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Consider a spring-mass system with spring constant ##k## and mass ##m##. Suppose I apply a force ##F_0 \cos(\omega t)## on the mass, but the frequency ##\omega## is very small, so small that it takes the system, say, a million years to reach a maximum and to go to 0 and so on. This is actually from the following lecture: by Walter Lewin. He argues that if the force goes so slowly then there must be equilibrium at all moment in time between the spring force and the force that we apply, and he does so without using the equation for amplitude of an undamped driven harmonic oscillator or the differential equation for such a motion. We should be able to guess it with common sense.

Equation: $$m\ddot{x} = -kx + F_0 \cos(\omega t)$$

Question: I suppose that there must be a common sense explanation of the driving force and the spring force cancelling each other or almost doing so when the frequency of the driving force is small or near zero. How does a force “going extremely slowly” make the spring force cancel or brings the system into equilibrium at all moments in time? What is the connection between this small frequency and the equilibrium?