Irodov Problem 4.101
Physics · mechanics
Problem
The free damped oscillations are maintained in a circuit, such that the voltage across the capacitor varies as V = Vif,e-O t cos cot. Find the moments of time when the modulus of the voltage across the capacitor reaches (a) peak values; (b) maximum (extremum) values.Intuitive explanation
The voltage here has two competing parts: an exponentially decaying envelope that shrinks everything over time, and a cosine oscillation that wiggles up and down. For peaks of the modulus (absolute value), you need to distinguish between where the oscillation itself peaks—which happens regularly at the cosine's maxima—versus where the magnitude of the decaying oscillation reaches local extrema, which requires calculus because the envelope is also changing. The key insight is that modulus peaks occur when the oscillation crosses zero or reaches its cosine extrema, while true extrema of happen when the derivative of the envelope and oscillation balance each other out.
Intuitive explanation
The voltage here has two competing parts: an exponentially decaying envelope that shrinks everything over time, and a cosine oscillation that wiggles up and down. For peaks of the modulus (absolute value), you need to distinguish between where the oscillation itself peaks—which happens regularly at the cosine's maxima—versus where the magnitude of the decaying oscillation reaches local extrema, which requires calculus because the envelope is also changing. The key insight is that modulus peaks occur when the oscillation crosses zero or reaches its cosine extrema, while true extrema of happen when the derivative of the envelope and oscillation balance each other out.