- #1
Hetware
- 125
- 1
I believe this is pretty standard.
Given a mass m on a spring with spring constant k, a solution to the second order differential equation of motion m[itex]\ddot{x}[/itex] = -kx, is x = cos ωot, and ωo = [itex]\sqrt{k/m}[/itex].
If that same oscillator is driven with a force F(t) = Fo cos ωt the equation of motion becomes m[itex]\ddot{x}[/itex] = -kx + F(t). If we try x = C cos ωt as a solution, we find it works (mathematically) if C = Fo/m(ωo2-ω2).
But this doesn't seem real. It says x is going to trace out a cos curve when plotted against time. If I'm applying a force to the oscillator at a frequency different form its natural frequency, sometimes the spring force and the driving force will be in the same direction, and other times they will be in opposition. That will lead to a more complex motion than C cos ωt.
The math gives us the solution, but it doesn't seem to make physical sense. I believe in resonance, and the general notion that the the amplitude will be greater as ωo→ω.
I've seen this developed in one form or another several times, and every time, my mind rebelled, but I just forced myself to accept it.
I cannot think of a situation in nature in which C cos ωt would be the actual motion of the oscillator. Supposedly this solution applies to a mass on a spring. Can anybody give me an instance in which the motion of the mass would actually conform to C cos ωt?
Given a mass m on a spring with spring constant k, a solution to the second order differential equation of motion m[itex]\ddot{x}[/itex] = -kx, is x = cos ωot, and ωo = [itex]\sqrt{k/m}[/itex].
If that same oscillator is driven with a force F(t) = Fo cos ωt the equation of motion becomes m[itex]\ddot{x}[/itex] = -kx + F(t). If we try x = C cos ωt as a solution, we find it works (mathematically) if C = Fo/m(ωo2-ω2).
But this doesn't seem real. It says x is going to trace out a cos curve when plotted against time. If I'm applying a force to the oscillator at a frequency different form its natural frequency, sometimes the spring force and the driving force will be in the same direction, and other times they will be in opposition. That will lead to a more complex motion than C cos ωt.
The math gives us the solution, but it doesn't seem to make physical sense. I believe in resonance, and the general notion that the the amplitude will be greater as ωo→ω.
I've seen this developed in one form or another several times, and every time, my mind rebelled, but I just forced myself to accept it.
I cannot think of a situation in nature in which C cos ωt would be the actual motion of the oscillator. Supposedly this solution applies to a mass on a spring. Can anybody give me an instance in which the motion of the mass would actually conform to C cos ωt?