It's often said that inertial mass is a measure of how hard it is to shake an object (as distinct from gravitational mass, which is how hard it is to lift an object in a gravitational field). Because this is so often said, one would think that at some point in my physics career I would have been asked to do a toy problem where I derive a formula for the inertial mass of an object based on how hard it is to shake. But I've never been assigned such a problem, so I sat down to try to perform the calculation myself.
It seems to me that "how hard it is to shake an object" could be made precise as the average power needed to induce oscillations at a certain amplitude with a certain period. So suppose we shake a mass so that its motion is given by $$ x(t) = A\sin \left(\frac{2\pi t}{T}\right) $$ This implies an applied force $$ F(t) = m\ddot{x}(t) = -\frac{4\pi^2 mA}{T^2}\sin \left(\frac{2\pi t}{T}\right) $$ And thus an instantaneous power $$ P(t) = F(t)\dot{x}(t) = -\frac{8\pi^3 mA^2}{T^3}\sin \left(\frac{2\pi t}{T}\right) \cos \left(\frac{2\pi t}{T}\right) $$ which can be simplified to $$ P(t) = -\frac{4\pi^3 mA^2}{T^3}\sin \left(\frac{4\pi t}{T}\right) $$ Averaging over a period, we obtain \begin{align} \langle P \rangle &= \frac{1}{T} \int_0^T-\frac{4\pi^3 mA^2}{T^3}\sin \left(\frac{4\pi t}{T}\right) \, dt \\&= -\frac{4\pi^3 mA^2}{T^4} \int_0^T \sin \left(\frac{4\pi t}{T}\right) \, dt \\&= \frac{\pi^2 mA^2}{T^3} \left[ \cos\left(\frac{4\pi t}{T}\right) \right]_0^T \\&= 0 \tag{!} \end{align} Disturbingly, we find that the average power is $0$, which makes it useless for calculating mass. This makes some sense, however, since we alternately do work with and against the motion of the mass.
I seem to recall from my time studying E&M that the thing to do in such situations is to instead calculate the RMS of the power, which yields \begin{align} P_{\text{RMS}} &= \frac{4\pi^3 mA^2}{T^3} \left[ \frac{1}{T} \int_0^T \sin^2 \left(\frac{4\pi t}{T}\right) \right]^{1/2} \\&= \frac{4\pi^3 mA^2}{T^3}\left[ \frac{1}{T} \cdot \frac{T}{2} \right]^{1/2} \\&= \frac{2\sqrt{2}\pi^3 mA^2}{T^3} \end{align} In terms of frequency $\omega$, we have $$ P_{\text{RMS}} = \frac{mA^2\omega^3}{2\sqrt{2}} $$ which seems like a pretty reasonable equation. But unlike the E&M case where we are very naturally led to consider RMS quantities (since power goes like $I^2$), the introduction of RMS here seems sort of artificial and like a trick to avoid an unpleasant $0$.
In essence, my question is (1) does this RMS equation for power make precise the notion that inertial mass is how hard it is to shake something, and (2) is there a physical motivation for looking at RMS power in this situation?
