# What are the conditions for different types of damping in oscillations?

• kraigandrews
In summary, we are given a mass m moving in one dimension x with a restoring force of −kx and a damping force of −γ[(x)\dot]. The conditions for underdamped oscillations, overdamped oscillations, and critical damping are determined. For a specific scenario with m=0.80 kg, γ=1.18 kg/s, and critically damped oscillations, the value of k is calculated to be 0.435 N/m. The solution for the differential equation for critical damping is x(t)=(A+Bt)e^(-\betat), and solving for A and B gives the values of a and \betaa. When solving for the time at displacement a/2, t=2
kraigandrews

## Homework Statement

A mass m moves in one dimension x subject to a restoring force −kx and a damping force −γ[(x)\dot]. What are the conditions for underdamped oscillations, overdamped oscillations, and critical damping?
Now, suppose m is 0.80 kg, γ is 1.18 kg/s, and the oscillations are critically damped. What is k?

The object starts at rest, displaced by some amount a. At what time is the displacement a/2?

## Homework Equations

$\beta$=$\gamma$/(2m)

## The Attempt at a Solution

Ok so I know k=0.435 N/m

and I know the solution for the diff eq for critical damping is:

x(t)=(A+Bt)e^(-$\beta$t)

my problem then becomes solving for A and B.
from the given info I would think x(0)=a and x'(t)=0
giving me A=a and B=$\beta$a.

I am not sure if this because when I solve for t at x(a/2) I do not get the right answer

How did you solve for t? I found t=2.28 s.

## 1. What are critically damped oscillations?

Critically damped oscillations refer to a type of harmonic motion where the damping force is equal to the restoring force, resulting in the system returning to equilibrium without any oscillations or overshooting.

## 2. How do critically damped oscillations differ from overdamped and underdamped oscillations?

In overdamped oscillations, the damping force is greater than the restoring force, causing the system to return to equilibrium slowly without any oscillations. In underdamped oscillations, the damping force is less than the restoring force, resulting in oscillations that gradually decrease in amplitude over time.

## 3. What factors affect the critical damping ratio?

The critical damping ratio is affected by the mass of the system, the spring constant, and the damping constant. A higher mass or damping constant will result in a higher critical damping ratio, while a higher spring constant will result in a lower critical damping ratio.

## 4. What are some real-life examples of critically damped oscillations?

One example of critically damped oscillations is the shock absorbers in a car's suspension system. The damping force of the shock absorbers is designed to be equal to the restoring force of the springs, resulting in a smooth and controlled ride without any bouncing or oscillations.

## 5. How is the critical damping ratio calculated?

The critical damping ratio can be calculated by dividing the damping constant by 2 times the square root of the mass of the system multiplied by the spring constant. This can also be expressed in terms of the natural frequency of the system, where the critical damping ratio is equal to the damping constant divided by 2 times the natural frequency.

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