For an object in damped harmonic motion with initial...

For an object in damped harmonic motion with initial amplitude $a,$ period $2 \pi / \omega,$ and damping constant $c,$ find an equation that models the displacement $y$ at time $t$ if

(a) $y=0$ at time $t=0 : y=$ __________
(b) $y=a$ at time $t=0 : y=$ __________

Key Concepts

Sinusoidal Functions in Oscillations

Sinusoidal functions, such as sine and cosine, are essential for modeling periodic phenomena like oscillations. These functions capture the regular, repeating behavior of the system. In a damped oscillator, the sinusoidal function defines the oscillatory behavior while the exponential factor controls the diminishing amplitude.

Initial Conditions and Phase Shift Determination

The choice between sine and cosine functions (or a phase-shifted sine function) in the model depends on the initial conditions of the system. For instance, if the displacement is zero at time zero, a sine function is appropriate, since sin(0) = 0. Conversely, if the displacement is at a maximum (equal to the initial amplitude) at time zero, a cosine function is used because cos(0) = 1. This adjustment ensures that the model accurately reflects the initial state of the system.

Damped Harmonic Motion

Damped harmonic motion describes an oscillatory system where the amplitude of oscillation decays exponentially over time due to the presence of non-conservative forces like friction or air resistance. This type of motion is modeled by combining a rapidly decreasing exponential factor with a periodic sinusoidal function, capturing both the periodic nature of the oscillation and the energy loss over time.

Exponential Decay

Exponential decay is a mathematical concept expressing how a quantity decreases at a rate proportional to its current value. In the context of damped oscillations, the exponential term (typically expressed as e^(–ct) where c is the damping coefficient) scales the amplitude of the oscillation, ensuring that the amplitude reduces continuously as time progresses.

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