What Constant c Achieves Critical Damping?

[rzn972]

Homework Statement

Suppose a mass- spring is modeled by x''+ cx'+100x. What should the damping constant c be in order that critical damping occurs?

Homework Equations

The Attempt at a Solution

I set the discriminant equal to zero so that there are real repeated roots.
B^2-4ac=0
c^2=400.
c=20 or c=-20.
The answer should only be c=20. For critical damping, the constant must be positive?

 

[pasmith]

Homework Statement

Suppose a mass- spring is modeled by x''+ cx'+100x = 0. What should the damping constant c be in order that critical damping occurs?

Homework Equations

The Attempt at a Solution

I set the discriminant equal to zero so that there are real repeated roots.
B^2-4ac=0
c^2=400.
c=20 or c=-20.
The answer should only be c=20. For critical damping, the constant must be positive?

What is the repeated root? You need it to be negative so that $x$ decays with time, do you not?

 

[AlephZero]

For critical damping, the constant must be positive?

For any level of damping, c must be positive.

If c is negative, you are putting energy into the system, not taking energy out.

 

[ChrisVer]

You could check the Damping ratio here:
http://en.wikipedia.org/wiki/Damping
and the critical damping condition

 

[BvU]

An other way to look at it: damping force must be opposite to x', which it is if c is positive:

x'' = -cx' -100x