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Thread: Equation of Motion for a Simple Pendulum

  1. #1
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    Equation of Motion for a Simple Pendulum

    I've already derived the equation, and simplified it to the linear case, now I have to solve it... The equation is:

    $\displaystyle \frac{d^2\theta}{dt^2} = -\frac{g}{l}\theta$

    Where l is length of the pendulum.

    Doing some research, I keep finding that the solution is:

    $\displaystyle \theta = \theta_0 cos\left(\sqrt{\frac{g}{l}}t\right)$

    Where $\displaystyle \theta_0$ is the initial condition at time t=0

    Now, here's my attempt at a solution:

    Since it is linear and homogeneous, we can assume a solution:

    $\displaystyle \theta = e^{kt}$

    Where k is an arbitrary constant.

    This leads to the indicial equation:

    $\displaystyle k^2 + \frac{g}{l} = 0$

    The solution to which is:

    $\displaystyle k = \pm i\sqrt{\frac{g}{l}}$

    This leads to two solutions:

    $\displaystyle \theta_1 = e^{i\sqrt{\frac{g}{l}}t}$
    $\displaystyle \theta_2 = e^{-i\sqrt{\frac{g}{l}}t}$

    A linear combination of these leads to the general homogeneous solution:

    $\displaystyle \theta_h = Ae^{i\sqrt{\frac{g}{l}}t} + Be^{-i\sqrt{\frac{g}{l}}t}$

    Expanding the exponentials into their trigonometric form, we get:

    $\displaystyle \theta_h = A\left[cos\left(\sqrt{\frac{g}{l}}t\right) + isin\left(\sqrt{\frac{g}{l}}t\right)\right]$ $\displaystyle + B\left[cos\left(\sqrt{\frac{g}{l}}t\right) - isin\left(\sqrt{\frac{g}{l}}t\right)\right]$

    $\displaystyle \theta_h = (A + B)cos\left(\sqrt{\frac{g}{l}}t\right) + i(A - B)sin\left(\sqrt{\frac{g}{l}}t\right)$

    From here I'm not sure how to proceed...
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  2. #2
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    Quote Originally Posted by Aryth View Post
    I've already derived the equation, and simplified it to the linear case, now I have to solve it... The equation is:

    $\displaystyle \frac{d^2\theta}{dt^2} = -\frac{g}{l}\theta$

    Where l is length of the pendulum.

    Doing some research, I keep finding that the solution is:

    $\displaystyle \theta = \theta_0 cos\left(\sqrt{\frac{g}{l}}t\right)$

    Where $\displaystyle \theta_0$ is the initial condition at time t=0

    Now, here's my attempt at a solution:

    Since it is linear and homogeneous, we can assume a solution:

    $\displaystyle \theta = e^{kt}$

    Where k is an arbitrary constant.

    This leads to the indicial equation:

    $\displaystyle k^2 + \frac{g}{l} = 0$

    The solution to which is:

    $\displaystyle k = \pm i\sqrt{\frac{g}{l}}$

    This leads to two solutions:

    $\displaystyle \theta_1 = e^{i\sqrt{\frac{g}{l}}t}$
    $\displaystyle \theta_2 = e^{-i\sqrt{\frac{g}{l}}t}$

    A linear combination of these leads to the general homogeneous solution:

    $\displaystyle \theta_h = Ae^{i\sqrt{\frac{g}{l}}t} + Be^{-i\sqrt{\frac{g}{l}}t}$

    Expanding the exponentials into their trigonometric form, we get:

    $\displaystyle \theta_h = A\left[cos\left(\sqrt{\frac{g}{l}}t\right) + isin\left(\sqrt{\frac{g}{l}}t\right)\right]$ $\displaystyle + B\left[cos\left(\sqrt{\frac{g}{l}}t\right) - isin\left(\sqrt{\frac{g}{l}}t\right)\right]$

    $\displaystyle \theta_h = (A + B)cos\left(\sqrt{\frac{g}{l}}t\right) + i(A - B)sin\left(\sqrt{\frac{g}{l}}t\right)$

    From here I'm not sure how to proceed...

    The equation for the simple pendulum SHOULD BE... $\displaystyle \ddot{\theta} = -\frac{g}{l} \sin(\theta) $. If you use the process I gave you with THAT equation, then you get $\displaystyle \theta = \theta_0 \cos(\sqrt{\frac{g}{l} t}) $. You won't get the same solution if you use the equation you started with, which is the small angle approximation of the proper equation of motion...

    EDIT: Perhaps I'm wrong about that actually..
    Last edited by Mush; Oct 30th 2009 at 11:19 AM.
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    Super Member Aryth's Avatar
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    Quote Originally Posted by Mush View Post
    The equation for the simple pendulum SHOULD BE... $\displaystyle \ddot{\theta} = -\frac{g}{l} \sin(\theta) $. If you use the process I gave you with THAT equation, then you get $\displaystyle \theta = \theta_0 \cos(\sqrt{\frac{g}{l} t}) $. You won't get the same solution if you use the equation you started with, which is the small angle approximation of the proper equation of motion...

    EDIT: Perhaps I'm wrong about that actually..
    You're somewhat right... I actually meant that the solution was:

    $\displaystyle \theta = \theta_0 cos\left(\sqrt{\frac{g}{l}}t + \phi\right)$

    How do you proceed to get that solution?
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    Quote Originally Posted by Aryth View Post
    You're somewhat right... I actually meant that the solution was:

    $\displaystyle \theta = \theta_0 cos\left(\sqrt{\frac{g}{l}}t + \phi\right)$

    How do you proceed to get that solution?

    Are you familiar with Laplace Tranforms? That's the method I would use, as it makes the problem so much easier.

    So we start from the equation with the small angle approximation $\displaystyle \ddot{\theta} = - \frac{g}{l} \theta $

    If we assume that the pendulum is going to be in free vibration when released from some angle $\displaystyle \theta_0 $, then the initial conditions of a pendulum as as follows: $\displaystyle \theta(0) = \theta_0 $, $\displaystyle \dot{\theta}(0) = 0 $.

    Then perform the Laplace Transform of both sides of the equation, noting that:

    $\displaystyle \mathcal{L} ( \ddot{\theta}) = s^2 \Theta(s) - s \theta(0) - \dot{\theta}(0) $

    $\displaystyle \mathcal{L}( k \theta )= k \Theta(s) $ for some constant $\displaystyle k $ (in this case $\displaystyle k = -\frac{g}{l} $

    Once you've done that, insert your initial conditions into the new transformed equation, and rearrange to make $\displaystyle \Theta(s) $ the subject. Then use the following rule to perform the inverse Laplace Transform to get $\displaystyle \theta(t) $:

    $\displaystyle \mathcal{L}^{-1} \bigg(K \frac{s}{s^2 + \omega^2}\bigg) = K \cos(\omega t) $, for some constant $\displaystyle K $. (In order to get your equation into the form you need to use this inverse laplace transform rule, it might be helpful to know that $\displaystyle \frac{g}{l} = \bigg(\sqrt{\frac{g}{l}}\bigg)^2 $ )
    Last edited by Mush; Oct 30th 2009 at 01:25 PM.
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    Quote Originally Posted by Aryth View Post
    I've already derived the equation, and simplified it to the linear case, now I have to solve it... The equation is:

    $\displaystyle \frac{d^2\theta}{dt^2} = -\frac{g}{l}\theta$

    Where l is length of the pendulum.

    Doing some research, I keep finding that the solution is:

    $\displaystyle \theta = \theta_0 cos\left(\sqrt{\frac{g}{l}}t\right)$

    Where $\displaystyle \theta_0$ is the initial condition at time t=0

    Now, here's my attempt at a solution:

    Since it is linear and homogeneous, we can assume a solution:

    $\displaystyle \theta = e^{kt}$

    Where k is an arbitrary constant.

    This leads to the indicial equation:

    $\displaystyle k^2 + \frac{g}{l} = 0$

    The solution to which is:

    $\displaystyle k = \pm i\sqrt{\frac{g}{l}}$

    This leads to two solutions:

    $\displaystyle \theta_1 = e^{i\sqrt{\frac{g}{l}}t}$
    $\displaystyle \theta_2 = e^{-i\sqrt{\frac{g}{l}}t}$

    A linear combination of these leads to the general homogeneous solution:

    $\displaystyle \theta_h = Ae^{i\sqrt{\frac{g}{l}}t} + Be^{-i\sqrt{\frac{g}{l}}t}$

    Expanding the exponentials into their trigonometric form, we get:

    $\displaystyle \theta_h = A\left[cos\left(\sqrt{\frac{g}{l}}t\right) + isin\left(\sqrt{\frac{g}{l}}t\right)\right]$ $\displaystyle + B\left[cos\left(\sqrt{\frac{g}{l}}t\right) - isin\left(\sqrt{\frac{g}{l}}t\right)\right]$

    $\displaystyle \theta_h = (A + B)cos\left(\sqrt{\frac{g}{l}}t\right) + i(A - B)sin\left(\sqrt{\frac{g}{l}}t\right)$

    From here I'm not sure how to proceed...
    Take C= A+ B and D= i(A- B) so your answer becomes $\displaystyle \theta_h= C cos(\left(\sqrt{\frac{g}{l}}}t\right)+ D sin\left(\sqrt{\frac{g}{l}}t\right)$

    Now there is a trig identity that says that cos(x+ y)= cos(x)cos(y)- sin(x)sin(y). That would be of this form if we could take [tex]x= \sqrt{\frac{g}{l}}}t\right[/itex] and y such that cos(y)= C, sin(y)= -D.

    Unfortunately, we must have $\displaystyle cos^2(y)+ sin^2(y)= 1$ and it may not happen that $\displaystyle C^2+ D^2= 1$. To fix that, multiply and divide by $\displaystyle \sqrt{C^2+ D^2}$. Then we have $\displaystyle \sqrt{C^2+ D^2}\left(\frac{C}{\sqrt{C^2+ D^2}}cos\left(\sqrt{\frac{g}{l}}}t\right)+ \frac{D}{\sqrt{C^2+ D^2}}sin\left(\sqrt{\frac{g}{l}}}t\right)\right)$.

    Now, we do have $\displaystyle \left(\frac{C}{\sqrt{C^2+ D^2}}\right)^2+ \left(\frac{D}{\sqrt{C^2+ D^2}}\right)^2= \frac{C^2+ D^2}{C^2+ D^2}= 1$.

    We have $\displaystyle y(t)= \sqrt{C^2+ D^2}cos\left(\sqrt{\frac{g}{l}}}t\right+ \phi)$ where $\displaystyle \phi= arccos\left(\frac{C}{\sqrt{C^2+ D^2}}\right)$
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