# The rate at which a damped, driven oscillator does work

## Homework Statement

Consider a damped oscillator, with natural frequency ω_naut and damping constant both fixed, that is driven by a force F(t)=F_naut*cos(ωt).

a) Find the rate P(t) at which F(t) does work and show that the average (P)avg over any number of complete cycles is mβω2A2. b) Verify that this is the same as the average rate at which energy is lost to the resistive force. c) Show that as ω is varied, (P)avg is maximum when ω=ω_naut; that is, the resonance of the power occurs at ω=ω_naut (exactly).

## Homework Equations

(P)avg= 1/τ ∫ Fv dt
F= F_naut*cos(ωt)
x(t)= A*cos(ωt)
v(t)= -Aωsin(ωt)
ω= √[(ω_naut)2 - β2 ]

## The Attempt at a Solution

I plugged in the equation for F(t) and v(t) in the integral ( P(t) = ∫Fv ) and used u substitution, making cos(ωt) the "u" and -ωsin(ωt)dt the "du".

So I then had ∫(F_naut)Au du

After integrating and plugging cos(ωt) back in for u, I had:

(P)avg = (1/2τ)*(F_naut)A*cos2(ωt), integrated from -τ/2 to τ/2.

This is where I hit a problem. My work doesn't seem like it will be equal to mβω2A2. Because I can't solve this, I can't figure out the next two parts either. Did I make a mathematical error in integrating, or did I set up the integral wrong? Thank you!

Last edited:

ehild
Homework Helper

## Homework Equations

(P)avg= 1/τ ∫ Fv dt
F= F_naut*cos(ωt)
x(t)= A*cos(ωt)
v(t)= -Aωsin(ωt)
ω= √[(ω_naut)2 - β2 ]

If the oscillator is driven by F=Fo cos(ωt) the oscillator will move with the frequency of the driving force, but with a phase difference, x(t)=A(cosωt+θ)
Both the phase difference θ and the amplitude A depend on the natural frequency and the driving frequency.

ehild

Yes, but how can I incorporate that into my set up?

ehild
Homework Helper
x(t)=A(cosωt+θ). v(t)= -Aωsin(ωt+θ). Find A and θ in terms of ω_naut, ω and F_naut, and calculate the integral 1/τ ∫ Fv dt.

ehild