Final answer:
The equations of motion for an underdamped oscillating spring can be represented by differential equations, with the specific case of underdamping where the angular frequency is greater than b/2m. Resonance increases the amplitude of oscillations when the frequency of the driving force matches the system's natural frequency. Calculations for force constants and stored energy involve formulas dependent on mass, period, and displacement.
Step-by-step explanation:
Oscillating Springs and Differential Equations
The subject being dealt with is the behavior of oscillating springs and the differential equations that describe this motion, especially within the context of a damped mass-spring system. The question pertains to the case of underdamping, where the angular frequency of the undamped spring is greater than b/2m (√k/m > b/2m). The differential equation for a damped oscillator is given by mx'' + bx' + kx = 0, where x represents the displacement, m the mass, b the damping coefficient, k the spring constant, and x' and x'' represent the first and second derivatives of x with respect to time, respectively.
The associated equations of motion for forced oscillations involve a driving force that is added to the equation, resulting in mx'' + bx' + kx = F(t), where F(t) represents the periodic driving force. Resonance occurs when the frequency of the driving force matches the natural frequency of the system, causing a significant increase in the amplitude of the oscillator.
A system oscillating at resonance has the following characteristics: Maximum energy transfer from the driving force to the oscillating system, an amplitude of oscillations that reaches a maximum, and a phase difference of π/2 between the driving force and the displacement of the oscillator.
The velocity of a mass oscillating in Simple Harmonic Motion (SHM) can be found by taking the derivative of the position equation, usually expressed as v = -ωxA cos(ωt + ϕ), where ω is the angular frequency, A the amplitude, and ϕ the phase constant.
In resonance, the amplitude of the mass oscillating on a spring in a viscous fluid decreases over time, typically represented by a cosine function enveloped by an exponential decay, indicative of the energy dissipation due to damping.
For an object like the one in New York City used to dampen wind-driven oscillations, calculating the effective force constant for a given period involves using the formula k = (4π²m)/T², and calculating the energy stored in the springs for a displacement x involves using the formula E = 1/2kx².