# Knowing Damping and Spring Constant Finding Time

1. Apr 28, 2013

### CalebtheCoward

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

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

3. The attempt at a solution

Not exactly sure how to get started.

2. Apr 28, 2013

### 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.

3. Apr 28, 2013

### 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?

4. Apr 28, 2013

### 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?

5. Apr 29, 2013

### 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.

6. Apr 29, 2013

### 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.

7. Apr 29, 2013

### 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.