Topic: Oscillatory Motion
Concepts: Newton’s laws, Simple harmonic motion

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Solution:

Suppose we displace the water so that the right side moves up by distance $x$ and the left side moves down by distance $x$. Then the extra weight on the right side is

where $A$ is the cross-sectional area of the U-tube. This force accelerates all the water so by Newton’s 2nd law,

where the minus sign accounts for the fact that the force is restoring. Since $m=\rho V=\rho AL$, we have

This is of simple harmonic form ($\ddot{z}=-\omega^2z$) so we can identify the angular frequency as

Thus, the frequency is given by

so the answer is A.

Topic: Fluids
Concepts: Pressure with depth

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Solution:

We compute the pressure at the bottom of the U-tube in two ways: through the left side and through the right side. From the left side,

From the right side,

Since these are equal, we have

We are given $\rho_{\text{oil}}=\rho_{\text{water}}/2$ so

Finally, the height difference $h$ between both sides is

so the answer is B.