# Forced oscillations without damping

## 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!

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

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...

sort of*

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