Steady state behavior for a particle undergoing damped forced oscillations

[elsamp123]

Homework Statement

consider a system with a damping force undergoing forced oscillations at an angular frequency ω
a) what is the instantaneous kinetic energy of the system?
b) what is the instantaneous potential energy of the system?
c) what is the ratio of the average kinetic energy to the average potential energy? express the answer in terms of the ratio ω/ω0
d) for what values of ω are the average kinetic energy and the average potential energy equal? what is the total energy of the system under these conditions?
e) how does the total energy of the system vary with time for an arbitrary value of ω? for what values of ω is the total energy constant in time?

Homework Equations

x = A cos(ωt-δ)
A(ω) = (F₀ /mω₀²) [(ω₀/ω)/[(ω₀/ω-ω/ω₀)²+(1/Q²)]^1/2)]

The Attempt at a Solution

i know that K.E = 1/2 m (dx/dt)²

and P.E = 1/2 kx²

so part and b I've got .. but how do u go about calculating the Avg K.E?

 

[vela]

You want to integrate the kinetic energy over one cycle:
$$\langle K \rangle = \frac{1}{T}\int_0^T \frac{1}{2}mv(t)^2\,dt$$where T is the period of oscillation.

 

[elsamp123]

So I would go:

ω/2∏ $\int^{ω/(2∏)}_{0}$ 1/2 m (dx/dt)²?

so would P.E go from 0 to A then?

Thanks a ton btw! you really helped me get started with this HW

 

[vela]

You'd still average the potential over one cycle by integrating with respect to time and dividing by the period.

 

[elsamp123]

Thank you so very much! You are very kind! :)