Water tower/spring system Diff Eq

1. Mar 3, 2013

SithsNGiggles

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. Mar 3, 2013

HallsofIvy

Staff Emeritus
What you have is that $y(t)= cos(\sqrt{1/10}t)$ and $y(t)= sin(\sqrt{1/10}t)$ are solutions to the associated homogeneous equation, 10000x''+ 1000x= 0. Can you find the general solution to the entire equation?

3. Mar 3, 2013

SithsNGiggles

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?