Find the time constant of a damped system

AI Thread Summary
To find the time constant of a damped system, the damping constant must first be determined from the reduction of amplitude to 5% over 25 seconds. The relevant equation for damped motion is xmax(t) = Ae^(-t/2τ), which relates amplitude and time. The initial amplitude was calculated to be 89 cm, and this value is crucial for determining the damping constant. Once the damping constant is found, it can be substituted into the formula τ = m/b to calculate the time constant. Understanding these relationships is essential for solving the problem effectively.
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Homework Statement



A 3.9kg block hangs from a spring with constant 2160 N/m. The block is pulled down 6.2 cm from the equilibrium position and given an initial velocity of 1.5 m/s back towards equilibrium. The mass and spring are now immersed in water to damp the motion, so that the amplitude is reduced to 5% of its original value in 25 seconds. What is the time constant?

Homework Equations


τ=m/b where b is damping constant
xmax(t)=Ae^-t/2τ


The Attempt at a Solution


In previous parts of the question I found that the Amplitude (before damping) is 89cm, but I'm not sure if that's useful in this problem. I thought there might be some way to find out what the damping constant is from the fact that it reduces to 5% of A in 25 seconds, and then just plug that into τ=m/b, but I'm not sure how to go about that. Any help would be really appreciated!
 
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I thought there might be some way to find out what the damping constant is from the fact that it reduces to 5% of A in 25 seconds,...
Sounds good to me - what does the equation of damped harmonic motion look like?
 

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