# Steady state behavior for a particle undergoing damped forced oscillations

1. Oct 24, 2011

### elsamp123

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
x = A cos(ωt-δ)
A(ω) = (F0 /mω02) [(ω0/ω)/[(ω0/ω-ω/ω0)2+(1/Q2)]1/2)]
3. 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 ive got .. but how do u go about calculating the Avg K.E?

2. Oct 24, 2011

### vela

Staff Emeritus
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.

3. Oct 24, 2011

### 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

4. Oct 24, 2011

### vela

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

5. Oct 24, 2011

### elsamp123

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