Proof of oscillation about the equilibrium

[Bonnie]

Homework Statement

The problem is question 2(a) in the attached pdf. I seem to find myself at a dead end and am not sure where to go from here - I will attach my working in a separate file, but basically I need to show that the oscillator passes/crosses over the x = 0 boundary at a positive time, but I don't seem to be able to do that (or have enough information to) without ending up with nothing, or an impossible equation, like an exponential term = 0.

Homework Equations

The Attempt at a Solution

 

[BvU]

Your picture is half illegible. Seems you do find an $\omega$ though.
However: your try with an intial displacement of 0 of course doesn't let the thing oscillate !

free tip: work with symbols, not with numbers. Only at the last moment, if/when a value is needed, subtitute values.

 

[Bonnie]

Your picture is half illegible. Seems you do find an $\omega$ though.
However: your try with an intial displacement of 0 of course doesn't let the thing oscillate !

free tip: work with symbols, not with numbers. Only at the last moment, if/when a value is needed, subtitute values.

ω is 20 rads^-1, from the equation (I have shown it only with values substituted):
p = -γ/2 +/- √[ω² - (γ/2)²]
where ω² = 400 and γ = 5\

And I set x(0) = D, not 0, as x is a function of t, is that incorrect?
Thanks

 

[BvU]

And I set x(0) = D, not 0, as x is a function of t, is that incorrect?

No, that's correct. But it looks a lot like a zero on my screen.

Your Solution

has $x=Ce^{-{5\over 2}t} e^{\rm illegible}$ when originally it was

( so probably $x=Ce^{pt}$ ?) and you found two $p$. What happened to the second ?

 

[Bonnie]

No, that's correct. But it looks a lot like a zero on my screen.

Your Solution View attachment 225681
has $x=Ce^{-{5\over 2}t} e^{\rm illegible}$ when originally it was View attachment 225682 ( so probably $x=Ce^{pt}$ ?) and you found two $p$. What happened to the second ?

I'll try to attach a better photo, but the second p is included in the e^{+/- 393.75j}

 

[Bonnie]

I'll try to attach a better photo, but the second p is included in the e^{+/- 393.75j}

Ah, I've just realized that the photo quality is significantly decreased by uploading it here. Apologies for that

 

[BvU]

You need to rethink your solution. A second order differential equation needs two integration constants (one for $x$ and one for $\dot x$ if you want to put it that way).

If there are two $p$ to solve the characteristic equation, then $C_1 \,e^{p_1 t}$ is a solution and so is $C_2 \,e^{p_2 t}$.