Physics extended problems (resonance framework)

Problem 4 – Resonant oscillator energy#

A resonant oscillator has energy

$$ E = \frac{1}{2} D_6 τ_r^2 + X. $$

1. If $$τ_r$$ increases by 15%, how does the first term change?
2. What is the qualitative effect on total energy?

Problem 5 – Triadic field strength#

A field strength is modeled as

$$ F = \frac{T_f + D_3}{\sqrt{ΛΘ + τ_r}}. $$

1. If $$τ_r$$ increases, how does the denominator change?
2. What is the qualitative effect on $$F$$?

Problem 6 – Resonant thermal radiation#

A simplified radiation model is

$$ R = X e^{-D_9 / τ_r}. $$

1. If $$τ_r$$ increases, does radiation increase or decrease?
2. Why?

Problem 7 – Momentum drift#

Momentum drift is modeled as

$$ p = D_6 τ_r - ΛΘ. $$

1. If $$τ_r$$ increases, how does $$p$$ change?
2. If $$ΛΘ$$ increases, what happens?

Problem 8 – Triadic wave packet spreading#

A wave packet spreads according to

$$ σ(t) = \sqrt{σ_0^2 + \frac{T_f^2 t^2}{D_9 + τ_r}}. $$

1. If $$τ_r$$ increases, how does the spreading rate change?
2. If $$T_f$$ increases, what is the effect?