# Damped oscillator energy dissipated per cycle.

• teroenza
In summary, the energy dissipated by the damping force of a damped oscillator with a sinusoidal driving force of equal natural and drive frequency is equal to 2*pi*m*B*w*A^2. This can be derived by equating the rate of work done by the force to the rate of energy dissipation, and integrating the full sinusoidal velocity. The negative sign can be disregarded as the problem asks for the energy dissipated.
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.

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

## 1. What is a damped oscillator?

A damped oscillator is a physical system that exhibits oscillatory motion, but gradually loses energy over time due to the presence of damping forces.

## 2. How is energy dissipated in a damped oscillator?

The energy in a damped oscillator is dissipated through the action of damping forces, such as friction or air resistance, which convert the kinetic energy of the oscillator into heat.

## 3. What is the energy dissipated per cycle in a damped oscillator?

The energy dissipated per cycle in a damped oscillator depends on the amount of damping present. In general, the energy dissipated per cycle is equal to the difference between the initial and final energies of the oscillator.

## 4. How is the energy dissipated per cycle related to the damping coefficient?

The energy dissipated per cycle is directly proportional to the damping coefficient. A larger damping coefficient indicates a greater amount of damping, leading to a larger amount of energy dissipated per cycle.

## 5. Can the energy dissipated per cycle be calculated mathematically?

Yes, the energy dissipated per cycle can be calculated using the formula: E_diss = (1/2) * k * x_0^2 * e^(-2Bt), where k is the spring constant, x_0 is the amplitude of the oscillator, B is the damping coefficient, and t is the time elapsed.

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