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SithsNGiggles
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Homework Statement
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.
Homework Equations
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."