Damped harmonic motion question

[aks_sky]

A damped harmonic motion starts from rest at time t=0 with displacement A₀ has the equation:

x(t) = A₀/cos (delta)*e^(-t/tau) *cos (w't + delta)

w' is the angular frequency, tau is the time constant and delta is given by:

tan (delta) = - (1/w' tau)

find the time when the maximum negative displacement occurs. express it in terms of period.



So, Can you just give me a hint on where to start and where to go from there.

Thank you

 

[kuruman]

Hint: Which term in your expression for x(t) can conceivably turn negative as time increases? Is it the exponential or is it the cosine?

 

[aks_sky]

That would be the exponential since we have -t in there. So i guess in this case i am supposed to differentiate with respect to t first?

 

[kuruman]

Just wait before you take any derivatives.

$$e^{-t}=\frac{1}{e^t}$$

For what value of t is the expression negative?

 

[aks_sky]

oh oops i dint see that part. The exponential is going to give me a positive value. How did i forget that. my bad

 

[kuruman]

Do you know where to go from here? The first thing to do is to find a value for the phase delta.

 

[aks_sky]

I don't know exactly where to go since i am confused about what it is asking for exactly. And to find the value for delta i will be using the tan (delta) = (1/w' tau) and then i guess i can substitute w' = 2*pi/ T' in there and go from there?

 

[kuruman]

You are given x(t) and you are told that at t = 0, the oscillator is at A₀. If you evaluate your expression at t = 0, i.e. find x(0), is it equal to A₀ or is it equal to something else?

 

[aks_sky]

ohhh i see.. no problemo.. thank you