# Critically Damped System - Viscous force

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1. Apr 15, 2017

### dumbdumNotSmart

1. The problem statement, all variables and given/known data
You got a plate hanging from a spring (hookes law: k) with a viscous force acting on it, -bv.

If we place a mass on the plate, gravity will cause it to oscillate.

Prove that if we want the plate to oscillate as little as possible (Crticial damping, no?), then $$b=2m \sqrt{(g/Δx)}$$
2. Relevant equations
$$F=ma$$
3. The attempt at a solution
I cannot for the life of me get the differential equation right. Given the conditions I assume they want me to find b for a critically damped system (also answer looks like so). This is my best try

$$ma=-bv +mg -kx$$

Since g is not accompanied by x, dx or d2x it will be part of the particular solution, not the homogenous. What am I missing here?

2. Apr 15, 2017

### Dr.D

Gravity is a constant, therefore it will contribute a constant, nonoscillatory terms to the solution. Why do you think you need to focus on the homogeneous solution only?

3. Apr 15, 2017

### vela

Staff Emeritus

Is this the actual wording of the problem? It's a bit misleading to say gravity will cause the system to oscillate since it's the spring and mass which results in the oscillation. Also, I wouldn't equate "oscillate as little as possible" with "critical damping." A highly overdamped system won't oscillate either, but a critically damped system could still allow some overshoot. That motion is arguably allowing "more oscillation" than the overdamped system.

4. Apr 15, 2017

### dumbdumNotSmart

Mainly because the condition for minimum time to reach equilibrium is given by a critically damped system. If this is so then it must be shown in the homogenous equation.

The question is not phrased that way. I just wanted to make it clear that the initial impulse was given by gravity. The question does state that the system has to reach equilibrium sooner than any other system. According to my reading (Tipler & Mosca Physics textbook) a critically damped system will return to it's resting state sooner than any other system. A overdamped system will settle on the resting state but will take more time.