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Water tower/spring system Diff Eq

  1. Mar 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support acts as the spring, where the induced vibrations are horizontal. Suppose that the container with water has a mass of 10,000 kg. It takes a force of 1000 N to displace the container 1 m. For simplicity, assume no friction. When the earthquake hits the water tower is at rest.

    Suppose that an earthquake induces an external force ##F(t)=mA\omega^2\cos(\omega t)##.

    What is the natural frequency of the water tower?

    Find a formula for the maximal amplitude of the resulting oscillations of the water container (the maximal deviation from the rest position). The motion will be a high frequency wave modulated by a low frequency wave, so simply find the constant in front of the sines.

    2. Relevant equations

    3. The attempt at a solution

    Here's the differential equation I set up:

    ##10,000x''+1,000x=mA\omega^2\cos(\omega t)##

    For the natural frequency, I used the formula ##\omega_0=\sqrt{\frac{k}{m}}##, which gives me ##\omega_0=\sqrt{\frac{1}{10}}\text{ rad/s}=\frac{1}{2\pi}\sqrt{\frac{1}{10}}\text{ Hz}##. Is this right?

    And for the second part, do I just solve this equation? I'm not sure what it means to find the "constant in front of the sines."
     
  2. jcsd
  3. Mar 3, 2013 #2

    HallsofIvy

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    What you have is that [itex]y(t)= cos(\sqrt{1/10}t)[/itex] and [itex]y(t)= sin(\sqrt{1/10}t)[/itex] are solutions to the associated homogeneous equation, 10000x''+ 1000x= 0. Can you find the general solution to the entire equation?
     
  4. Mar 3, 2013 #3
    Yup, I've found that the general solution is
    ##\displaystyle x(t)=C_1\cos\left(\sqrt{\frac{1}{10}}t\right)+C_2 \sin \left(\sqrt{\frac{1}{10}}t\right)+\frac{mA \omega ^2}{1000-10000\omega^2}\cos(\omega t)##

    We're also assuming ##\omega\not=\omega_0## for the second part. I forgot to put that in my first post.

    By the way, is the ##m## in the solution the same as the mass of the water tower?
     
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