# Critically Damped System - Viscous force

• dumbdumNotSmart
In summary, the problem asks to prove that for a critically damped system, the coefficient of viscous force (b) can be calculated as b=2m√(g/Δx) in order to minimize oscillation and reach equilibrium in the shortest amount of time.
dumbdumNotSmart

## Homework Statement

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 $$b=2m \sqrt{(g/Δx)}$$

## Homework Equations

$$F=ma$$

## 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?

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?

dumbdumNotSmart said:

## Homework Statement

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 $$b=2m \sqrt{(g/Δx)}$$

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.

dumbdumNotSmart
Dr.D said:
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?

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.

vela said:
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.

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.

## 1. What is a critically damped system?

A critically damped system is a type of mechanical system that is designed to return to its equilibrium position as quickly as possible without any oscillations. This is achieved by balancing the damping force and the restoring force, resulting in a critically damped response.

## 2. What is the role of viscous force in a critically damped system?

The viscous force in a critically damped system is responsible for dissipating energy through friction. This helps to dampen any oscillations and bring the system back to equilibrium quickly.

## 3. How is the damping ratio related to the viscous force in a critically damped system?

The damping ratio is a measure of the amount of damping in a system. In a critically damped system, the damping ratio is equal to 1, indicating that the viscous force is equal to the restoring force.

## 4. What are the advantages of a critically damped system?

A critically damped system has several advantages, including a quick response time, stability, and minimal oscillations. This makes it ideal for applications where precision and stability are crucial, such as in control systems.

## 5. How is a critically damped system different from an overdamped or underdamped system?

An overdamped system has a higher damping ratio and a slower response time, while an underdamped system has a lower damping ratio and tends to oscillate before reaching equilibrium. A critically damped system strikes a balance between these two and has the quickest response time without any oscillations.

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