# 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

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?