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Amplitude of two carts attached by a spring

  1. Apr 15, 2015 #1
    1. The problem statement, all variables and given/known data
    #1. [Coupled Oscillators] A cart of
    mass m (car #1) and another cart of mass 2m (car #2) on a
    horizontal surface are connected by an
    ideal massless spring of spring constant k.
    The two carts are set into oscillation about
    their center-of-mass (CM) in Simple
    Harmonic Motion (SHM).
    (a) What is the ratio of their oscillation amplitudes A1/A2 ? Please explain your reasoning.
    (b) If the CM remains at rest, what is the effective spring constant of the part of the spring (k1
    and k2) between each cart and their CM in terms of k? Please give your reasoning.
    (c) Let's check your answers to (b). When cart #2 moves a distance x to the right (→), how far
    does cart #1 move to the left (←)? How much does the spring stretch? What is the effective
    spring constant for each cart, i.e., the tension force it feels due to the spring divided by the
    distance it has been displaced from its equilibrium position? Please show your calculations.
    How do your findings compare with part (b)?
    (d) What is the frequency of each cart's SHM oscillation? How should these frequencies
    compare? Do they? Please show your calculations.
    2. equations
    a = -k/m • x
    F = ma = -kx
    1/2mv2 = 1/2kA2

    although im not sure if any of these will help


    3. The attempt at a solution
    for part (a) i wrote that the ratio is 2:1 because since the center of mass is 2/3 from car #1 i did 2/3:1/3.

    for part (b) i did a simpaler thing that k1=2/3k and k2=1/3k

    for part (c) im stuck but i feel that if if it went x to the right the #1 car would just follow but that doesnt make sense based on the next part of c
     
  2. jcsd
  3. Apr 15, 2015 #2

    haruspex

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    For part b), you need some actual logic. What you have posted appears to be a wild guess.
    In part a), you found the ratio of displacements. Suppose mass 2m is displaced x relative to mass centre. What is the total extension of the spring? What is the tension in the spring? What does that tell you about the effective spring constant in relation to 2m's displacements?
     
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