What is the general form of the equation for a damped harmonic oscillator?
Practice Questions
Q1
What is the general form of the equation for a damped harmonic oscillator?
x(t) = A cos(ωt)
x(t) = A e^(-bt) cos(ωt)
x(t) = A sin(ωt)
x(t) = A e^(bt) cos(ωt)
Questions & Step-by-Step Solutions
What is the general form of the equation for a damped harmonic oscillator?
Step 1: Understand what a damped harmonic oscillator is. It is a system that experiences oscillations (like a swinging pendulum) but loses energy over time due to damping (like friction).
Step 2: Identify the components of the equation. The equation x(t) = A e^(-bt) cos(ωt) has several parts: A is the amplitude (the maximum distance from the center), e^(-bt) represents the damping effect, and cos(ωt) describes the oscillation.
Step 3: Recognize that 'x(t)' represents the position of the oscillator at time 't'.
Step 4: Note that 'A' is a constant that determines how far the oscillator moves from its rest position at the start.
Step 5: Understand that 'b' is the damping coefficient, which affects how quickly the oscillations decrease in amplitude over time.
Step 6: Realize that 'ω' (omega) is the angular frequency, which determines how fast the oscillations occur.
Step 7: Combine all these parts to see that the equation describes how the position of the damped harmonic oscillator changes over time.
