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

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SUMMARY

The discussion focuses on the conditions for underdamped, overdamped, and critically damped oscillations in a mass-spring-damper system. The mass m is 0.80 kg, and the damping coefficient γ is 1.18 kg/s, leading to a calculated spring constant k of 0.435 N/m for critical damping. The solution for the displacement x(t) under critical damping is given by x(t) = (A + Bt)e^(-βt), where β = γ/(2m). The user seeks clarification on solving for time t when the displacement is a/2, having initially calculated t as 2.28 seconds.

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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^(-\betat)

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=\betaa.

I am not sure if this because when I solve for t at x(a/2) I do not get the right answer
 
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How did you solve for t? I found t=2.28 s.
 

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