Overdamped and critically damped oscillator

[Aneleh]

Can you help me start on this one:
Show that an overdamped or critically damped oscillator can cross the origin at most once.

 

[hotvette]

Can you help me start on this one

Do you have definitions for 'overdamped' and 'critically damped'? If so, then it seems to me a prudent course of action would be to get equations of each and calculate the time it takes to cross the origin. A plot might be helpful also.

 

[quicknote]

An overdamped or critically damped oscillator is a type of system that experiences a resistive force that is greater than or equal to the restoring force. This results in the oscillator's motion being damped, meaning that it gradually decreases in amplitude over time.

To understand why an overdamped or critically damped oscillator can only cross the origin once, we need to consider the equation of motion for such a system. This equation can be written as:

m(d^2x/dt^2) + c(dx/dt) + kx = 0

where m is the mass of the oscillator, c is the damping coefficient, k is the spring constant, and x is the displacement from equilibrium.

For an overdamped or critically damped oscillator, the damping coefficient c is greater than or equal to the critical damping value, which is given by c_critical = 2√(mk). This means that the damping force is strong enough to prevent the oscillator from oscillating and it will eventually come to rest at the equilibrium position.

Now, let us consider the case where the oscillator starts at a non-zero initial displacement and is moving towards the origin. As the oscillator approaches the origin, the damping force will increase and eventually become equal to the restoring force (given by -kx). At this point, the net force acting on the oscillator will be zero and it will come to a stop at the origin.

If the oscillator were to continue moving past the origin, the damping force would become greater than the restoring force. This would cause the oscillator to move away from the origin, towards its equilibrium position. However, since the oscillator has already come to a stop at the origin, it cannot move in the opposite direction and therefore cannot cross the origin again.

In the case of an overdamped oscillator, the damping force is always greater than the restoring force, meaning that the oscillator will never even reach the origin and will come to a stop before it can cross it.

In conclusion, an overdamped or critically damped oscillator can only cross the origin once due to the balance between the damping and restoring forces. This is an important characteristic of these types of oscillators and can have implications in various fields such as engineering and physics.