Derive the expression of the mechanical energy for under damped system.

Answer: the expression of the mechanical energy for under damped system is;

x (t) = Ae-γ/2tcos⁡ (ωdt+ϕ), where ωd=ω02-γ2/4

γ = damping rate, and

ω0 = the angular frequency of the oscillator without damping.

Explanation:

The physical situation in mechanical energy defined through out the world has three possible results depending on the value of a (which is a constant value), which depends on the value of what is under our radical. This expression can either be positive, negative, or equal to zero which will result in overdamping, underdamping, and critical damping, as the case may be.

γ2 >4ω²0 This is the Over Damped case. Here, the system returns to equilibrium by exponentially decaying towards zero, and the system will not pass that equilibrium position more than once.

γ² < 4ω²0 this is the Under Damped case. Here, the system moves back and forth as it slowly returns to equilibrium and the amplitude of the system decreases over time.

Finally, γ² = 4ω²0

This is the Critically Damped case. Here, the system returns to equilibrium very fast without moving back and forth and without passing the equilibrium position at all.