Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Coupled pendulums problem please help

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data
    two coupled pendulums are used at positions x,1 and x,2

    Newton’s equation for the forces leads to the two equations:

    m,1 * (second derivative of x,1 with respect to t) = -k,1x,1 + k(x,2 - x,1)

    and m,2 * (second derivative of x,2 with respect to t) = -k,2x,2 - k(x,2 - x,1)

    This leads to the two solutions:

    x,1(t) = A,1*sin(ω,1*t + α,1) + A,2*sin(ω,2*t + α,2) (equation 3)


    x,2(t) = A,1*sin(ω,1*t + α,1) - A,2*sin(ω,2*t + α,2) (equation 4)


    A,1 = A,2 and α,1 = α,2

    rewrite equations (3) and (4) in the very interesting form:

    x,1(t) = 2A,1*cos(((ω,1 - ω,2)/2)*t)sin(((ω,1 + ω,2)/2) (equation 3a)


    x,2(t) = 2A,1*sin(((ω,1 - ω,2)/2)*t)cos(((ω,1 + ω,2)/2) (equation 4a)

    Basically i have to derive (3a) and (4a) from equations 3 and 4 using A,1 = A,2 and α,1 = α,2

    2. Relevant equations


    3. The attempt at a solution

    all ive managed to do is expand out the brackets and thats where i get stuck, is there anyone that can help get me in the right direction as i have no idea where to go from here or how it changes from sin to a cos,

  2. jcsd
  3. Nov 13, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi 8614smith! Welcome to PF! :smile:

    You need to learn the four equations for sinA ± sinB and cosA ± cosB.

    In this case, use sinA + sin B = 2.sin((A+B)/2).cos((A-B)/2) :wink:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook