# Is Archimedes' principle valid with moving objects?

I was solving a classic application of Archimedes' principle: a body partially submerged that is made to oscillate vertically and perform a simple harmonic oscillations. The equations turn out to be the right ones for a sho but then it struck me: How is it that the liquid remains static? The body is moving vertically and the liquid should be affected.

Is this an approximation? Because I've seen this "problem" in many lecture notes. If so, how can I strictly justify the approximation? Does Buoyant force=rho gV still hold exactly?

If the object were bobbing up and down in the water at high enough speed, then there would at least be an appreciable drag force which would, to first approximation, be \begin{align} \mathbf F_d = -b\mathbf v \end{align} where $b$ is a constant and $\mathbf v$ is the velocity of the object relative to the water. If the speed $|\mathbf v|$ of the object is small, then so is the magnitude of the drag force, and simple harmonic motion becomes a good approximation.