How damped harmonic oscillator affects period

[p3forlife]

I have trouble understanding how damping affects the period (of a torsion pendulum). I know that damping affects the amplitude of the oscillator, however how would damping change the period then?

I have a feeling this has to do with angular frequency, w, given by:

w = sqrt( (k/m) - (b^2/4m^2) )
where k is the torsion constant
b is the damping constant
m is the mass on the pendulum

Since the period is the inverse of frequency, would the inverse of the above equation answer my question?

 

[naresh]

Yes, as long as you ensure that the units are right.

 

[thomate1]

The damped harmonic oscillator refers to a system that experiences damping, or a gradual loss of energy, as it oscillates. This can be caused by various factors such as friction, air resistance, or electrical resistance. Damping affects the behavior of the oscillator, including its amplitude and period.

To understand how damping affects the period of a torsion pendulum, we must first consider the equation for the period of a simple harmonic oscillator, given by T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. This equation assumes no damping present.

When damping is introduced, the equation for the angular frequency, as mentioned in the question, becomes w = √[(k/m) - (b^2/4m^2)]. This shows that the angular frequency is affected by the damping constant, b. As the damping constant increases, the angular frequency decreases, resulting in a longer period of oscillation.

This can be understood intuitively by considering the effect of damping on the amplitude of the oscillator. As the oscillator experiences damping, its amplitude decreases over time. This means that the pendulum takes longer to complete one full oscillation, resulting in a longer period.

To answer your question, the inverse of the equation for angular frequency does not directly give the period. Instead, you would need to solve for T in the equation T = 2π/w, using the damped angular frequency, to find the period of the damped oscillator.

In summary, damping affects the period of a torsion pendulum by decreasing the angular frequency, which in turn results in a longer period of oscillation. This is due to the gradual loss of energy in the system caused by damping.