1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two Identical Pendulums Connected by a Light Spring

  1. Feb 5, 2017 #1
    1. The problem statement, all variables and given/known data

    Two identical pendulums of the same mass m are connected by a light spring. The displacements of the two masses are given, respectively, by xa = Acos( (w2-w1)t/2 )cos( (w2 + w1)t/2 ), xb = Asin( (w2-w1)t/2 )sin( (w2 + w1)t/2 ).

    Assume that the sprint is sufficiently weak that its potential energy can be neglected and that the energy of each pendulum can be considered to be constant over a cycle of its oscillation.

    Show that the energies of the two masses are are:

    Ea = 1/2 m * A^2 ( (w2 + w1)/2 )^2 cos^2( (w2 - w1)t/2 )
    Eb = 1/2 m * A^2 ( (w2 + w1)/2 )^2 sin^2( (w2 - w1)t/2 )

    2. Relevant equations

    Energy of simple harmonic oscillator = (1/2)mw^2 (amplitude)^2

    3. The attempt at a solution

    In the book's solution, it says that amplitude = A cos[ (w2 - w1)t/ 2] or A sin[ (w2 - w1)t/2 ]. Where does it get this from? What happened to the other cosine/sine term? Why is it using this particular term?
    Last edited: Feb 5, 2017
  2. jcsd
  3. Feb 5, 2017 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Those are constants. Should there be some occurrences of t in there?
  4. Feb 5, 2017 #3
    Whoops, thanks for catching that. I edited my post.
  5. Feb 5, 2017 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If the two frequencies are similar, you can view the displacement equations as a rapid oscillation (the average of the frequencies) modulated in amplitude by a beat frequency (half the difference). In that view, we can treat the beat frequency factor as a time-dependent amplitude, A(t). Applying the standard formula should then yield the result.
    That said, there is nothing in the statement of the question to justify the assumption that the frequencies are so close.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted