Damped oscillator energy dissipated per cycle.

[teroenza]

Homework Statement

Sinusoidal driving force driving a damped oscillator (mass = m).
Natural frequency is assumed to equal the drive frequency = w
Time has elapsed to the point any transients have dissipated.

Show that the energy dissipated by the damping force [F=-bv] during one cycle is equal to

2*pi*m*B*w*A^2

Homework Equations

Total energy = (1/2)*m*w^2*A^2

A=amplitude

x=Acos(w*t-d)

v=A*w*sin(w*t-d)

d=delta=phase shift

B=Beta=Damping coefficient = b/(2*m)

The Attempt at a Solution

I am explicitly told in a hint that the rate at which the force does work is F*v.

My reasoning is,

Rate of work done by force = rate of energy dissipation. Work done by the non-conservative damping force must be = the change in energy of the oscillator. This rate times the time period desired will give the energy loss in in that amount of time. The time in question is one period (2*pi)/w .

(F*v)*t

(-bv*v)*(2*pi)/w

(-b(A*w)^2)*(2*pi)/w where I have said v^2=(A*w)^2

-2*pi*(2*m*B)*w*A^2 substituting b=B*2*m , the result at which I am stuck.

The negative sign may be disregarded I believe, because the problem asks for the energy dissipated, and the oscillator is losing energy. Bu the extra factor of 2 confuses me. I believe am close.

 

[teroenza]

Found the mistake. Don't multiply by time, you must integrate the full sinusoidal velocity to get an (pi/2) factor.