Power of a driven oscillating spring

[jdc15]

Homework Statement

a) Consider a driven mass-spring system with viscous friction using the notation of the lecture of Oct. 29, available on Vista. [The driving frequency is ω, the natural frequency is ω0, the friction force is -cv, the mass is m, the spring constant is k, the driving force is kD sin (ω t). Note that the phase Φ is negative, between 0 and π .] Write a formula for the rate of energy loss due to the friction force, once the steady state has been reached, as a function of time, t.

b) Write a formula for the rate at which the driving force is doing work on the mass-spring system, once the steady state has been reached, as a function of time, t.

c) Find the total energy loss due to friction over one period of the oscillations and also the total work done by the driving force over one period. Check whether or not they are equal. To do this problem you need to use the fact that the integral of a sine or cosine function over one period is zero. This integral is simply the average value over one period divided by the period. This is clearly zero if you think about what a cosine or sine function looks like.

Homework Equations

We have the equation for work: $$W=\int{F}\cdot dr$$
Power: $$dW/dt$$
Position: $$r=Asin(\omega t + \phi )$$
Velocity: $$v=dr/dt$$
Frictional force: $$F_f=-cv$$
Driving force: $$F_D=kDsin(\omega t)$$
Amplitude: A is a complex formula involving $$\omega, \omega 0, k, m, c$$ (will post later if needed)

The Attempt at a Solution

For a, I went r=Asin(wt+phi) so dr=Awcos(wt+phi)dt and v=Awcos(wt+phi). Thus work is $$\int{-cvA\omega cos(\omega t + \phi )} dt$$, so power is $$p=-cA^2\omega^2cos^2(\omega t + phi)$$. Now this answer makes sense to me, I know I'm missing a phi term that I need to solve for, but it seems to match the simulations we've been shown. However, my TA told me I should separate the integral $$\int{F}dr=F\int{}dr=Fr$$ but this doesn't make sense to me because r and F both depend on time. Does anyone have an explanation?

Thanks in advance.

PS. This is a continuation of another post that originally was just a calc question: https://www.physicsforums.com/showthread.php?p=2967731&posted=1#post2967731"

 

[jdc15]

Nevermind I figured it out.