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Springs in series and in parallel

  1. Jun 11, 2017 #1
    1. The problem statement, all variables and given/known data

    A mass ##m## hangs from a combination of two springs, each with spring constant ##k##, connected in series. If the mass is doubled to ##2m## the mass will hang lower by a distance ##h##. If three such springs are arranged in parallel to support a mass of ##5m## what will be the frequency of small oscillations if the system is perturbed?

    2. Relevant equations

    3. The attempt at a solution

    A combination of two springs in series, each with spring constant ##k##, is equivalent to a system of one spring with spring constant
    \begin{align}
    \frac{1}{k_{\text{eq}}} &= \frac{1}{k} + \frac{1}{k}\\
    k_{\text{eq}} &= k/2.
    \end{align}
    For a mass ##m## hanging from the system of two springs, let the extension of the springs be ##x## m. Then, if the mass is doubled to ##2m##, the extension of the springs is ##(x + h)## m. Therefore, we find that
    \begin{align}
    mg &= (k/2)x,\\
    2mg &= (k/2)(x + h),
    \end{align}
    so that
    \begin{align}
    \frac{x + h}{x} &= 2\\
    x &= h.
    \end{align}
    Therefore, the spring constant of each spring, in terms of the mass ##m## and distance ##h##, is
    \begin{align}
    k = 2mg/h.
    \end{align}
    A combination of three springs in parallel, each with spring constant ##2mg/h##, is equivalent to a system of one spring with spring constant
    \begin{align}
    k_{\text{eq}} &= 2mg/h + 2mg/h + 2mg/h\\
    k_{\text{eq}} &= 6mg/h.
    \end{align}
    If the springs are arranged in parallel to support a mass of ##5m## and perturbed from equilibrium, then the frequency of small oscillations is
    \begin{align}
    \omega = \sqrt{k_{\text{eq}}/m} = \sqrt{6g/h}.
    \end{align}

    Is my solution correct?
     
  2. jcsd
  3. Jun 11, 2017 #2

    Doc Al

    User Avatar

    Staff: Mentor

    The mass is 5m, not m. (Otherwise good.)
     
  4. Jun 11, 2017 #3
    Ah, right! So, the answer is

    $$\omega = \sqrt{k_{\text{eq}}/5m} = \sqrt{6g/5m}.$$
     
  5. Jun 11, 2017 #4

    Doc Al

    User Avatar

    Staff: Mentor

    Looks good.
     
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