Knowing Damping and Spring Constant Finding Time

[CalebtheCoward]

Homework Statement

A spring with a force constant of 1.50 N/m is attached to a mass of 120 g. The system has a damping constant of 0.0180 Ns/m. How long does it take the amplitude of the oscillations to decrease from 10.0 mm to 5.00 mm?

Homework Equations

(double dot)x +(c/m)(one dot)x+(k/m)x=0

The Attempt at a Solution

Not exactly sure how to get started.

 

[UVW]

How do we usually get started with equations of the form a*y'' + b*y' + c*y = 0? Here's a hint: try to find a characteristic equation.

 

[CalebtheCoward]

I think I understand what you're saying. So start with a standard equation, then take the derivative producing two other equations, and substitute the given values in front of each of the respective equations. Is this correct?

 

[UVW]

It might be, but I'll need you to be a little more specific. Can you show us the work that you are talking about?

 

[CalebtheCoward]

Well. I found the equation that works.
final amplitude=original amplitude times e raised to the quantity of -(bt/2m) where b represents the damping constant, t the amount of time, and m the mass of said device. And after that it's pretty basic math. If this could be derived from the characteristic equation, I would hope so, but I must admit that I couldn't do such a thing.

 

[CalebtheCoward]

And for others,
First 5=10e^-(bt/2m), 5/10=e^-(bt/2m), ln(1/2)=-(bt/2m), solve for t, t=(2mln(1/2))/-b, substitute given values, [2(0.120)ln(1/2)]/-(0.0180)=t=9.24 seconds.

 

[UVW]

The equation looks great.

We know from the characteristic equation that the solution is a linear combination of exponentials and sines/cosines whose values we can determine from the characteristic equation. After this, we can find the amplitude of the linear combination with the Pythagorean theorem. If all works out, the sines and cosines will go to 1 (remember your trig identities), and the exponential will slip out of the amplitude.

Give that a try. The equation should come out as you got it.