Which of the following describes the motion of a damped oscillator mathematicall
Practice Questions
Q1
Which of the following describes the motion of a damped oscillator mathematically?
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
Which of the following describes the motion of a damped oscillator mathematically?
Step 1: Understand what a damped oscillator is. It is a system that oscillates (moves back and forth) but loses energy over time due to damping forces like friction.
Step 2: Identify the key components of the motion equation. The equation x(t) = A e^(-bt) cos(ωt) has three main parts: A, e^(-bt), and cos(ωt).
Step 3: Recognize what each part means: A is the initial amplitude (how far it starts from the center), e^(-bt) represents the damping effect (how the motion decreases over time), and cos(ωt) describes the oscillation pattern (the back and forth movement).
Step 4: Note that 'b' is the damping coefficient, which tells us how quickly the energy is lost. A larger 'b' means faster damping.
Step 5: Combine all parts to understand that the equation shows how the position x changes over time t for a damped oscillator.
