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halfoflessthan5
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Homework Statement
the equation of motion for a damped harmonic oscillator is
d^2x/dt^2 + 2(gamma)dx/dt +[(omega0)^2]x =0
...
show that
x(t) = Ae^(mt) + Be^(pt)
where
m= -(gamma) + [(gamma)^2 - (omega0)^2 ]^1/2
p =-(gamma) - [(gamma)^2 - (omega0)^2 ]^1/2
If x=x0 and dx/dt =v0 at t=0. show that
A= v0 - px0
m - p
B =mx0 - v0
m - p
In the case of very strong damping (i.e gamma >> omega0) show that
p (approx)= -2(gamma)
*************************************
m (approx)= -(omega0)^2
2(gamma)
Hence show that if v0 = 0, the displacement of the oscillator is given approximately by
x(t) = x0 e^(q)
where q = (t(omega0)^2) / 2(gamma)
Homework Equations
The Attempt at a Solution
Im okay up until the asterixes. Dont understand how you get the approximation for p in the limit gamma>>omega0. It just tends towards 0 as far as i can see. I tried l'hopitals, isolating the dominant term etc, but couldn't get anywhere
Dont quite get the last bit either, but that might be because i don't get the step before.
(PS I am going to learn latex soon sorry for all the mess)
EDIT: Yes, youre right Aleph. thankyou
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