Crossing the brachistochrone- Critical damping

An interesting case arises if the damping angular frequency ω΄ is equal to zero (critical damping). Taking the condition,

and setting ω΄=0, we have,

so that from the equation of motion

we have the following solution,

At time t=0, we have

Accordingly, the energies at time t=0 will be,

Therefore, the total energy will be

where the last term refers to the mass μ₀ of the oscillator.

We may notice that if the final speed v₀ of the object (‘hanging’ from the oscillator) is equal to the speed of light c (assuming that this is the speed of the oscillator) then it will be

so that the moving object will have consumed a mass equal to the mass per wavelength of the oscillator, as soon as it reaches the speed of light.

At any time t, the energies will be

The last equation can be written simply as

Thus when critical damping is applied, the energy of the system is unified by a common expression for the different forms of energies.

Notes:

This is a comparison between the energy of the damped oscillator, and the energy in relativity.

In the ‘wet sponge’ example, we saw that if the damping factor γ changes with time,

then we take for the mass of the object an expression of the form

Here we can make a similar assumption with respect to the energy of the damped oscillator,

so that setting, for example,

we have for the energy that

In this equation, the energy has the form of a Gaussian function.

If we raise both terms to the power of two (in order to make the ½ factor disappear), we have

The linear (or better the second order) approximation of the previous exponential, at v≈0, v<<c, is

so that

and we take back the relativistic expression for the energy.

This is a related graph:

The red line is the relativistic energy in the form of

while the green line is the exponential energy in the Gaussian form

where E₀≈c≈1 in the graph.

The basic aspect here is that while the relativistic energy will be zero if the speed v of the object is equal to the speed of light c, the energy in the Gaussian form will be zero if the speed v of the object is infinite.

This is another way to suggest a correction to the relativistic expression.