Driven oscillator amplitude steady state X(t) = ##Asin(\omega t + \delta)##

[Redwaves]

Homework Statement

Finding the amplitude of a driven oscillator

Relevant Equations

Oscillator without driven force at $t_0$, $x(t_0) = 5.0e^{-7t_0} cos (24t_0 + \frac{\pi}{4})$
at $t_2$ an external force is applied
Force, $F(t_2) = 35.0sin(25t_2 + \frac{\pi}{3})$
m = 2.5kg

I found $\frac{\gamma}{2} = 7$, $\gamma = 14$
$\omega_0^2 = \omega_d^2 + \frac{\gamma^2}{4} = 25$
$\omega_0 = \omega = 25$, thus $\delta = \frac{\pi}{2}$

$A = \frac{\frac{F_0}{m}}{\sqrt((\omega_0^2 - \omega^2)+ \gamma^2\omega^2)} = 0.04$
Thus, $X(t) = 0.04sin(25t + \frac{\pi}{3} - \frac{\pi}{2})$
However, the correct answer in my book is
$X(t) = 4sin(25t + \frac{4\pi}{3})$

I don't see why my amplitude and the phase isn't correct.

 

[haruspex]

The question as stated does not really make sense. $t_0, t_2$ are presented as constants, specific instants in time, then treated as variables. That gets really confusing given the phase specifications.
I assume it means...
A certain oscillator would obey $x(t)=5e^{-7t}\cos(24t+\frac\pi 4)$, were it not for the applied force $F(t)=35\sin(25t+\frac\pi 3)$.
Is that how you interpreted it?

 

[Redwaves]

The question as stated does not really make sense. $t_0, t_2$ are presented as constants, specific instants in time, then treated as variables. That gets really confusing given the phase specifications.
I assume it means...
A certain oscillator would obey $x(t)=5e^{-7t}\cos(24t+\frac\pi 4)$, were it not for the applied force $F(t)=35\sin(25t+\frac\pi 3)$.
Is that how you interpreted it?

In the book they use t and t' I used t_0 and t_2 to avoid confusion

 

[TSny]

I was also wondering about the wording of the question. @Redwaves, Can you give a full statement of the problem exactly as given in the text?

 

[Redwaves]

The question is in french, so I try to translate the best I can, sorry.
First the oscillator oscillate like $x(t) = 5.0e^{-7t} cos (24t + \frac{\pi}{4})$, then

with the same oscillator on a different experience we apply a force $F(t') = 35.0sin(25t' + \frac{\pi}{3})$

 

[PeroK]

The question is in french, so I try to translate the best I can, sorry.
First the oscillator oscillate like $x(t) = 5.0e^{-7t} cos (24t + \frac{\pi}{4})$, then

with the same oscillator on a different experience we apply a force $F(t') = 35.0sin(25t' + \frac{\pi}{3})$

Does it specify a relationship between $t$ and $t'$?

 

[Redwaves]

Does it specify a relationship between $t$ and $t'$?

There's no relation between t and t'.
Basically, I could replace t' by t, I think they use t' just to show that it is a new experience.

 

[TSny]

Regarding the 4.0 vs .04 for the amplitude, what are the units?

 

[Redwaves]

Regarding the 4.0 vs .04 for the amplitude, what are the units?

I don't know for 4.0, however Should I know the units of .04 using $A = \frac{\frac{F_0}{m}}{\sqrt((\omega_0^2 - \omega^2)+ \gamma^2\omega^2)} = 0.04$ , where m = 2.5kg

The issue could possibly be the units. It makes sense.

 

[TSny]

Since the mass is given in kg, I assume that the units are SI units. The SI unit for amplitude would then be meters. It looks to me like your answer of 0.04 m is correct. However, maybe the textbook answer expresses the amplitude in centimeters. I'm just speculating.

Your result of $\pi/3 - \pi/2$ for the phase of the steady-state solution also appears to me to be correct.

 

[Redwaves]

All right, thanks. I'll try to investigate. I thought my answer was just bad.

 

[haruspex]

There's no relation between t and t'.
Basically, I could replace t' by t, I think they use t' just to show that it is a new experience.

As I wrote, that creates a problem for how to interpret the stated phases, $\pi/3$ and $\pi/4$.

 

[Redwaves]

As I wrote, that creates a problem for how to interpret the stated phases, $\pi/3$ and $\pi/4$.

I'm not sure if I was clear. Or just the statement.

 

[TSny]

Maybe it would help if you quote the original question in its entirety in French.

 

[haruspex]

I'm not sure if I was clear. Or just the statement.

If the equations are expressed in independent time bases, t, t', the phase constants in those equations become meaningless. If you shift the origin of one time base relative to the other then the phase relationship shifts.

 

[Redwaves]

Here's the statement in french.
"Lors d'une expérience se déroulant au temps t' un oscillateur de masse m libre de toute force externe et auquel ont été donnés une vitesse et un déplacement initiaux, oscille suivant un déplacement x(t'). Dans une autre expérience, se déroulant au temps t, n'ayant aucun rapport avec le temps t', une force externe F(t) est appliquée à cet oscillateur. Déterminer aussi précisément que possible, l'expression du déplacement de l'oscillateur en régime permanent, X(t)."

x(t') and F(t) was as above.

 

[TSny]

OK, thanks for stating the entire problem.

"Google Translate" gives the following translation:

During an experiment taking place at time t', an oscillator of mass m free of any external force and to which an initial speed and an initial displacement have been given, oscillates according to a displacement x (t'). In another experiment, taking place at time t, having no relation to time t', an external force F(t) is applied to this oscillator. Determine as precisely as possible the expression of the displacement of the oscillator in steady state, X(t).

As you pointed out in post #7, there is no relation between $t'$ and $t$.
I don't see any mistakes in your work.

 

[haruspex]

Here's the statement in french.
"Lors d'une expérience se déroulant au temps t' un oscillateur de masse m libre de toute force externe et auquel ont été donnés une vitesse et un déplacement initiaux, oscille suivant un déplacement x(t'). Dans une autre expérience, se déroulant au temps t, n'ayant aucun rapport avec le temps t', une force externe F(t) est appliquée à cet oscillateur. Déterminer aussi précisément que possible, l'expression du déplacement de l'oscillateur en régime permanent, X(t)."

x(t') and F(t) was as above.

Did the given phase constants, $\pi/3$ and $\pi/4$, play any part in your calculations? If they did, you must have made some assumption regarding the relationship between t and t', perhaps without realising.

 

[Redwaves]

Did the given phase constants, $\pi/3$ and $\pi/4$, play any part in your calculations? If they did, you must have made some assumption regarding the relationship between t and t', perhaps without realising.

No, I didn't use the phases in my calculations. I think I typed all my calculations.