Checking whether my Diff EQ limit problem is correct

[lilmul123]

figured it out, thanks

Figured it out, thanks.

 

[AEM]

I believe I have solved the problem, I just want to make sure of its correctness.

I was to solve a spring differential equation that involved a damping force and to create a equation of motion. I did. Now, I need to figure out whether the weight ever passes through its equilibrium. The equation I ended up with is

x = e^(-8t) * (.5 - 2t) where x is displacement and t is time.

Now, to check whether weight passes through the equilibrium, what I did was set up the limit as x approaches infinity of (.5 - 2t)/(e^8t). Then, I used l'hopital's rule to find the limit as x approaches infinity of (-2)/(8e^8t). This comes out to be that the limit is 0, and therefore, the spring never passes through the equilibrium and there is no t-value.


Is my logic sound and my math correct?

I think you wrote the wrong thing. Don't you want to take the limit as t goes to infinity?

 

[LCKurtz]

I believe I have solved the problem, I just want to make sure of its correctness.

I was to solve a spring differential equation that involved a damping force and to create a equation of motion. I did. Now, I need to figure out whether the weight ever passes through its equilibrium. The equation I ended up with is

x = e^(-8t) * (.5 - 2t) where x is displacement and t is time.

Now, to check whether weight passes through the equilibrium, what I did was set up the limit as t approaches infinity of (.5 - 2t)/(e^8t). Then, I used l'hopital's rule to find the limit as t approaches infinity of (-2)/(8e^8t). This comes out to be that the limit is 0, and therefore, the spring never passes through the equilibrium and there is no t-value.


Is my logic sound and my math correct?

No, your logic isn't sound, assuming your solution for x is correct. Presuming the equilibrium position is x = 0 the question is asking whether that happens for t > 0.

 

[LCKurtz]

Right, and as t goes to infinity, it seems x gets closer and closer to x=0, but will never hit zero, is that correct or am I still lost?

It isn't asking you about the limit as t goes to infinity. It is asking whether there are any finite values of t where x hits the equilibrium position.

 

[LCKurtz]

Stop with the infinitely large bit. The question has nothing to do with infinity. The question is whether x = 0 for any value of t. It isn't a rocket science question. Look at your equation.

 

[LCKurtz]

You're welcome. The point of questions like that is that it depends on the damping. If the system is damped strongly enough it might just ooze down towards equilibrium. Otherwise it might do as in this problem, cross equilibrium once and settle towards equilibrium. Or if it is underdamped, it might oscillate around equilibrium as it settles.