    En = hw(n+1/2)

    Where h is h-bar (Planck’s constant over 2pi) and w is the angular frequency (2*pi*frequency). n stays the same, so En/w must be conserved.

    Guess I’ll have to start finding puzzles that don’t have solutions hidden in wikipedia :).

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