Forced oscillations without damping

[Dixanadu]

Homework Statement

We are to develop the equations of motion for an undamped horizontal spring system, the mass of which is being driven by a periodic force: F=F0 cos wt. I know how to do it but my teacher has defined an odd term, the meaning of which I want to be clarified.

Homework Equations

The potential energy of the system is V = 1/2 kx^2. So the force is -kx.
The driving force is F0 cos wt.
The natural frequency of the system is w0^2 = k/m

The Attempt at a Solution

So, here's my solution:
$m\ddot{x}+kx = F_{0} cos(ωt)$

My teacher has the same thing. But what he does next is that he says
$a = \frac{F_{0}}{k}$

And then:
$\ddot{x}+ω^{2}_{0}x = ω^{2}_{0} a cos(ωt)$

Which is all fine...but what the heck is $a$? is it just some random thing or is it something of physical significance?

Thanks a lot guys!

 

[haruspex]

Suppose that the applied force were constant, at its max value F₀. What would F₀/k represent?

 

[Dixanadu]

By constant, do you mean it's not periodic? That means it has sort infinite time period. So it's like moving it in a straight line. Logically I'd think that F0/k would be the extension of the spring due to the force but I'm not sure if that's correct...

 

[Dixanadu]

sort of*

 

[Sentin3l]

I think my reasoning for this is correct.

Consider first that the frequency of the driving force goes to zero.

$lim_{ω\rightarrow0}\:F_{0}cos(ωt) = ...$

The driving force then goes to it's max value of $F_{0}$
This can now be described by Hooke's Law (considering only the magnitude, not direction):

$F_{0} = kx → x = F_{0}/k$

So essentially $a$ can be thought of a non-periodic driving force amplitude, where the spring constant effectively determines the value of it.

My textbook literally reads: "At very low frequencies the amplitude of oscillations tends to the value of the amplitude a of the point of suspension. Under these conditions the motion is governed by the spring constant of the spring."

 

[haruspex]

I'd think that F0/k would be the extension of the spring due to the force

Quite so.