Frequency of Undamped Driven Oscillator near Zero

[Samama Fahim]

Description of the Problem:
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?

 

[olivermsun]

I don't know if this agrees with your common sense and/or helps, but here's one practical example that you might consider:
Suppose you are hanging a mass by a spring (or a rubber band or something similar), and the system is at equilibrium, with the spring stretched to whatever (additional) length is needed to counteract the force of gravity on the mass. Now imagine that you exert an oscillating force on the weight, in addition to the "balanced" force of the spring against gravity, by moving the support of the spring (or the rubber band) up and down. If you do this quickly, near the natural oscillating frequency of the spring, you can get the spring-mass system oscillating up and down a huge distance, nearly out of phase with your forcing. On the other hand, if you move the support up and down extremely slowly, the spring-mass hardly even "feels" the springiness. Instead, the mass moves up and down more or less exactly when you move the support up and down. The spring remains nearly in equilibrium at all times because of the huge mismatch in frequency between the spring-mass oscillation frequency and the external force.