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Thread: mass between two springs equation help

  1. #1
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    mass between two springs equation help

    can some one please help with the equation of motion, I have attached the question. its confusing me cause there are two springs attached to one mass,

    what I get is

    $\displaystyle m \frac{d^{2}x}{dt^{2}} = -k_{1}(x-x_{0}) + k_{2} (l-x_{0}) $

    but than when I solve for the equilibrium position I get

    $\displaystyle x = \frac{k_{1}-k_{2})x_{0} + k_{2}l}{k_{1}} $

    but the correct answer is

    $\displaystyle x = \frac{k_{1}-k_{2})x_{0} + k_{2}l}{k_{1} + k_{2}} $

    I dont understand how it is $\displaystyle k_{1} + k_{2} $

    so my this equation $\displaystyle -k_{1}(x-x_{0}) + k_{2} (l-x_{0}) = 0 $ must be wrong

    can anyone suggest how to get the correct equation for the equilibrium position?



    thanks
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  2. #2
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    Re: mass between two springs equation help

    Let $\displaystyle F_1$ be the force exerted by the spring with Hooke's constant $\displaystyle k_1$ and $\displaystyle F_2$ be the force exerted by the spring with Hooke's constant $\displaystyle k_2$. At $\displaystyle x=x_0$, you know $\displaystyle F_1=0$. At $\displaystyle x = l-x_0$, you know $\displaystyle F_2 = 0$, so $\displaystyle F_1 = -(x-x_0)k_1$. For $\displaystyle F_2$, when $\displaystyle x<l-x_0$, $\displaystyle F_2>0$ and when $\displaystyle x>l-x_0$, $\displaystyle F_2<0$, so $\displaystyle F_2 = -(x-(l-x_0))k_2 = -(x-l+x_0)k_2$

    Then, $\displaystyle F_1+F_2=0$ when $\displaystyle -(x-x_0)k_1-(x-l+x_0)k_2=0$, so $\displaystyle (k_1+k_2)x=(l-x_0)k_2+x_0k_1$. Then the forces are equal when

    $\displaystyle x = \dfrac{x_0k_1+(l-x_0)k_2}{k_1+k_2} = \dfrac{(k_1-k_2)x_0+k_2l}{k_1+k_2}$ just as expected.
    Thanks from Tweety
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