# Solving for time with an Overdamped Oscillator

In summary, the equation x(t) = A1*e-(γ-q)*t + A2*e-(γ+q)*t is used to find when the mass is within 10% of its equilibrium. A1, A2, γ, and q are given values. To solve for t, the equation was manipulated by dividing the right side by x and rearranging the terms. However, this resulted in an unsolvable equation.

## Homework Statement

How long will it take until the mass is within 10% of its equilibrium?

I already solved most of what is needed in previous parts of the question. I just need help solving for t because it is in two exponents in the equation.

## Homework Equations

This is the equation
x(t) = A1*e-(γ-q)*t + A2*e-(γ+q)*t
A1 = 3.61
A2 = -0.61
γ = 0.9
q = 0.64

## The Attempt at a Solution

First I found where x is .1 from its equilibrium
x = .1A = .1(3m) = 0.3m
Then I plugged in the A values. I divided the right side of the equation by x and moved the whole A2 term to the left of the equation to have an exponential on both sides.
Then I took the ln of both terms to have an equation with t on both sides.
-(γ+q)*(A2/x )*t = -(γ-q)*(A1/x)*t

If I plug these variables in, I am left with
number*t = number*t and that isn't really solvable...

Everytime I come up with an answer, it is negative but time can't be..

## 1. What is an overdamped oscillator?

An overdamped oscillator is a type of oscillator system that experiences damping forces that are greater than the restoring forces, causing the system to return to its equilibrium position slowly without oscillating.

## 2. How do you solve for time with an overdamped oscillator?

To solve for time with an overdamped oscillator, you can use the equation t = ln(x/x0)/λ, where t is time, x is the displacement from equilibrium at time t, x0 is the initial displacement from equilibrium, and λ is the damping coefficient.

## 3. What is the relationship between time and damping coefficient in an overdamped oscillator?

The time it takes for an overdamped oscillator to return to equilibrium is directly proportional to the damping coefficient. As the damping coefficient increases, the time required for the system to reach equilibrium also increases.

## 4. Can an overdamped oscillator exhibit oscillatory motion?

No, an overdamped oscillator does not exhibit oscillatory motion. The damping forces are strong enough to prevent the system from oscillating, causing it to return to equilibrium slowly without any oscillations.

## 5. How does the initial displacement affect the time it takes for an overdamped oscillator to return to equilibrium?

The initial displacement does not affect the time it takes for an overdamped oscillator to return to equilibrium. The equation for time does not include the initial displacement, and the system will always return to equilibrium at the same rate regardless of the initial displacement.

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